% This program is copyright (C) 1984 by D. E. Knuth; all rights are reserved. % Copying of this file is authorized only if (1) you are D. E. Knuth, or if % (2) you make absolutely no changes to your copy. (The WEB system provides % for alterations via an auxiliary file; the master file should stay intact.) % In other words, METAFONT is under essentially the same ground rules as TeX. % TeX is a trademark of the American Mathematical Society. % METAFONT is a trademark of Addison-Wesley Publishing Company. % Version 0 was completed on July 28, 1984. % Version 1 was completed on January 4, 1986; it corresponds to "Volume D". % Version 1.1 trivially corrected the punctuation in one message (June 1986). % Version 1.2 corrected an arithmetic overflow problem (July 1986). % Version 1.3 improved rounding when elliptical pens are made (November 1986). % Version 1.4 corrected scan_declared_variable timing (May 1988). % Version 1.5 fixes negative halving in allocator when mem_min<0 (June 1988). % Version 1.6 keeps open_log_file from calling fatal_error (November 1988). % Version 1.7 solves that problem a better way (December 1988). % A few "harmless" optimizations have been made without changing versions. % A reward of $20.48 will be paid to the first finder of any remaining bug. % (This amount will double again in 1989.) % Although considerable effort has been expended to make the METAFONT program % correct and reliable, no warranty is implied; the author disclaims any % obligation or liability for damages, including but not limited to % special, indirect, or consequential damages arising out of or in % connection with the use or performance of this software. This work has % been a ``labor of love'' and the author hopes that users enjoy it. % Here is TeX material that gets inserted after \input webmac \def\hang{\hangindent 3em\noindent\ignorespaces} \def\textindent#1{\hangindent2.5em\noindent\hbox to2.5em{\hss#1 }\ignorespaces} \font\ninerm=cmr9 \let\mc=\ninerm % medium caps for names like SAIL \def\PASCAL{Pascal} \def\ph{\hbox{Pascal-H}} \def\psqrt#1{\sqrt{\mathstrut#1}} \def\k{_{k+1}} \font\tenlogo=logo10 % font used for the METAFONT logo \font\logos=logosl10 \font\eightlogo=logo8 \def\MF{{\tenlogo META}\-{\tenlogo FONT}} \def\<#1>{$\langle#1\rangle$} \def\section{\mathhexbox278} \chardef\%=`\% \let\swap=\leftrightarrow \def\round{\mathop{\rm round}\nolimits} \def\(#1){} % this is used to make section names sort themselves better \def\9#1{} % this is used for sort keys in the index via @@:sort key}{entry@@> \outer\def\N#1. \[#2]#3.{\MN#1.\vfil\eject % begin starred section \def\rhead{PART #2:\uppercase{#3}} % define running headline \message{*\modno} % progress report \edef\next{\write\cont{\Z{\?#2]#3}{\modno}{\the\pageno}}}\next \ifon\startsection{\bf\ignorespaces#3.\quad}\ignorespaces} \let\?=\relax % we want to be able to \write a \? \def\title{{\eightlogo METAFONT}} \def\topofcontents{\hsize 5.5in \vglue -30pt plus 1fil minus 1.5in \def\?##1]{\hbox to 1in{\hfil##1.\ }} } \def\botofcontents{\vskip 0pt plus 1fil minus 1.5in} \pageno=3 \def\glob{13} % this should be the section number of "" \def\gglob{20, 26} % this should be the next two sections of "" @* \[1] Introduction. This is \MF, a font compiler intended to produce typefaces of high quality. The \PASCAL\ program that follows is the definition of \MF84, a standard @:PASCAL}{\PASCAL@> @!@:METAFONT84}{\MF84@> version of \MF\ that is designed to be highly portable so that identical output will be obtainable on a great variety of computers. The conventions of \MF84 are the same as those of \TeX82. The main purpose of the following program is to explain the algorithms of \MF\ as clearly as possible. As a result, the program will not necessarily be very efficient when a particular \PASCAL\ compiler has translated it into a particular machine language. However, the program has been written so that it can be tuned to run efficiently in a wide variety of operating environments by making comparatively few changes. Such flexibility is possible because the documentation that follows is written in the \.{WEB} language, which is at a higher level than \PASCAL; the preprocessing step that converts \.{WEB} to \PASCAL\ is able to introduce most of the necessary refinements. Semi-automatic translation to other languages is also feasible, because the program below does not make extensive use of features that are peculiar to \PASCAL. A large piece of software like \MF\ has inherent complexity that cannot be reduced below a certain level of difficulty, although each individual part is fairly simple by itself. The \.{WEB} language is intended to make the algorithms as readable as possible, by reflecting the way the individual program pieces fit together and by providing the cross-references that connect different parts. Detailed comments about what is going on, and about why things were done in certain ways, have been liberally sprinkled throughout the program. These comments explain features of the implementation, but they rarely attempt to explain the \MF\ language itself, since the reader is supposed to be familiar with {\sl The {\logos METAFONT\/}book}. @.WEB@> @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> @ The present implementation has a long ancestry, beginning in the spring of~1977, when its author wrote a prototype set of subroutines and macros @^Knuth, Donald Ervin@> that were used to develop the first Computer Modern fonts. This original proto-\MF\ required the user to recompile a {\mc SAIL} program whenever any character was changed, because it was not a ``language'' for font design; the language was {\mc SAIL}. After several hundred characters had been designed in that way, the author developed an interpretable language called \MF, in which it was possible to express the Computer Modern programs less cryptically. A complete \MF\ processor was designed and coded by the author in 1979. This program, written in {\mc SAIL}, was adapted for use with a variety of typesetting equipment and display terminals by Leo Guibas, Lyle Ramshaw, and David Fuchs. @^Guibas, Leonidas Ioannis@> @^Ramshaw, Lyle Harold@> @^Fuchs, David Raymond@> Major improvements to the design of Computer Modern fonts were made in the spring of 1982, after which it became clear that a new language would better express the needs of letterform designers. Therefore an entirely new \MF\ language and system were developed in 1984; the present system retains the name and some of the spirit of \MF79, but all of the details have changed. No doubt there still is plenty of room for improvement, but the author is firmly committed to keeping \MF84 ``frozen'' from now on; stability and reliability are to be its main virtues. On the other hand, the \.{WEB} description can be extended without changing the core of \MF84 itself, and the program has been designed so that such extensions are not extremely difficult to make. The |banner| string defined here should be changed whenever \MF\ undergoes any modifications, so that it will be clear which version of \MF\ might be the guilty party when a problem arises. @^extensions to \MF@> @^system dependencies@> If this program is changed, the resulting system should not be called `\MF\kern.5pt'; the official name `\MF\kern.5pt' by itself is reserved for software systems that are fully compatible with each other. A special test suite called the ``\.{TRAP} test'' is available for helping to determine whether a particular implementation deserves to be known as `\MF\kern.5pt' [cf.~Stanford Computer Science report CS1095, January 1986]. @d banner=='This is METAFONT, Version 1.7' {printed when \MF\ starts} @ Different \PASCAL s have slightly different conventions, and the present @!@:PASCAL H}{\ph@> program expresses \MF\ in terms of the \PASCAL\ that was available to the author in 1984. Constructions that apply to this particular compiler, which we shall call \ph, should help the reader to see how to make an appropriate interface for other systems if necessary. (\ph\ is Charles Hedrick's modification of a compiler @^Hedrick, Charles Locke@> for the DECsystem-10 that was originally developed at the University of Hamburg; cf.\ {\sl SOFTWARE---Practice \AM\ Experience \bf6} (1976), 29--42. The \MF\ program below is intended to be adaptable, without extensive changes, to most other versions of \PASCAL, so it does not fully use the admirable features of \ph. Indeed, a conscious effort has been made here to avoid using several idiosyncratic features of standard \PASCAL\ itself, so that most of the code can be translated mechanically into other high-level languages. For example, the `\&{with}' and `\\{new}' features are not used, nor are pointer types, set types, or enumerated scalar types; there are no `\&{var}' parameters, except in the case of files; there are no tag fields on variant records; there are no |real| variables; no procedures are declared local to other procedures.) The portions of this program that involve system-dependent code, where changes might be necessary because of differences between \PASCAL\ compilers and/or differences between operating systems, can be identified by looking at the sections whose numbers are listed under `system dependencies' in the index. Furthermore, the index entries for `dirty \PASCAL' list all places where the restrictions of \PASCAL\ have not been followed perfectly, for one reason or another. @!@^system dependencies@> @!@^dirty \PASCAL@> @ The program begins with a normal \PASCAL\ program heading, whose components will be filled in later, using the conventions of \.{WEB}. @.WEB@> For example, the portion of the program called `\X\glob:Global variables\X' below will be replaced by a sequence of variable declarations that starts in $\section\glob$ of this documentation. In this way, we are able to define each individual global variable when we are prepared to understand what it means; we do not have to define all of the globals at once. Cross references in $\section\glob$, where it says ``See also sections \gglob, \dots,'' also make it possible to look at the set of all global variables, if desired. Similar remarks apply to the other portions of the program heading. Actually the heading shown here is not quite normal: The |program| line does not mention any |output| file, because \ph\ would ask the \MF\ user to specify a file name if |output| were specified here. @^system dependencies@> @d mtype==t@&y@&p@&e {this is a \.{WEB} coding trick:} @f mtype==type {`\&{mtype}' will be equivalent to `\&{type}'} @f type==true {but `|type|' will not be treated as a reserved word} @p @t\4@>@@/ program MF; {all file names are defined dynamically} label @@/ const @@/ mtype @@/ var @@/ @# procedure initialize; {this procedure gets things started properly} var @@/ begin @@/ end;@# @t\4@>@@/ @t\4@>@@/ @ The overall \MF\ program begins with the heading just shown, after which comes a bunch of procedure declarations and function declarations. Finally we will get to the main program, which begins with the comment `|start_here|'. If you want to skip down to the main program now, you can look up `|start_here|' in the index. But the author suggests that the best way to understand this program is to follow pretty much the order of \MF's components as they appear in the \.{WEB} description you are now reading, since the present ordering is intended to combine the advantages of the ``bottom up'' and ``top down'' approaches to the problem of understanding a somewhat complicated system. @ Three labels must be declared in the main program, so we give them symbolic names. @d start_of_MF=1 {go here when \MF's variables are initialized} @d end_of_MF=9998 {go here to close files and terminate gracefully} @d final_end=9999 {this label marks the ending of the program} @= start_of_MF@t\hskip-2pt@>, end_of_MF@t\hskip-2pt@>,@,final_end; {key control points} @ Some of the code below is intended to be used only when diagnosing the strange behavior that sometimes occurs when \MF\ is being installed or when system wizards are fooling around with \MF\ without quite knowing what they are doing. Such code will not normally be compiled; it is delimited by the codewords `$|debug|\ldots|gubed|$', with apologies to people who wish to preserve the purity of English. Similarly, there is some conditional code delimited by `$|stat|\ldots|tats|$' that is intended for use when statistics are to be kept about \MF's memory usage. The |stat| $\ldots$ |tats| code also implements special diagnostic information that is printed when $\\{tracingedges}>1$. @^debugging@> @d debug==@{ {change this to `$\\{debug}\equiv\null$' when debugging} @d gubed==@t@>@} {change this to `$\\{gubed}\equiv\null$' when debugging} @f debug==begin @f gubed==end @# @d stat==@{ {change this to `$\\{stat}\equiv\null$' when gathering usage statistics} @d tats==@t@>@} {change this to `$\\{tats}\equiv\null$' when gathering usage statistics} @f stat==begin @f tats==end @ This program has two important variations: (1) There is a long and slow version called \.{INIMF}, which does the extra calculations needed to @.INIMF@> initialize \MF's internal tables; and (2)~there is a shorter and faster production version, which cuts the initialization to a bare minimum. Parts of the program that are needed in (1) but not in (2) are delimited by the codewords `$|init|\ldots|tini|$'. @d init== {change this to `$\\{init}\equiv\.{@@\{}$' in the production version} @d tini== {change this to `$\\{tini}\equiv\.{@@\}}$' in the production version} @f init==begin @f tini==end @ If the first character of a \PASCAL\ comment is a dollar sign, \ph\ treats the comment as a list of ``compiler directives'' that will affect the translation of this program into machine language. The directives shown below specify full checking and inclusion of the \PASCAL\ debugger when \MF\ is being debugged, but they cause range checking and other redundant code to be eliminated when the production system is being generated. Arithmetic overflow will be detected in all cases. @^system dependencies@> @^Overflow in arithmetic@> @= @{@&$C-,A+,D-@} {no range check, catch arithmetic overflow, no debug overhead} @!debug @{@&$C+,D+@}@+ gubed {but turn everything on when debugging} @ This \MF\ implementation conforms to the rules of the {\sl Pascal User @:PASCAL}{\PASCAL@> @^system dependencies@> Manual} published by Jensen and Wirth in 1975, except where system-dependent @^Wirth, Niklaus@> @^Jensen, Kathleen@> code is necessary to make a useful system program, and except in another respect where such conformity would unnecessarily obscure the meaning and clutter up the code: We assume that |case| statements may include a default case that applies if no matching label is found. Thus, we shall use constructions like $$\vbox{\halign{\ignorespaces#\hfil\cr |case x of|\cr 1: $\langle\,$code for $x=1\,\rangle$;\cr 3: $\langle\,$code for $x=3\,\rangle$;\cr |othercases| $\langle\,$code for |x<>1| and |x<>3|$\,\rangle$\cr |endcases|\cr}}$$ since most \PASCAL\ compilers have plugged this hole in the language by incorporating some sort of default mechanism. For example, the \ph\ compiler allows `|others|:' as a default label, and other \PASCAL s allow syntaxes like `\&{else}' or `\&{otherwise}' or `\\{otherwise}:', etc. The definitions of |othercases| and |endcases| should be changed to agree with local conventions. Note that no semicolon appears before |endcases| in this program, so the definition of |endcases| should include a semicolon if the compiler wants one. (Of course, if no default mechanism is available, the |case| statements of \MF\ will have to be laboriously extended by listing all remaining cases. People who are stuck with such \PASCAL s have, in fact, done this, successfully but not happily!) @d othercases == others: {default for cases not listed explicitly} @d endcases == @+end {follows the default case in an extended |case| statement} @f othercases == else @f endcases == end @ The following parameters can be changed at compile time to extend or reduce \MF's capacity. They may have different values in \.{INIMF} and in production versions of \MF. @.INIMF@> @^system dependencies@> @= @!mem_max=30000; {greatest index in \MF's internal |mem| array; must be strictly less than |max_halfword|; must be equal to |mem_top| in \.{INIMF}, otherwise |>=mem_top|} @!max_internal=100; {maximum number of internal quantities} @!buf_size=500; {maximum number of characters simultaneously present in current lines of open files; must not exceed |max_halfword|} @!error_line=72; {width of context lines on terminal error messages} @!half_error_line=42; {width of first lines of contexts in terminal error messages; should be between 30 and |error_line-15|} @!max_print_line=79; {width of longest text lines output; should be at least 60} @!screen_width=768; {number of pixels in each row of screen display} @!screen_depth=1024; {number of pixels in each column of screen display} @!stack_size=30; {maximum number of simultaneous input sources} @!max_strings=2000; {maximum number of strings; must not exceed |max_halfword|} @!string_vacancies=8000; {the minimum number of characters that should be available for the user's identifier names and strings, after \MF's own error messages are stored} @!pool_size=32000; {maximum number of characters in strings, including all error messages and help texts, and the names of all identifiers; must exceed |string_vacancies| by the total length of \MF's own strings, which is currently about 22000} @!move_size=5000; {space for storing moves in a single octant} @!max_wiggle=300; {number of autorounded points per cycle} @!gf_buf_size=800; {size of the output buffer, must be a multiple of 8} @!file_name_size=40; {file names shouldn't be longer than this} @!pool_name='MFbases:MF.POOL '; {string of length |file_name_size|; tells where the string pool appears} @.MFbases@> @!path_size=300; {maximum number of knots between breakpoints of a path} @!bistack_size=785; {size of stack for bisection algorithms; should probably be left at this value} @!header_size=100; {maximum number of \.{TFM} header words, times~4} @!lig_table_size=300; {maximum number of ligature/kern steps} @!max_font_dimen=50; {maximum number of \&{fontdimen} parameters} @ Like the preceding parameters, the following quantities can be changed at compile time to extend or reduce \MF's capacity. But if they are changed, it is necessary to rerun the initialization program \.{INIMF} @.INIMF@> to generate new tables for the production \MF\ program. One can't simply make helter-skelter changes to the following constants, since certain rather complex initialization numbers are computed from them. They are defined here using \.{WEB} macros, instead of being put into \PASCAL's |const| list, in order to emphasize this distinction. @d mem_min=0 {smallest index in the |mem| array, must not be less than |min_halfword|} @d mem_top==30000 {largest index in the |mem| array dumped by \.{INIMF}; must be substantially larger than |mem_min| and not greater than |mem_max|} @d hash_size=2100 {maximum number of symbolic tokens, must be less than |max_halfword-3*param_size|} @d hash_prime=1777 {a prime number equal to about 85\% of |hash_size|} @d max_in_open=6 {maximum number of input files and error insertions that can be going on simultaneously} @d param_size=150 {maximum number of simultaneous macro parameters} @^system dependencies@> @ In case somebody has inadvertently made bad settings of the ``constants,'' \MF\ checks them using a global variable called |bad|. This is the first of many sections of \MF\ where global variables are defined. @= @!bad:integer; {is some ``constant'' wrong?} @ Later on we will say `\ignorespaces|if mem_max>=max_halfword then bad:=10|', or something similar. (We can't do that until |max_halfword| has been defined.) @= bad:=0; if (half_error_line<30)or(half_error_line>error_line-15) then bad:=1; if max_print_line<60 then bad:=2; if gf_buf_size mod 8<>0 then bad:=3; if mem_min+1100>mem_top then bad:=4; if hash_prime>hash_size then bad:=5; if header_size mod 4 <> 0 then bad:=6; @ Labels are given symbolic names by the following definitions, so that occasional |goto| statements will be meaningful. We insert the label `|exit|:' just before the `\ignorespaces|end|\unskip' of a procedure in which we have used the `|return|' statement defined below; the label `|restart|' is occasionally used at the very beginning of a procedure; and the label `|reswitch|' is occasionally used just prior to a |case| statement in which some cases change the conditions and we wish to branch to the newly applicable case. Loops that are set up with the |loop| construction defined below are commonly exited by going to `|done|' or to `|found|' or to `|not_found|', and they are sometimes repeated by going to `|continue|'. If two or more parts of a subroutine start differently but end up the same, the shared code may be gathered together at `|common_ending|'. Incidentally, this program never declares a label that isn't actually used, because some fussy \PASCAL\ compilers will complain about redundant labels. @d exit=10 {go here to leave a procedure} @d restart=20 {go here to start a procedure again} @d reswitch=21 {go here to start a case statement again} @d continue=22 {go here to resume a loop} @d done=30 {go here to exit a loop} @d done1=31 {like |done|, when there is more than one loop} @d done2=32 {for exiting the second loop in a long block} @d done3=33 {for exiting the third loop in a very long block} @d done4=34 {for exiting the fourth loop in an extremely long block} @d done5=35 {for exiting the fifth loop in an immense block} @d done6=36 {for exiting the sixth loop in a block} @d found=40 {go here when you've found it} @d found1=41 {like |found|, when there's more than one per routine} @d found2=42 {like |found|, when there's more than two per routine} @d not_found=45 {go here when you've found nothing} @d common_ending=50 {go here when you want to merge with another branch} @ Here are some macros for common programming idioms. @d incr(#) == #:=#+1 {increase a variable by unity} @d decr(#) == #:=#-1 {decrease a variable by unity} @d negate(#) == #:=-# {change the sign of a variable} @d double(#) == #:=#+# {multiply a variable by two} @d loop == @+ while true do@+ {repeat over and over until a |goto| happens} @f loop == xclause {\.{WEB}'s |xclause| acts like `\ignorespaces|while true do|\unskip'} @d do_nothing == {empty statement} @d return == goto exit {terminate a procedure call} @f return == nil {\.{WEB} will henceforth say |return| instead of \\{return}} @* \[2] The character set. In order to make \MF\ readily portable to a wide variety of computers, all of its input text is converted to an internal seven-bit code that is essentially standard ASCII, the ``American Standard Code for Information Interchange.'' This conversion is done immediately when each character is read in. Conversely, characters are converted from ASCII to the user's external representation just before they are output to a text file. @^ASCII code@> Such an internal code is relevant to users of \MF\ only with respect to the \&{char} and \&{ASCII} operations, and the comparison of strings. @ Characters of text that have been converted to \MF's internal form are said to be of type |ASCII_code|, which is a subrange of the integers. @= @!ASCII_code=0..127; {seven-bit numbers} @ The original \PASCAL\ compiler was designed in the late 60s, when six-bit character sets were common, so it did not make provision for lowercase letters. Nowadays, of course, we need to deal with both capital and small letters in a convenient way, especially in a program for font design; so the present specification of \MF\ has been written under the assumption that the \PASCAL\ compiler and run-time system permit the use of text files with more than 64 distinguishable characters. More precisely, we assume that the character set contains at least the letters and symbols associated with ASCII codes @'40 through @'176; all of these characters are now available on most computer terminals. Since we are dealing with more characters than were present in the first \PASCAL\ compilers, we have to decide what to call the associated data type. Some \PASCAL s use the original name |char| for the characters in text files, even though there now are more than 64 such characters, while other \PASCAL s consider |char| to be a 64-element subrange of a larger data type that has some other name. In order to accommodate this difference, we shall use the name |text_char| to stand for the data type of the characters that are converted to and from |ASCII_code| when they are input and output. We shall also assume that |text_char| consists of the elements |chr(first_text_char)| through |chr(last_text_char)|, inclusive. The following definitions should be adjusted if necessary. @^system dependencies@> @d text_char == char {the data type of characters in text files} @d first_text_char=0 {ordinal number of the smallest element of |text_char|} @d last_text_char=127 {ordinal number of the largest element of |text_char|} @= @!i:0..last_text_char; @ The \MF\ processor converts between ASCII code and the user's external character set by means of arrays |xord| and |xchr| that are analogous to \PASCAL's |ord| and |chr| functions. @= @!xord: array [text_char] of ASCII_code; {specifies conversion of input characters} @!xchr: array [ASCII_code] of text_char; {specifies conversion of output characters} @ Since we are assuming that our \PASCAL\ system is able to read and write the visible characters of standard ASCII (although not necessarily using the ASCII codes to represent them), the following assignment statements initialize most of the |xchr| array properly, without needing any system-dependent changes. On the other hand, it is possible to implement \MF\ with less complete character sets, and in such cases it will be necessary to change something here. @^system dependencies@> @= xchr[@'40]:=' '; xchr[@'41]:='!'; xchr[@'42]:='"'; xchr[@'43]:='#'; xchr[@'44]:='$'; xchr[@'45]:='%'; xchr[@'46]:='&'; xchr[@'47]:='''';@/ xchr[@'50]:='('; xchr[@'51]:=')'; xchr[@'52]:='*'; xchr[@'53]:='+'; xchr[@'54]:=','; xchr[@'55]:='-'; xchr[@'56]:='.'; xchr[@'57]:='/';@/ xchr[@'60]:='0'; xchr[@'61]:='1'; xchr[@'62]:='2'; xchr[@'63]:='3'; xchr[@'64]:='4'; xchr[@'65]:='5'; xchr[@'66]:='6'; xchr[@'67]:='7';@/ xchr[@'70]:='8'; xchr[@'71]:='9'; xchr[@'72]:=':'; xchr[@'73]:=';'; xchr[@'74]:='<'; xchr[@'75]:='='; xchr[@'76]:='>'; xchr[@'77]:='?';@/ xchr[@'100]:='@@'; xchr[@'101]:='A'; xchr[@'102]:='B'; xchr[@'103]:='C'; xchr[@'104]:='D'; xchr[@'105]:='E'; xchr[@'106]:='F'; xchr[@'107]:='G';@/ xchr[@'110]:='H'; xchr[@'111]:='I'; xchr[@'112]:='J'; xchr[@'113]:='K'; xchr[@'114]:='L'; xchr[@'115]:='M'; xchr[@'116]:='N'; xchr[@'117]:='O';@/ xchr[@'120]:='P'; xchr[@'121]:='Q'; xchr[@'122]:='R'; xchr[@'123]:='S'; xchr[@'124]:='T'; xchr[@'125]:='U'; xchr[@'126]:='V'; xchr[@'127]:='W';@/ xchr[@'130]:='X'; xchr[@'131]:='Y'; xchr[@'132]:='Z'; xchr[@'133]:='['; xchr[@'134]:='\'; xchr[@'135]:=']'; xchr[@'136]:='^'; xchr[@'137]:='_';@/ xchr[@'140]:='`'; xchr[@'141]:='a'; xchr[@'142]:='b'; xchr[@'143]:='c'; xchr[@'144]:='d'; xchr[@'145]:='e'; xchr[@'146]:='f'; xchr[@'147]:='g';@/ xchr[@'150]:='h'; xchr[@'151]:='i'; xchr[@'152]:='j'; xchr[@'153]:='k'; xchr[@'154]:='l'; xchr[@'155]:='m'; xchr[@'156]:='n'; xchr[@'157]:='o';@/ xchr[@'160]:='p'; xchr[@'161]:='q'; xchr[@'162]:='r'; xchr[@'163]:='s'; xchr[@'164]:='t'; xchr[@'165]:='u'; xchr[@'166]:='v'; xchr[@'167]:='w';@/ xchr[@'170]:='x'; xchr[@'171]:='y'; xchr[@'172]:='z'; xchr[@'173]:='{'; xchr[@'174]:='|'; xchr[@'175]:='}'; xchr[@'176]:='~';@/ xchr[0]:=' '; xchr[@'177]:=' '; {ASCII codes 0 and |@'177| do not appear in text} @ The ASCII code is ``standard'' only to a certain extent, since many computer installations have found it advantageous to have ready access to more than 94 printing characters. If \MF\ is being used on a garden-variety \PASCAL\ for which only standard ASCII codes will appear in the input and output files, it doesn't really matter what codes are specified in |xchr[1..@'37]|, but the safest policy is to blank everything out by using the code shown below. However, other settings of |xchr| will make \MF\ more friendly on computers that have an extended character set, so that users can type things like `\.^^Z' instead of `\.{<>}'. People with extended character sets can assign codes arbitrarily, giving an |xchr| equivalent to whatever characters the users of \MF\ are allowed to have in their input files. Appropriate changes to \MF's |char_class| table should then be made. (Unlike \TeX, each installation of \MF\ has a fixed assignment of category codes, called the |char_class|.) Such changes make portability of programs more difficult, so they should be introduced cautiously if at all. @^character set dependencies@> @^system dependencies@> @= for i:=1 to @'37 do xchr[i]:=' '; @ The following system-independent code makes the |xord| array contain a suitable inverse to the information in |xchr|. Note that if |xchr[i]=xchr[j]| where |i= for i:=first_text_char to last_text_char do xord[chr(i)]:=@'177; for i:=1 to @'176 do xord[xchr[i]]:=i; @* \[3] Input and output. The bane of portability is the fact that different operating systems treat input and output quite differently, perhaps because computer scientists have not given sufficient attention to this problem. People have felt somehow that input and output are not part of ``real'' programming. Well, it is true that some kinds of programming are more fun than others. With existing input/output conventions being so diverse and so messy, the only sources of joy in such parts of the code are the rare occasions when one can find a way to make the program a little less bad than it might have been. We have two choices, either to attack I/O now and get it over with, or to postpone it until near the end. Neither prospect is very attractive, so let's get it over with. The basic operations we need to do are (1)~inputting and outputting of text, to or from a file or the user's terminal; (2)~inputting and outputting of eight-bit bytes, to or from a file; (3)~instructing the operating system to initiate (``open'') or to terminate (``close'') input or output from a specified file; (4)~testing whether the end of an input file has been reached; (5)~display of bits on the user's screen. The bit-display operation will be discussed in a later section; we shall deal here only with more traditional kinds of I/O. \MF\ needs to deal with two kinds of files. We shall use the term |alpha_file| for a file that contains textual data, and the term |byte_file| for a file that contains eight-bit binary information. These two types turn out to be the same on many computers, but sometimes there is a significant distinction, so we shall be careful to distinguish between them. Standard protocols for transferring such files from computer to computer, via high-speed networks, are now becoming available to more and more communities of users. The program actually makes use also of a third kind of file, called a |word_file|, when dumping and reloading base information for its own initialization. We shall define a word file later; but it will be possible for us to specify simple operations on word files before they are defined. @= @!eight_bits=0..255; {unsigned one-byte quantity} @!alpha_file=packed file of text_char; {files that contain textual data} @!byte_file=packed file of eight_bits; {files that contain binary data} @ Most of what we need to do with respect to input and output can be handled by the I/O facilities that are standard in \PASCAL, i.e., the routines called |get|, |put|, |eof|, and so on. But standard \PASCAL\ does not allow file variables to be associated with file names that are determined at run time, so it cannot be used to implement \MF; some sort of extension to \PASCAL's ordinary |reset| and |rewrite| is crucial for our purposes. We shall assume that |name_of_file| is a variable of an appropriate type such that the \PASCAL\ run-time system being used to implement \MF\ can open a file whose external name is specified by |name_of_file|. @^system dependencies@> @= @!name_of_file:packed array[1..file_name_size] of char;@;@/ {on some systems this may be a \&{record} variable} @!name_length:0..file_name_size;@/{this many characters are actually relevant in |name_of_file| (the rest are blank)} @ The \ph\ compiler with which the present version of \MF\ was prepared has extended the rules of \PASCAL\ in a very convenient way. To open file~|f|, we can write $$\vbox{\halign{#\hfil\qquad&#\hfil\cr |reset(f,@t\\{name}@>,'/O')|&for input;\cr |rewrite(f,@t\\{name}@>,'/O')|&for output.\cr}}$$ The `\\{name}' parameter, which is of type `\ignorespaces|packed array[@t\<\\{any}>@>] of text_char|', stands for the name of the external file that is being opened for input or output. Blank spaces that might appear in \\{name} are ignored. The `\.{/O}' parameter tells the operating system not to issue its own error messages if something goes wrong. If a file of the specified name cannot be found, or if such a file cannot be opened for some other reason (e.g., someone may already be trying to write the same file), we will have |@!erstat(f)<>0| after an unsuccessful |reset| or |rewrite|. This allows \MF\ to undertake appropriate corrective action. @:PASCAL H}{\ph@> @^system dependencies@> \MF's file-opening procedures return |false| if no file identified by |name_of_file| could be opened. @d reset_OK(#)==erstat(#)=0 @d rewrite_OK(#)==erstat(#)=0 @p function a_open_in(var @!f:alpha_file):boolean; {open a text file for input} begin reset(f,name_of_file,'/O'); a_open_in:=reset_OK(f); end; @# function a_open_out(var @!f:alpha_file):boolean; {open a text file for output} begin rewrite(f,name_of_file,'/O'); a_open_out:=rewrite_OK(f); end; @# function b_open_out(var @!f:byte_file):boolean; {open a binary file for output} begin rewrite(f,name_of_file,'/O'); b_open_out:=rewrite_OK(f); end; @# function w_open_in(var @!f:word_file):boolean; {open a word file for input} begin reset(f,name_of_file,'/O'); w_open_in:=reset_OK(f); end; @# function w_open_out(var @!f:word_file):boolean; {open a word file for output} begin rewrite(f,name_of_file,'/O'); w_open_out:=rewrite_OK(f); end; @ Files can be closed with the \ph\ routine `|close(f)|', which @^system dependencies@> should be used when all input or output with respect to |f| has been completed. This makes |f| available to be opened again, if desired; and if |f| was used for output, the |close| operation makes the corresponding external file appear on the user's area, ready to be read. @p procedure a_close(var @!f:alpha_file); {close a text file} begin close(f); end; @# procedure b_close(var @!f:byte_file); {close a binary file} begin close(f); end; @# procedure w_close(var @!f:word_file); {close a word file} begin close(f); end; @ Binary input and output are done with \PASCAL's ordinary |get| and |put| procedures, so we don't have to make any other special arrangements for binary~I/O. Text output is also easy to do with standard \PASCAL\ routines. The treatment of text input is more difficult, however, because of the necessary translation to |ASCII_code| values. \MF's conventions should be efficient, and they should blend nicely with the user's operating environment. @ Input from text files is read one line at a time, using a routine called |input_ln|. This function is defined in terms of global variables called |buffer|, |first|, and |last| that will be described in detail later; for now, it suffices for us to know that |buffer| is an array of |ASCII_code| values, and that |first| and |last| are indices into this array representing the beginning and ending of a line of text. @= @!buffer:array[0..buf_size] of ASCII_code; {lines of characters being read} @!first:0..buf_size; {the first unused position in |buffer|} @!last:0..buf_size; {end of the line just input to |buffer|} @!max_buf_stack:0..buf_size; {largest index used in |buffer|} @ The |input_ln| function brings the next line of input from the specified field into available positions of the buffer array and returns the value |true|, unless the file has already been entirely read, in which case it returns |false| and sets |last:=first|. In general, the |ASCII_code| numbers that represent the next line of the file are input into |buffer[first]|, |buffer[first+1]|, \dots, |buffer[last-1]|; and the global variable |last| is set equal to |first| plus the length of the line. Trailing blanks are removed from the line; thus, either |last=first| (in which case the line was entirely blank) or |buffer[last-1]<>" "|. @^inner loop@> An overflow error is given, however, if the normal actions of |input_ln| would make |last>=buf_size|; this is done so that other parts of \MF\ can safely look at the contents of |buffer[last+1]| without overstepping the bounds of the |buffer| array. Upon entry to |input_ln|, the condition |first=max_buf_stack then begin max_buf_stack:=last+1; if max_buf_stack=buf_size then overflow("buffer size",buf_size); @:METAFONT capacity exceeded buffer size}{\quad buffer size@> end; buffer[last]:=xord[f^]; get(f); incr(last); if buffer[last-1]<>" " then last_nonblank:=last; end; last:=last_nonblank; input_ln:=true; end; end; @ The user's terminal acts essentially like other files of text, except that it is used both for input and for output. When the terminal is considered an input file, the file variable is called |term_in|, and when it is considered an output file the file variable is |term_out|. @^system dependencies@> @= @!term_in:alpha_file; {the terminal as an input file} @!term_out:alpha_file; {the terminal as an output file} @ Here is how to open the terminal files in \ph. The `\.{/I}' switch suppresses the first |get|. @^system dependencies@> @d t_open_in==reset(term_in,'TTY:','/O/I') {open the terminal for text input} @d t_open_out==rewrite(term_out,'TTY:','/O') {open the terminal for text output} @ Sometimes it is necessary to synchronize the input/output mixture that happens on the user's terminal, and three system-dependent procedures are used for this purpose. The first of these, |update_terminal|, is called when we want to make sure that everything we have output to the terminal so far has actually left the computer's internal buffers and been sent. The second, |clear_terminal|, is called when we wish to cancel any input that the user may have typed ahead (since we are about to issue an unexpected error message). The third, |wake_up_terminal|, is supposed to revive the terminal if the user has disabled it by some instruction to the operating system. The following macros show how these operations can be specified in \ph: @^system dependencies@> @d update_terminal == break(term_out) {empty the terminal output buffer} @d clear_terminal == break_in(term_in,true) {clear the terminal input buffer} @d wake_up_terminal == do_nothing {cancel the user's cancellation of output} @ We need a special routine to read the first line of \MF\ input from the user's terminal. This line is different because it is read before we have opened the transcript file; there is sort of a ``chicken and egg'' problem here. If the user types `\.{input cmr10}' on the first line, or if some macro invoked by that line does such an \.{input}, the transcript file will be named `\.{cmr10.log}'; but if no \.{input} commands are performed during the first line of terminal input, the transcript file will acquire its default name `\.{mfput.log}'. (The transcript file will not contain error messages generated by the first line before the first \.{input} command.) The first line is even more special if we are lucky enough to have an operating system that treats \MF\ differently from a run-of-the-mill \PASCAL\ object program. It's nice to let the user start running a \MF\ job by typing a command line like `\.{MF cmr10}'; in such a case, \MF\ will operate as if the first line of input were `\.{cmr10}', i.e., the first line will consist of the remainder of the command line, after the part that invoked \MF. @ Different systems have different ways to get started. But regardless of what conventions are adopted, the routine that initializes the terminal should satisfy the following specifications: \yskip\textindent{1)}It should open file |term_in| for input from the terminal. (The file |term_out| will already be open for output to the terminal.) \textindent{2)}If the user has given a command line, this line should be considered the first line of terminal input. Otherwise the user should be prompted with `\.{**}', and the first line of input should be whatever is typed in response. \textindent{3)}The first line of input, which might or might not be a command line, should appear in locations |first| to |last-1| of the |buffer| array. \textindent{4)}The global variable |loc| should be set so that the character that \MF\ reads next is in |buffer[loc]|. This character should not be blank, and we should have |loc @p function init_terminal:boolean; {gets the terminal input started} label exit; begin t_open_in; loop@+begin wake_up_terminal; write(term_out,'**'); update_terminal; @.**@> if not input_ln(term_in,true) then {this shouldn't happen} begin write_ln(term_out); write(term_out,'! End of file on the terminal... why?'); @.End of file on the terminal@> init_terminal:=false; return; end; loc:=first; while (loc which converts single-character strings into the ASCII code number of the single character involved, while it converts other strings into integers and builds a string pool file. Thus, when the string constant \.{"."} appears in the program below, \.{WEB} converts it into the integer 46, which is the ASCII code for a period, while \.{WEB} will convert a string like \.{"hello"} into some integer greater than~127. String number 46 will presumably be the single character `\..'\thinspace; but some ASCII codes have no standard visible representation, and \MF\ may need to be able to print an arbitrary ASCII character, so the first 128 strings are used to specify exactly what should be printed for each of the 128 possibilities. Elements of the |str_pool| array must be ASCII codes that can actually be printed; i.e., they must have an |xchr| equivalent in the local character set. (This restriction applies only to preloaded strings, not to those generated dynamically by the user.) @= @!pool_pointer = 0..pool_size; {for variables that point into |str_pool|} @!str_number = 0..max_strings; {for variables that point into |str_start|} @ @= @!str_pool:packed array[pool_pointer] of ASCII_code; {the characters} @!str_start : array[str_number] of pool_pointer; {the starting pointers} @!pool_ptr : pool_pointer; {first unused position in |str_pool|} @!str_ptr : str_number; {number of the current string being created} @!init_pool_ptr : pool_pointer; {the starting value of |pool_ptr|} @!init_str_ptr : str_number; {the starting value of |str_ptr|} @!max_pool_ptr : pool_pointer; {the maximum so far of |pool_ptr|} @!max_str_ptr : str_number; {the maximum so far of |str_ptr|} @ Several of the elementary string operations are performed using \.{WEB} macros instead of \PASCAL\ procedures, because many of the operations are done quite frequently and we want to avoid the overhead of procedure calls. For example, here is a simple macro that computes the length of a string. @.WEB@> @d length(#)==(str_start[#+1]-str_start[#]) {the number of characters in string number \#} @ The length of the current string is called |cur_length|: @d cur_length == (pool_ptr - str_start[str_ptr]) @ Strings are created by appending character codes to |str_pool|. The macro called |append_char|, defined here, does not check to see if the value of |pool_ptr| has gotten too high; this test is supposed to be made before |append_char| is used. To test if there is room to append |l| more characters to |str_pool|, we shall write |str_room(l)|, which aborts \MF\ and gives an apologetic error message if there isn't enough room. @d append_char(#) == {put |ASCII_code| \# at the end of |str_pool|} begin str_pool[pool_ptr]:=#; incr(pool_ptr); end @d str_room(#) == {make sure that the pool hasn't overflowed} begin if pool_ptr+# > max_pool_ptr then begin if pool_ptr+# > pool_size then overflow("pool size",pool_size-init_pool_ptr); @:METAFONT capacity exceeded pool size}{\quad pool size@> max_pool_ptr:=pool_ptr+#; end; end @ \MF's string expressions are implemented in a brute-force way: Every new string or substring that is needed is simply copied into the string pool. Such a scheme can be justified because string expressions aren't a big deal in \MF\ applications; strings rarely need to be saved from one statement to the next. But it would waste space needlessly if we didn't try to reclaim the space of strings that are going to be used only once. Therefore a simple reference count mechanism is provided: If there are @^reference counts@> no references to a certain string from elsewhere in the program, and if there are no references to any strings created subsequent to it, then the string space will be reclaimed. The number of references to string number |s| will be |str_ref[s]|. The special value |str_ref[s]=max_str_ref=127| is used to denote an unknown positive number of references; such strings will never be recycled. If a string is ever referred to more than 126 times, simultaneously, we put it in this category. Hence a single byte suffices to store each |str_ref|. @d max_str_ref=127 {``infinite'' number of references} @d add_str_ref(#)==begin if str_ref[#]= @!str_ref:array[str_number] of 0..max_str_ref; @ Here's what we do when a string reference disappears: @d delete_str_ref(#)== begin if str_ref[#]1 then decr(str_ref[#])@+else flush_string(#); end @= procedure flush_string(@!s:str_number); begin if s0; pool_ptr:=str_start[str_ptr]; end; @ Once a sequence of characters has been appended to |str_pool|, it officially becomes a string when the function |make_string| is called. This function returns the identification number of the new string as its value. @p function make_string : str_number; {current string enters the pool} begin if str_ptr=max_str_ptr then begin if str_ptr=max_strings then overflow("number of strings",max_strings-init_str_ptr); @:METAFONT capacity exceeded number of strings}{\quad number of strings@> incr(max_str_ptr); end; str_ref[str_ptr]:=1; incr(str_ptr); str_start[str_ptr]:=pool_ptr; make_string:=str_ptr-1; end; @ The following subroutine compares string |s| with another string of the same length that appears in |buffer| starting at position |k|; the result is |true| if and only if the strings are equal. @p function str_eq_buf(@!s:str_number;@!k:integer):boolean; {test equality of strings} label not_found; {loop exit} var @!j: pool_pointer; {running index} @!result: boolean; {result of comparison} begin j:=str_start[s]; while jbuffer[k] then begin result:=false; goto not_found; end; incr(j); incr(k); end; result:=true; not_found: str_eq_buf:=result; end; @ Here is a similar routine, but it compares two strings in the string pool, and it does not assume that they have the same length. If the first string is lexicographically greater than, less than, or equal to the second, the result is respectively positive, negative, or zero. @p function str_vs_str(@!s,@!t:str_number):integer; {test equality of strings} label exit; var @!j,@!k: pool_pointer; {running indices} @!ls,@!lt:integer; {lengths} @!l:integer; {length remaining to test} begin ls:=length(s); lt:=length(t); if ls<=lt then l:=ls@+else l:=lt; j:=str_start[s]; k:=str_start[t]; while l>0 do begin if str_pool[j]<>str_pool[k] then begin str_vs_str:=str_pool[j]-str_pool[k]; return; end; incr(j); incr(k); decr(l); end; str_vs_str:=ls-lt; exit:end; @ The initial values of |str_pool|, |str_start|, |pool_ptr|, and |str_ptr| are computed by the \.{INIMF} program, based in part on the information that \.{WEB} has output while processing \MF. @.INIMF@> @^string pool@> @p @!init function get_strings_started:boolean; {initializes the string pool, but returns |false| if something goes wrong} label done,exit; var @!k,@!l:0..127; {small indices or counters} @!m,@!n:text_char; {characters input from |pool_file|} @!g:str_number; {garbage} @!a:integer; {accumulator for check sum} @!c:boolean; {check sum has been checked} begin pool_ptr:=0; str_ptr:=0; max_pool_ptr:=0; max_str_ptr:=0; str_start[0]:=0; @; @; exit:end; tini @ @= for k:=0 to 127 do begin if (@) then begin append_char("^"); append_char("^"); if k<@'100 then append_char(k+@'100) else append_char(k-@'100); end else append_char(k); g:=make_string; str_ref[g]:=max_str_ref; end @ The first 128 strings will contain 95 standard ASCII characters, and the other 33 characters will be printed in three-symbol form like `\.{\^\^A}' unless a system-dependent change is made here. Installations that have an extended character set, where for example |xchr[@'32]=@t\.{\'^^Z\'}@>|, would like string @'32 to be the single character @'32 instead of the three characters @'136, @'136, @'132 (\.{\^\^Z}). On the other hand, even people with an extended character set will want to represent string @'15 by \.{\^\^M}, since @'15 is ASCII's ``carriage return'' code; the idea is to produce visible strings instead of tabs or line-feeds or carriage-returns or bell-rings or characters that are treated anomalously in text files. The boolean expression defined here should be |true| unless \MF\ internal code number~|k| corresponds to a non-troublesome visible symbol in the local character set. At MIT, for example, the appropriate formula would be `|k in [0,@'10..@'12,@'14,@'15,@'33,@'177]|'. If character |k| cannot be printed, then character |k+@'100| or |k-@'100| must be printable; thus, at least 64 printable characters are needed. @^character set dependencies@> @^system dependencies@> @= (k<" ")or(k>"~") @ When the \.{WEB} system program called \.{TANGLE} processes the \.{MF.WEB} description that you are now reading, it outputs the \PASCAL\ program \.{MF.PAS} and also a string pool file called \.{MF.POOL}. The \.{INIMF} @.WEB@>@.INIMF@> program reads the latter file, where each string appears as a two-digit decimal length followed by the string itself, and the information is recorded in \MF's string memory. @= @!init @!pool_file:alpha_file; {the string-pool file output by \.{TANGLE}} tini @ @d bad_pool(#)==begin wake_up_terminal; write_ln(term_out,#); a_close(pool_file); get_strings_started:=false; return; end @= name_of_file:=pool_name; {we needn't set |name_length|} if a_open_in(pool_file) then begin c:=false; repeat @; until c; a_close(pool_file); get_strings_started:=true; end else bad_pool('! I can''t read MF.POOL.') @.I can't read MF.POOL@> @ @= begin if eof(pool_file) then bad_pool('! MF.POOL has no check sum.'); @.MF.POOL has no check sum@> read(pool_file,m,n); {read two digits of string length} if m='*' then @ else begin if (xord[m]<"0")or(xord[m]>"9")or@| (xord[n]<"0")or(xord[n]>"9") then bad_pool('! MF.POOL line doesn''t begin with two digits.'); @.MF.POOL line doesn't...@> l:=xord[m]*10+xord[n]-"0"*11; {compute the length} if pool_ptr+l+string_vacancies>pool_size then bad_pool('! You have to increase POOLSIZE.'); @.You have to increase POOLSIZE@> for k:=1 to l do begin if eoln(pool_file) then m:=' '@+else read(pool_file,m); append_char(xord[m]); end; read_ln(pool_file); g:=make_string; str_ref[g]:=max_str_ref; end; end @ The \.{WEB} operation \.{@@\$} denotes the value that should be at the end of this \.{MF.POOL} file; any other value means that the wrong pool file has been loaded. @^check sum@> @= begin a:=0; k:=1; loop@+ begin if (xord[n]<"0")or(xord[n]>"9") then bad_pool('! MF.POOL check sum doesn''t have nine digits.'); @.MF.POOL check sum...@> a:=10*a+xord[n]-"0"; if k=9 then goto done; incr(k); read(pool_file,n); end; done: if a<>@$ then bad_pool('! MF.POOL doesn''t match; TANGLE me again.'); @.MF.POOL doesn't match@> c:=true; end @* \[5] On-line and off-line printing. Messages that are sent to a user's terminal and to the transcript-log file are produced by several `|print|' procedures. These procedures will direct their output to a variety of places, based on the setting of the global variable |selector|, which has the following possible values: \yskip \hang |term_and_log|, the normal setting, prints on the terminal and on the transcript file. \hang |log_only|, prints only on the transcript file. \hang |term_only|, prints only on the terminal. \hang |no_print|, doesn't print at all. This is used only in rare cases before the transcript file is open. \hang |pseudo|, puts output into a cyclic buffer that is used by the |show_context| routine; see that routine for the reasoning behind this curious mode. \hang |new_string|, appends the output to the current string in the string pool. \yskip \noindent The symbolic names `|term_and_log|', etc., have been assigned numeric codes that satisfy the convenient relations |no_print+1=term_only|, |no_print+2=log_only|, |term_only+2=log_only+1=term_and_log|. Three additional global variables, |tally| and |term_offset| and |file_offset|, record the number of characters that have been printed since they were most recently cleared to zero. We use |tally| to record the length of (possibly very long) stretches of printing; |term_offset| and |file_offset|, on the other hand, keep track of how many characters have appeared so far on the current line that has been output to the terminal or to the transcript file, respectively. @d no_print=0 {|selector| setting that makes data disappear} @d term_only=1 {printing is destined for the terminal only} @d log_only=2 {printing is destined for the transcript file only} @d term_and_log=3 {normal |selector| setting} @d pseudo=4 {special |selector| setting for |show_context|} @d new_string=5 {printing is deflected to the string pool} @d max_selector=5 {highest selector setting} @= @!log_file : alpha_file; {transcript of \MF\ session} @!selector : 0..max_selector; {where to print a message} @!dig : array[0..22] of 0..15; {digits in a number being output} @!tally : integer; {the number of characters recently printed} @!term_offset : 0..max_print_line; {the number of characters on the current terminal line} @!file_offset : 0..max_print_line; {the number of characters on the current file line} @!trick_buf:array[0..error_line] of ASCII_code; {circular buffer for pseudoprinting} @!trick_count: integer; {threshold for pseudoprinting, explained later} @!first_count: integer; {another variable for pseudoprinting} @ @= selector:=term_only; tally:=0; term_offset:=0; file_offset:=0; @ Macro abbreviations for output to the terminal and to the log file are defined here for convenience. Some systems need special conventions for terminal output, and it is possible to adhere to those conventions by changing |wterm|, |wterm_ln|, and |wterm_cr| here. @^system dependencies@> @d wterm(#)==write(term_out,#) @d wterm_ln(#)==write_ln(term_out,#) @d wterm_cr==write_ln(term_out) @d wlog(#)==write(log_file,#) @d wlog_ln(#)==write_ln(log_file,#) @d wlog_cr==write_ln(log_file) @ To end a line of text output, we call |print_ln|. @= procedure print_ln; {prints an end-of-line} begin case selector of term_and_log: begin wterm_cr; wlog_cr; term_offset:=0; file_offset:=0; end; log_only: begin wlog_cr; file_offset:=0; end; term_only: begin wterm_cr; term_offset:=0; end; no_print,pseudo,new_string: do_nothing; end; {there are no other cases} end; {note that |tally| is not affected} @ The |print_char| procedure sends one character to the desired destination, using the |xchr| array to map it into an external character compatible with |input_ln|. All printing comes through |print_ln| or |print_char|. @= procedure print_char(@!s:ASCII_code); {prints a single character} begin case selector of term_and_log: begin wterm(xchr[s]); wlog(xchr[s]); incr(term_offset); incr(file_offset); if term_offset=max_print_line then begin wterm_cr; term_offset:=0; end; if file_offset=max_print_line then begin wlog_cr; file_offset:=0; end; end; log_only: begin wlog(xchr[s]); incr(file_offset); if file_offset=max_print_line then print_ln; end; term_only: begin wterm(xchr[s]); incr(term_offset); if term_offset=max_print_line then print_ln; end; no_print: do_nothing; pseudo: if tally @= procedure print(@!s:integer); {prints string |s|} var @!j:pool_pointer; {current character code position} begin if (s<0)or(s>=str_ptr) then s:="???"; {this can't happen} @.???@> j:=str_start[s]; while j= procedure slow_print(@!s:integer); {prints string |s|} var @!j:pool_pointer; {current character code position} begin if (s<0)or(s>=str_ptr) then s:="???"; {this can't happen} @.???@> j:=str_start[s]; while j= wterm(banner); if base_ident=0 then wterm_ln(' (no base preloaded)') else begin print(base_ident); print_ln; end; update_terminal; @ The procedure |print_nl| is like |print|, but it makes sure that the string appears at the beginning of a new line. @= procedure print_nl(@!s:str_number); {prints string |s| at beginning of line} begin if ((term_offset>0)and(odd(selector)))or@| ((file_offset>0)and(selector>=log_only)) then print_ln; print(s); end; @ An array of digits in the range |0..9| is printed by |print_the_digs|. @= procedure print_the_digs(@!k:eight_bits); {prints |dig[k-1]|$\,\ldots\,$|dig[0]|} begin while k>0 do begin decr(k); print_char("0"+dig[k]); end; end; @ The following procedure, which prints out the decimal representation of a given integer |n|, has been written carefully so that it works properly if |n=0| or if |(-n)| would cause overflow. It does not apply |mod| or |div| to negative arguments, since such operations are not implemented consistently by all \PASCAL\ compilers. @= procedure print_int(@!n:integer); {prints an integer in decimal form} var k:0..23; {index to current digit; we assume that $|n|<10^{23}$} @!m:integer; {used to negate |n| in possibly dangerous cases} begin k:=0; if n<0 then begin print_char("-"); if n>-100000000 then negate(n) else begin m:=-1-n; n:=m div 10; m:=(m mod 10)+1; k:=1; if m<10 then dig[0]:=m else begin dig[0]:=0; incr(n); end; end; end; repeat dig[k]:=n mod 10; n:=n div 10; incr(k); until n=0; print_the_digs(k); end; @ \MF\ also makes use of a trivial procedure to print two digits. The following subroutine is usually called with a parameter in the range |0<=n<=99|. @p procedure print_dd(@!n:integer); {prints two least significant digits} begin n:=abs(n) mod 100; print_char("0"+(n div 10)); print_char("0"+(n mod 10)); end; @ Here is a procedure that asks the user to type a line of input, assuming that the |selector| setting is either |term_only| or |term_and_log|. The input is placed into locations |first| through |last-1| of the |buffer| array, and echoed on the transcript file if appropriate. This procedure is never called when |interaction term_offset:=0; {the user's line ended with \<\rm return>} decr(selector); {prepare to echo the input} if last<>first then for k:=first to last-1 do print(buffer[k]); print_ln; buffer[last]:="%"; incr(selector); {restore previous status} end; @* \[6] Reporting errors. When something anomalous is detected, \MF\ typically does something like this: $$\vbox{\halign{#\hfil\cr |print_err("Something anomalous has been detected");|\cr |help3("This is the first line of my offer to help.")|\cr |("This is the second line. I'm trying to")|\cr |("explain the best way for you to proceed.");|\cr |error;|\cr}}$$ A two-line help message would be given using |help2|, etc.; these informal helps should use simple vocabulary that complements the words used in the official error message that was printed. (Outside the U.S.A., the help messages should preferably be translated into the local vernacular. Each line of help is at most 60 characters long, in the present implementation, so that |max_print_line| will not be exceeded.) The |print_err| procedure supplies a `\.!' before the official message, and makes sure that the terminal is awake if a stop is going to occur. The |error| procedure supplies a `\..' after the official message, then it shows the location of the error; and if |interaction=error_stop_mode|, it also enters into a dialog with the user, during which time the help message may be printed. @^system dependencies@> @ The global variable |interaction| has four settings, representing increasing amounts of user interaction: @d batch_mode=0 {omits all stops and omits terminal output} @d nonstop_mode=1 {omits all stops} @d scroll_mode=2 {omits error stops} @d error_stop_mode=3 {stops at every opportunity to interact} @d print_err(#)==begin if interaction=error_stop_mode then wake_up_terminal; print_nl("! "); print(#); @.!\relax@> end @= @!interaction:batch_mode..error_stop_mode; {current level of interaction} @ @=interaction:=error_stop_mode; @ \MF\ is careful not to call |error| when the print |selector| setting might be unusual. The only possible values of |selector| at the time of error messages are \yskip\hang|no_print| (when |interaction=batch_mode| and |log_file| not yet open); \hang|term_only| (when |interaction>batch_mode| and |log_file| not yet open); \hang|log_only| (when |interaction=batch_mode| and |log_file| is open); \hang|term_and_log| (when |interaction>batch_mode| and |log_file| is open). @= if interaction=batch_mode then selector:=no_print@+else selector:=term_only @ A global variable |deletions_allowed| is set |false| if the |get_next| routine is active when |error| is called; this ensures that |get_next| will never be called recursively. @^recursion@> The global variable |history| records the worst level of error that has been detected. It has four possible values: |spotless|, |warning_issued|, |error_message_issued|, and |fatal_error_stop|. Another global variable, |error_count|, is increased by one when an |error| occurs without an interactive dialog, and it is reset to zero at the end of every statement. If |error_count| reaches 100, \MF\ decides that there is no point in continuing further. @d spotless=0 {|history| value when nothing has been amiss yet} @d warning_issued=1 {|history| value when |begin_diagnostic| has been called} @d error_message_issued=2 {|history| value when |error| has been called} @d fatal_error_stop=3 {|history| value when termination was premature} @= @!deletions_allowed:boolean; {is it safe for |error| to call |get_next|?} @!history:spotless..fatal_error_stop; {has the source input been clean so far?} @!error_count:-1..100; {the number of scrolled errors since the last statement ended} @ The value of |history| is initially |fatal_error_stop|, but it will be changed to |spotless| if \MF\ survives the initialization process. @= deletions_allowed:=true; error_count:=0; {|history| is initialized elsewhere} @ Since errors can be detected almost anywhere in \MF, we want to declare the error procedures near the beginning of the program. But the error procedures in turn use some other procedures, which need to be declared |forward| before we get to |error| itself. It is possible for |error| to be called recursively if some error arises when |get_next| is being used to delete a token, and/or if some fatal error occurs while \MF\ is trying to fix a non-fatal one. But such recursion @^recursion@> is never more than two levels deep. @= procedure@?normalize_selector; forward;@t\2@>@/ procedure@?get_next; forward;@t\2@>@/ procedure@?term_input; forward;@t\2@>@/ procedure@?show_context; forward;@t\2@>@/ procedure@?begin_file_reading; forward;@t\2@>@/ procedure@?open_log_file; forward;@t\2@>@/ procedure@?close_files_and_terminate; forward;@t\2@>@/ procedure@?clear_for_error_prompt; forward;@t\2@>@/ @t\4\hskip-\fontdimen2\font@>@;@+@!debug@+procedure@?debug_help; forward;@;@+gubed@;@/ @t\4@>@ @ Individual lines of help are recorded in the array |help_line|, which contains entries in positions |0..(help_ptr-1)|. They should be printed in reverse order, i.e., with |help_line[0]| last. @d hlp1(#)==help_line[0]:=#;@+end @d hlp2(#)==help_line[1]:=#; hlp1 @d hlp3(#)==help_line[2]:=#; hlp2 @d hlp4(#)==help_line[3]:=#; hlp3 @d hlp5(#)==help_line[4]:=#; hlp4 @d hlp6(#)==help_line[5]:=#; hlp5 @d help0==help_ptr:=0 {sometimes there might be no help} @d help1==@+begin help_ptr:=1; hlp1 {use this with one help line} @d help2==@+begin help_ptr:=2; hlp2 {use this with two help lines} @d help3==@+begin help_ptr:=3; hlp3 {use this with three help lines} @d help4==@+begin help_ptr:=4; hlp4 {use this with four help lines} @d help5==@+begin help_ptr:=5; hlp5 {use this with five help lines} @d help6==@+begin help_ptr:=6; hlp6 {use this with six help lines} @= @!help_line:array[0..5] of str_number; {helps for the next |error|} @!help_ptr:0..6; {the number of help lines present} @!use_err_help:boolean; {should the |err_help| string be shown?} @!err_help:str_number; {a string set up by \&{errhelp}} @ @= help_ptr:=0; use_err_help:=false; err_help:=0; @ The |jump_out| procedure just cuts across all active procedure levels and goes to |end_of_MF|. This is the only nonlocal |@!goto| statement in the whole program. It is used when there is no recovery from a particular error. Some \PASCAL\ compilers do not implement non-local |goto| statements. @^system dependencies@> In such cases the body of |jump_out| should simply be `|close_files_and_terminate|;\thinspace' followed by a call on some system procedure that quietly terminates the program. @= procedure jump_out; begin goto end_of_MF; end; @ Here now is the general |error| routine. @= procedure error; {completes the job of error reporting} label continue,exit; var @!c:ASCII_code; {what the user types} @!s1,@!s2,@!s3:integer; {used to save global variables when deleting tokens} @!j:pool_pointer; {character position being printed} begin if history; incr(error_count); if error_count=100 then begin print_nl("(That makes 100 errors; please try again.)"); @.That makes 100 errors...@> history:=fatal_error_stop; jump_out; end; @; exit:end; @ @= loop@+begin continue: clear_for_error_prompt; prompt_input("? "); @.?\relax@> if last=first then return; c:=buffer[first]; if c>="a" then c:=c+"A"-"a"; {convert to uppercase} @; end @ It is desirable to provide an `\.E' option here that gives the user an easy way to return from \MF\ to the system editor, with the offending line ready to be edited. But such an extension requires some system wizardry, so the present implementation simply types out what file should be edited and the relevant line number. @^system dependencies@> There is a secret `\.D' option available when the debugging routines have not been commented out. @^debugging@> @= case c of "0","1","2","3","4","5","6","7","8","9": if deletions_allowed then @; @t\4\4@>@;@+@!debug "D":begin debug_help;goto continue;@+end;@+gubed@/ "E": if file_ptr>0 then begin print_nl("You want to edit file "); @.You want to edit file x@> print(input_stack[file_ptr].name_field); print(" at line "); print_int(line); interaction:=scroll_mode; jump_out; end; "H": @; "I":@; "Q","R","S":@; "X":begin interaction:=scroll_mode; jump_out; end; othercases do_nothing endcases;@/ @ @ @= begin print("Type to proceed, S to scroll future error messages,");@/ @.Type to proceed...@> print_nl("R to run without stopping, Q to run quietly,");@/ print_nl("I to insert something, "); if file_ptr>0 then print("E to edit your file,"); if deletions_allowed then print_nl("1 or ... or 9 to ignore the next 1 to 9 tokens of input,"); print_nl("H for help, X to quit."); end @ Here the author of \MF\ apologizes for making use of the numerical relation between |"Q"|, |"R"|, |"S"|, and the desired interaction settings |batch_mode|, |nonstop_mode|, |scroll_mode|. @^Knuth, Donald Ervin@> @= begin error_count:=0; interaction:=batch_mode+c-"Q"; print("OK, entering "); case c of "Q":begin print("batchmode"); decr(selector); end; "R":print("nonstopmode"); "S":print("scrollmode"); end; {there are no other cases} print("..."); print_ln; update_terminal; return; end @ When the following code is executed, |buffer[(first+1)..(last-1)]| may contain the material inserted by the user; otherwise another prompt will be given. In order to understand this part of the program fully, you need to be familiar with \MF's input stacks. @= begin begin_file_reading; {enter a new syntactic level for terminal input} if last>first+1 then begin loc:=first+1; buffer[first]:=" "; end else begin prompt_input("insert>"); loc:=first; @.insert>@> end; first:=last+1; cur_input.limit_field:=last; return; end @ We allow deletion of up to 99 tokens at a time. @= begin s1:=cur_cmd; s2:=cur_mod; s3:=cur_sym; OK_to_interrupt:=false; if (last>first+1) and (buffer[first+1]>="0")and(buffer[first+1]<="9") then c:=c*10+buffer[first+1]-"0"*11 else c:=c-"0"; while c>0 do begin get_next; {one-level recursive call of |error| is possible} @; decr(c); end; cur_cmd:=s1; cur_mod:=s2; cur_sym:=s3; OK_to_interrupt:=true; help2("I have just deleted some text, as you asked.")@/ ("You can now delete more, or insert, or whatever."); show_context; goto continue; end @ @= begin if use_err_help then begin @; use_err_help:=false; end else begin if help_ptr=0 then help2("Sorry, I don't know how to help in this situation.")@/ @t\kern1em@>("Maybe you should try asking a human?"); repeat decr(help_ptr); print(help_line[help_ptr]); print_ln; until help_ptr=0; end; help4("Sorry, I already gave what help I could...")@/ ("Maybe you should try asking a human?")@/ ("An error might have occurred before I noticed any problems.")@/ ("``If all else fails, read the instructions.''");@/ goto continue; end @ @= j:=str_start[err_help]; while j"%" then print(str_pool[j]) else if j+1=str_start[err_help+1] then print_ln else if str_pool[j+1]<>"%" then print_ln else begin incr(j); print_char("%"); end; incr(j); end @ @= if interaction>batch_mode then decr(selector); {avoid terminal output} if use_err_help then begin print_nl(""); @; end else while help_ptr>0 do begin decr(help_ptr); print_nl(help_line[help_ptr]); end; print_ln; if interaction>batch_mode then incr(selector); {re-enable terminal output} print_ln @ In anomalous cases, the print selector might be in an unknown state; the following subroutine is called to fix things just enough to keep running a bit longer. @p procedure normalize_selector; begin if log_opened then selector:=term_and_log else selector:=term_only; if job_name=0 then open_log_file; if interaction=batch_mode then decr(selector); end; @ The following procedure prints \MF's last words before dying. @d succumb==begin if interaction=error_stop_mode then interaction:=scroll_mode; {no more interaction} if log_opened then error; @!debug if interaction>batch_mode then debug_help;@+gubed@;@/ history:=fatal_error_stop; jump_out; {irrecoverable error} end @= procedure fatal_error(@!s:str_number); {prints |s|, and that's it} begin normalize_selector;@/ print_err("Emergency stop"); help1(s); succumb; @.Emergency stop@> end; @ Here is the most dreaded error message. @= procedure overflow(@!s:str_number;@!n:integer); {stop due to finiteness} begin normalize_selector; print_err("METAFONT capacity exceeded, sorry ["); @.METAFONT capacity exceeded ...@> print(s); print_char("="); print_int(n); print_char("]"); help2("If you really absolutely need more capacity,")@/ ("you can ask a wizard to enlarge me."); succumb; end; @ The program might sometime run completely amok, at which point there is no choice but to stop. If no previous error has been detected, that's bad news; a message is printed that is really intended for the \MF\ maintenance person instead of the user (unless the user has been particularly diabolical). The index entries for `this can't happen' may help to pinpoint the problem. @^dry rot@> @= procedure confusion(@!s:str_number); {consistency check violated; |s| tells where} begin normalize_selector; if history help1("I'm broken. Please show this to someone who can fix can fix"); end else begin print_err("I can't go on meeting you like this"); @.I can't go on...@> help2("One of your faux pas seems to have wounded me deeply...")@/ ("in fact, I'm barely conscious. Please fix it and try again."); end; succumb; end; @ Users occasionally want to interrupt \MF\ while it's running. If the \PASCAL\ runtime system allows this, one can implement a routine that sets the global variable |interrupt| to some nonzero value when such an interrupt is signalled. Otherwise there is probably at least a way to make |interrupt| nonzero using the \PASCAL\ debugger. @^system dependencies@> @^debugging@> @d check_interrupt==begin if interrupt<>0 then pause_for_instructions; end @= @!interrupt:integer; {should \MF\ pause for instructions?} @!OK_to_interrupt:boolean; {should interrupts be observed?} @ @= interrupt:=0; OK_to_interrupt:=true; @ When an interrupt has been detected, the program goes into its highest interaction level and lets the user have the full flexibility of the |error| routine. \MF\ checks for interrupts only at times when it is safe to do this. @p procedure pause_for_instructions; begin if OK_to_interrupt then begin interaction:=error_stop_mode; if (selector=log_only)or(selector=no_print) then incr(selector); print_err("Interruption"); @.Interruption@> help3("You rang?")@/ ("Try to insert some instructions for me (e.g.,`I show x'),")@/ ("unless you just want to quit by typing `X'."); deletions_allowed:=false; error; deletions_allowed:=true; interrupt:=0; end; end; @ Many of \MF's error messages state that a missing token has been inserted behind the scenes. We can save string space and program space by putting this common code into a subroutine. @p procedure missing_err(@!s:str_number); begin print_err("Missing `"); print(s); print("' has been inserted"); @.Missing...inserted@> end; @* \[7] Arithmetic with scaled numbers. The principal computations performed by \MF\ are done entirely in terms of integers less than $2^{31}$ in magnitude; thus, the arithmetic specified in this program can be carried out in exactly the same way on a wide variety of computers, including some small ones. @^small computers@> But \PASCAL\ does not define the @!|div| operation in the case of negative dividends; for example, the result of |(-2*n-1) div 2| is |-(n+1)| on some computers and |-n| on others. There are two principal types of arithmetic: ``translation-preserving,'' in which the identity |(a+q*b)div b=(a div b)+q| is valid; and ``negation-preserving,'' in which |(-a)div b=-(a div b)|. This leads to two \MF s, which can produce different results, although the differences should be negligible when the language is being used properly. The \TeX\ processor has been defined carefully so that both varieties of arithmetic will produce identical output, but it would be too inefficient to constrain \MF\ in a similar way. @d el_gordo == @'17777777777 {$2^{31}-1$, the largest value that \MF\ likes} @ One of \MF's most common operations is the calculation of $\lfloor{a+b\over2}\rfloor$, the midpoint of two given integers |a| and~|b|. The only decent way to do this in \PASCAL\ is to write `|(a+b) div 2|'; but on most machines it is far more efficient to calculate `|(a+b)| right shifted one bit'. Therefore the midpoint operation will always be denoted by `|half(a+b)|' in this program. If \MF\ is being implemented with languages that permit binary shifting, the |half| macro should be changed to make this operation as efficient as possible. @d half(#)==(#) div 2 @ A single computation might use several subroutine calls, and it is desirable to avoid producing multiple error messages in case of arithmetic overflow. So the routines below set the global variable |arith_error| to |true| instead of reporting errors directly to the user. @= @!arith_error:boolean; {has arithmetic overflow occurred recently?} @ @= arith_error:=false; @ At crucial points the program will say |check_arith|, to test if an arithmetic error has been detected. @d check_arith==begin if arith_error then clear_arith;@+end @p procedure clear_arith; begin print_err("Arithmetic overflow"); @.Arithmetic overflow@> help4("Uh, oh. A little while ago one of the quantities that I was")@/ ("computing got too large, so I'm afraid your answers will be")@/ ("somewhat askew. You'll probably have to adopt different")@/ ("tactics next time. But I shall try to carry on anyway."); error; arith_error:=false; end; @ Addition is not always checked to make sure that it doesn't overflow, but in places where overflow isn't too unlikely the |slow_add| routine is used. @p function slow_add(@!x,@!y:integer):integer; begin if x>=0 then if y<=el_gordo-x then slow_add:=x+y else begin arith_error:=true; slow_add:=el_gordo; end else if -y<=el_gordo+x then slow_add:=x+y else begin arith_error:=true; slow_add:=-el_gordo; end; end; @ Fixed-point arithmetic is done on {\sl scaled integers\/} that are multiples of $2^{-16}$. In other words, a binary point is assumed to be sixteen bit positions from the right end of a binary computer word. @d quarter_unit == @'40000 {$2^{14}$, represents 0.250000} @d half_unit == @'100000 {$2^{15}$, represents 0.50000} @d three_quarter_unit == @'140000 {$3\cdot2^{14}$, represents 0.75000} @d unity == @'200000 {$2^{16}$, represents 1.00000} @d two == @'400000 {$2^{17}$, represents 2.00000} @d three == @'600000 {$2^{17}+2^{16}$, represents 3.00000} @= @!scaled = integer; {this type is used for scaled integers} @!small_number=0..63; {this type is self-explanatory} @ The following function is used to create a scaled integer from a decimal fraction $(.d_0d_1\ldots d_{k-1})$, where |0<=k<=17|. The digit $d_i$ is given in |dig[i]|, and the calculation produces a correctly rounded result. @p function round_decimals(@!k:small_number) : scaled; {converts a decimal fraction} var @!a:integer; {the accumulator} begin a:=0; while k>0 do begin decr(k); a:=(a+dig[k]*two) div 10; end; round_decimals:=half(a+1); end; @ Conversely, here is a procedure analogous to |print_int|. If the output of this procedure is subsequently read by \MF\ and converted by the |round_decimals| routine above, it turns out that the original value will be reproduced exactly. A decimal point is printed only if the value is not an integer. If there is more than one way to print the result with the optimum number of digits following the decimal point, the closest possible value is given. The invariant relation in the \&{repeat} loop is that a sequence of decimal digits yet to be printed will yield the original number if and only if they form a fraction~$f$ in the range $s-\delta\L10\cdot2^{16}f= procedure print_scaled(@!s:scaled); {prints scaled real, rounded to five digits} var @!delta:scaled; {amount of allowable inaccuracy} begin if s<0 then begin print_char("-"); negate(s); {print the sign, if negative} end; print_int(s div unity); {print the integer part} s:=10*(s mod unity)+5; if s<>5 then begin delta:=10; print_char("."); repeat if delta>unity then s:=s+@'100000-(delta div 2); {round the final digit} print_char("0"+(s div unity)); s:=10*(s mod unity); delta:=delta*10; until s<=delta; end; end; @ We often want to print two scaled quantities in parentheses, separated by a comma. @= procedure print_two(@!x,@!y:scaled); {prints `|(x,y)|'} begin print_char("("); print_scaled(x); print_char(","); print_scaled(y); print_char(")"); end; @ The |scaled| quantities in \MF\ programs are generally supposed to be less than $2^{12}$ in absolute value, so \MF\ does much of its internal arithmetic with 28~significant bits of precision. A |fraction| denotes a scaled integer whose binary point is assumed to be 28 bit positions from the right. @d fraction_half==@'1000000000 {$2^{27}$, represents 0.50000000} @d fraction_one==@'2000000000 {$2^{28}$, represents 1.00000000} @d fraction_two==@'4000000000 {$2^{29}$, represents 2.00000000} @d fraction_three==@'6000000000 {$3\cdot2^{28}$, represents 3.00000000} @d fraction_four==@'10000000000 {$2^{30}$, represents 4.00000000} @= @!fraction=integer; {this type is used for scaled fractions} @ In fact, the two sorts of scaling discussed above aren't quite sufficient; \MF\ has yet another, used internally to keep track of angles in units of $2^{-20}$ degrees. @d forty_five_deg==@'264000000 {$45\cdot2^{20}$, represents $45^\circ$} @d ninety_deg==@'550000000 {$90\cdot2^{20}$, represents $90^\circ$} @d one_eighty_deg==@'1320000000 {$180\cdot2^{20}$, represents $180^\circ$} @d three_sixty_deg==@'2640000000 {$360\cdot2^{20}$, represents $360^\circ$} @= @!angle=integer; {this type is used for scaled angles} @ The |make_fraction| routine produces the |fraction| equivalent of |p/q|, given integers |p| and~|q|; it computes the integer $f=\lfloor2^{28}p/q+{1\over2}\rfloor$, when $p$ and $q$ are positive. If |p| and |q| are both of the same scaled type |t|, the ``type relation'' |make_fraction(t,t)=fraction| is valid; and it's also possible to use the subroutine ``backwards,'' using the relation |make_fraction(t,fraction)=t| between scaled types. If the result would have magnitude $2^{31}$ or more, |make_fraction| sets |arith_error:=true|. Most of \MF's internal computations have been designed to avoid this sort of error. If this subroutine were programmed in assembly language on a typical machine, we could simply compute |(@t$2^{28}$@>*p)div q|, since a double-precision product can often be input to a fixed-point division instruction. But when we are restricted to \PASCAL\ arithmetic it is necessary either to resort to multiple-precision maneuvering or to use a simple but slow iteration. The multiple-precision technique would be about three times faster than the code adopted here, but it would be comparatively long and tricky, involving about sixteen additional multiplications and divisions. This operation is part of \MF's ``inner loop''; indeed, it will consume nearly 10\% of the running time (exclusive of input and output) if the code below is left unchanged. A machine-dependent recoding will therefore make \MF\ run faster. The present implementation is highly portable, but slow; it avoids multiplication and division except in the initial stage. System wizards should be careful to replace it with a routine that is guaranteed to produce identical results in all cases. @^system dependencies@> As noted below, a few more routines should also be replaced by machine-dependent code, for efficiency. But when a procedure is not part of the ``inner loop,'' such changes aren't advisable; simplicity and robustness are preferable to trickery, unless the cost is too high. @^inner loop@> @p function make_fraction(@!p,@!q:integer):fraction; var @!f:integer; {the fraction bits, with a leading 1 bit} @!n:integer; {the integer part of $\vert p/q\vert$} @!negative:boolean; {should the result be negated?} @!be_careful:integer; {disables certain compiler optimizations} begin if p>=0 then negative:=false else begin negate(p); negative:=true; end; if q<=0 then begin debug if q=0 then confusion("/");@;@+gubed@;@/ @:this can't happen /}{\quad \./@> negate(q); negative:=not negative; end; n:=p div q; p:=p mod q; if n>=8 then begin arith_error:=true; if negative then make_fraction:=-el_gordo@+else make_fraction:=el_gordo; end else begin n:=(n-1)*fraction_one; @; if negative then make_fraction:=-(f+n)@+else make_fraction:=f+n; end; end; @ The |repeat| loop here preserves the following invariant relations: (i)~|0<=p @= f:=1; repeat be_careful:=p-q; p:=be_careful+p; if p>=0 then f:=f+f+1 else begin double(f); p:=p+q; end; until f>=fraction_one; be_careful:=p-q; if be_careful+p>=0 then incr(f) @ The dual of |make_fraction| is |take_fraction|, which multiplies a given integer~|q| by a fraction~|f|. When the operands are positive, it computes $p=\lfloor qf/2^{28}+{1\over2}\rfloor$, a symmetric function of |q| and~|f|. This routine is even more ``inner loopy'' than |make_fraction|; the present implementation consumes almost 20\% of \MF's computation time during typical jobs, so a machine-language substitute is advisable. @^inner loop@> @^system dependencies@> @p function take_fraction(@!q:integer;@!f:fraction):integer; var @!p:integer; {the fraction so far} @!negative:boolean; {should the result be negated?} @!n:integer; {additional multiple of $q$} @!be_careful:integer; {disables certain compiler optimizations} begin @=0| and |q>0|@>; if f; be_careful:=n-el_gordo; if be_careful+p>0 then begin arith_error:=true; n:=el_gordo-p; end; if negative then take_fraction:=-(n+p) else take_fraction:=n+p; end; @ @=0| and |q>0|@>= if f>=0 then negative:=false else begin negate(f); negative:=true; end; if q<0 then begin negate(q); negative:=not negative; end; @ The invariant relations in this case are (i)~$\lfloor(qf+p)/2^k\rfloor =\lfloor qf_0/2^{28}+{1\over2}\rfloor$, where $k$ is an integer and $f_0$ is the original value of~$f$; (ii)~$2^k\L f<2^{k+1}$. @^inner loop@> @= p:=fraction_half; {that's $2^{27}$; the invariants hold now with $k=28$} if q @p function take_scaled(@!q:integer;@!f:scaled):integer; var @!p:integer; {the fraction so far} @!negative:boolean; {should the result be negated?} @!n:integer; {additional multiple of $q$} @!be_careful:integer; {disables certain compiler optimizations} begin @=0| and |q>0|@>; if f; be_careful:=n-el_gordo; if be_careful+p>0 then begin arith_error:=true; n:=el_gordo-p; end; if negative then take_scaled:=-(n+p) else take_scaled:=n+p; end; @ @= p:=half_unit; {that's $2^{15}$; the invariants hold now with $k=16$} @^inner loop@> if q=0 then negative:=false else begin negate(p); negative:=true; end; if q<=0 then begin debug if q=0 then confusion("/");@+gubed@;@/ @:this can't happen /}{\quad \./@> negate(q); negative:=not negative; end; n:=p div q; p:=p mod q; if n>=@'100000 then begin arith_error:=true; if negative then make_scaled:=-el_gordo@+else make_scaled:=el_gordo; end else begin n:=(n-1)*unity; @; if negative then make_scaled:=-(f+n)@+else make_scaled:=f+n; end; end; @ @= f:=1; repeat be_careful:=p-q; p:=be_careful+p; if p>=0 then f:=f+f+1 else begin double(f); p:=p+q; end; until f>=unity; be_careful:=p-q; if be_careful+p>=0 then incr(f) @ Here is a typical example of how the routines above can be used. It computes the function $${1\over3\tau}f(\theta,\phi)= {\tau^{-1}\bigl(2+\sqrt2\,(\sin\theta-{1\over16}\sin\phi) (\sin\phi-{1\over16}\sin\theta)(\cos\theta-\cos\phi)\bigr)\over 3\,\bigl(1+{1\over2}(\sqrt5-1)\cos\theta+{1\over2}(3-\sqrt5\,)\cos\phi\bigr)},$$ where $\tau$ is a |scaled| ``tension'' parameter. This is \MF's magic fudge factor for placing the first control point of a curve that starts at an angle $\theta$ and ends at an angle $\phi$ from the straight path. (Actually, if the stated quantity exceeds 4, \MF\ reduces it to~4.) The trigonometric quantity to be multiplied by $\sqrt2$ is less than $\sqrt2$. (It's a sum of eight terms whose absolute values can be bounded using relations such as $\sin\theta\cos\theta\L{1\over2}$.) Thus the numerator is positive; and since the tension $\tau$ is constrained to be at least $3\over4$, the numerator is less than $16\over3$. The denominator is nonnegative and at most~6. Hence the fixed-point calculations below are guaranteed to stay within the bounds of a 32-bit computer word. The angles $\theta$ and $\phi$ are given implicitly in terms of |fraction| arguments |st|, |ct|, |sf|, and |cf|, representing $\sin\theta$, $\cos\theta$, $\sin\phi$, and $\cos\phi$, respectively. @p function velocity(@!st,@!ct,@!sf,@!cf:fraction;@!t:scaled):fraction; var @!acc,@!num,@!denom:integer; {registers for intermediate calculations} begin acc:=take_fraction(st-(sf div 16), sf-(st div 16)); acc:=take_fraction(acc,ct-cf); num:=fraction_two+take_fraction(acc,379625062); {$2^{28}\sqrt2\approx379625062.497$} denom:=fraction_three+take_fraction(ct,497706707)+take_fraction(cf,307599661); {$3\cdot2^{27}\cdot(\sqrt5-1)\approx497706706.78$ and $3\cdot2^{27}\cdot(3-\sqrt5\,)\approx307599661.22$} if t<>unity then num:=make_scaled(num,t); {|make_scaled(fraction,scaled)=fraction|} if num div 4>=denom then velocity:=fraction_four else velocity:=make_fraction(num,denom); end; @ The following somewhat different subroutine tests rigorously if $ab$ is greater than, equal to, or less than~$cd$, given integers $(a,b,c,d)$. In most cases a quick decision is reached. The result is $+1$, 0, or~$-1$ in the three respective cases. @d return_sign(#)==begin ab_vs_cd:=#; return; end @p function ab_vs_cd(@!a,b,c,d:integer):integer; label exit; var @!q,@!r:integer; {temporary registers} begin @=0|, |b,d>0|@>; loop@+ begin q := a div d; r := c div b; if q<>r then if q>r then return_sign(1)@+else return_sign(-1); q := a mod d; r := c mod b; if r=0 then if q=0 then return_sign(0)@+else return_sign(1); if q=0 then return_sign(-1); a:=b; b:=q; c:=d; d:=r; end; {now |a>d>0| and |c>b>0|} exit:end; @ @= if a<0 then begin negate(a); negate(b); end; if c<0 then begin negate(c); negate(d); end; if d<=0 then begin if b>=0 then if ((a=0)or(b=0))and((c=0)or(d=0)) then return_sign(0) else return_sign(1); if d=0 then if a=0 then return_sign(0)@+else return_sign(-1); q:=a; a:=c; c:=q; q:=-b; b:=-d; d:=q; end else if b<=0 then begin if b<0 then if a>0 then return_sign(-1); if c=0 then return_sign(0) else return_sign(-1); end @ We conclude this set of elementary routines with some simple rounding and truncation operations that are coded in a machine-independent fashion. The routines are slightly complicated because we want them to work without overflow whenever $-2^{31}\L x<2^{31}$. @p function floor_scaled(@!x:scaled):scaled; {$2^{16}\lfloor x/2^{16}\rfloor$} var @!be_careful:integer; {temporary register} begin if x>=0 then floor_scaled:=x-(x mod unity) else begin be_careful:=x+1; floor_scaled:=x+((-be_careful) mod unity)+1-unity; end; end; @# function floor_unscaled(@!x:scaled):integer; {$\lfloor x/2^{16}\rfloor$} var @!be_careful:integer; {temporary register} begin if x>=0 then floor_unscaled:=x div unity else begin be_careful:=x+1; floor_unscaled:=-(1+((-be_careful) div unity)); end; end; @# function round_unscaled(@!x:scaled):integer; {$\lfloor x/2^{16}+.5\rfloor$} var @!be_careful:integer; {temporary register} begin if x>=half_unit then round_unscaled:=1+((x-half_unit) div unity) else if x>=-half_unit then round_unscaled:=0 else begin be_careful:=x+1; round_unscaled:=-(1+((-be_careful-half_unit) div unity)); end; end; @# function round_fraction(@!x:fraction):scaled; {$\lfloor x/2^{12}+.5\rfloor$} var @!be_careful:integer; {temporary register} begin if x>=2048 then round_fraction:=1+((x-2048) div 4096) else if x>=-2048 then round_fraction:=0 else begin be_careful:=x+1; round_fraction:=-(1+((-be_careful-2048) div 4096)); end; end; @* \[8] Algebraic and transcendental functions. \MF\ computes all of the necessary special functions from scratch, without relying on |real| arithmetic or system subroutines for sines, cosines, etc. @ To get the square root of a |scaled| number |x|, we want to calculate $s=\lfloor 2^8\!\sqrt x +{1\over2}\rfloor$. If $x>0$, this is the unique integer such that $2^{16}x-s\L s^2<2^{16}x+s$. The following subroutine determines $s$ by an iterative method that maintains the invariant relations $x=2^{46-2k}x_0\bmod 2^{30}$, $0 else begin k:=23; q:=2; while x|\unskip} begin decr(k); x:=x+x+x+x; end; if x; until k=0; square_rt:=half(q); end; end; @ @= begin if x<0 then begin print_err("Square root of "); @.Square root...replaced by 0@> print_scaled(x); print(" has been replaced by 0"); help2("Since I don't take square roots of negative numbers,")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); error; end; square_rt:=0; end @ @= double(x); double(y); if x>=fraction_four then {note that |fraction_four=@t$2^{30}$@>|} begin x:=x-fraction_four; incr(y); end; double(x); y:=y+y-q; double(q); if x>=fraction_four then begin x:=x-fraction_four; incr(y); end; if y>q then begin y:=y-q; q:=q+2; end else if y<=0 then begin q:=q-2; y:=y+q; end; decr(k) @ Pythagorean addition $\psqrt{a^2+b^2}$ is implemented by an elegant iterative scheme due to Cleve Moler and Donald Morrison [{\sl IBM Journal @^Moler, Cleve Barry@> @^Morrison, Donald Ross@> of Research and Development\/ \bf27} (1983), 577--581]. It modifies |a| and~|b| in such a way that their pythagorean sum remains invariant, while the smaller argument decreases. @p function pyth_add(@!a,@!b:integer):integer; label done; var @!r:fraction; {register used to transform |a| and |b|} @!big:boolean; {is the result dangerously near $2^{31}$?} begin a:=abs(a); b:=abs(b); if a0 then begin if a; if big then if a= loop@+ begin r:=make_fraction(b,a); r:=take_fraction(r,r); {now $r\approx b^2/a^2$} if r=0 then goto done; r:=make_fraction(r,fraction_four+r); a:=a+take_fraction(a+a,r); b:=take_fraction(b,r); end; done: @ Here is a similar algorithm for $\psqrt{a^2-b^2}$. It converges slowly when $b$ is near $a$, but otherwise it works fine. @p function pyth_sub(@!a,@!b:integer):integer; label done; var @!r:fraction; {register used to transform |a| and |b|} @!big:boolean; {is the input dangerously near $2^{31}$?} begin a:=abs(a); b:=abs(b); if a<=b then @ else begin if a; if big then a:=a+a; end; pyth_sub:=a; end; @ @= loop@+ begin r:=make_fraction(b,a); r:=take_fraction(r,r); {now $r\approx b^2/a^2$} if r=0 then goto done; r:=make_fraction(r,fraction_four-r); a:=a-take_fraction(a+a,r); b:=take_fraction(b,r); end; done: @ @= begin if a help2("Since I don't take square roots of negative numbers,")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); error; end; a:=0; end @ The subroutines for logarithm and exponential involve two tables. The first is simple: |two_to_the[k]| equals $2^k$. The second involves a bit more calculation, which the author claims to have done correctly: |spec_log[k]| is $2^{27}$ times $\ln\bigl(1/(1-2^{-k})\bigr)= 2^{-k}+{1\over2}2^{-2k}+{1\over3}2^{-3k}+\cdots\,$, rounded to the nearest integer. @= @!two_to_the:array[0..30] of integer; {powers of two} @!spec_log:array[1..28] of integer; {special logarithms} @ @= @!k:integer; {all-purpose loop index} @ @= two_to_the[0]:=1; for k:=1 to 30 do two_to_the[k]:=2*two_to_the[k-1]; spec_log[1]:=93032640; spec_log[2]:=38612034; spec_log[3]:=17922280; spec_log[4]:=8662214; spec_log[5]:=4261238; spec_log[6]:=2113709; spec_log[7]:=1052693; spec_log[8]:=525315; spec_log[9]:=262400; spec_log[10]:=131136; spec_log[11]:=65552; spec_log[12]:=32772; spec_log[13]:=16385; for k:=14 to 27 do spec_log[k]:=two_to_the[27-k]; spec_log[28]:=1; @ Here is the routine that calculates $2^8$ times the natural logarithm of a |scaled| quantity; it is an integer approximation to $2^{24}\ln(x/2^{16})$, when |x| is a given positive integer. The method is based on exercise 1.2.2--25 in {\sl The Art of Computer Programming\/}: During the main iteration we have $1\L 2^{-30}x<1/(1-2^{1-k})$, and the logarithm of $2^{30}x$ remains to be added to an accumulator register called~$y$. Three auxiliary bits of accuracy are retained in~$y$ during the calculation, and sixteen auxiliary bits to extend |y| are kept in~|z| during the initial argument reduction. (We add $100\cdot2^{16}=6553600$ to~|z| and subtract 100 from~|y| so that |z| will not become negative; also, the actual amount subtracted from~|y| is~96, not~100, because we want to add~4 for rounding before the final division by~8.) @p function m_log(@!x:scaled):scaled; var @!y,@!z:integer; {auxiliary registers} @!k:integer; {iteration counter} begin if x<=0 then @ else begin y:=1302456956+4-100; {$14\times2^{27}\ln2\approx1302456956.421063$} z:=27595+6553600; {and $2^{16}\times .421063\approx 27595$} while xfraction_four+4 do @; m_log:=y div 8; end; end; @ @= begin z:=((x-1) div two_to_the[k])+1; {$z=\lceil x/2^k\rceil$} while x= begin print_err("Logarithm of "); @.Logarithm...replaced by 0@> print_scaled(x); print(" has been replaced by 0"); help2("Since I don't take logs of non-positive numbers,")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); error; m_log:=0; end @ Conversely, the exponential routine calculates $\exp(x/2^8)$, when |x| is |scaled|. The result is an integer approximation to $2^{16}\exp(x/2^{24})$, when |x| is regarded as an integer. @p function m_exp(@!x:scaled):scaled; var @!k:small_number; {loop control index} @!y,@!z:integer; {auxiliary registers} begin if x>174436200 then {$2^{24}\ln((2^{31}-1)/2^{16})\approx 174436199.51$} begin arith_error:=true; m_exp:=el_gordo; end else if x<-197694359 then m_exp:=0 {$2^{24}\ln(2^{-1}/2^{16})\approx-197694359.45$} else begin if x<=0 then begin z:=-8*x; y:=@'4000000; {$y=2^{20}$} end else begin if x<=127919879 then z:=1023359037-8*x {$2^{27}\ln((2^{31}-1)/2^{20})\approx 1023359037.125$} else z:=8*(174436200-x); {|z| is always nonnegative} y:=el_gordo; end; @; if x<=127919879 then m_exp:=(y+8) div 16@+else m_exp:=y; end; end; @ The idea here is that subtracting |spec_log[k]| from |z| corresponds to multiplying |y| by $1-2^{-k}$. A subtle point (which had to be checked) was that if $x=127919879$, the value of~|y| will decrease so that |y+8| doesn't overflow. In fact, $z$ will be 5 in this case, and |y| will decrease by~64 when |k=25| and by~16 when |k=27|. @= k:=1; while z>0 do begin while z>=spec_log[k] do begin z:=z-spec_log[k]; y:=y-1-((y-two_to_the[k-1]) div two_to_the[k]); end; incr(k); end @ The trigonometric subroutines use an auxiliary table such that |spec_atan[k]| contains an approximation to the |angle| whose tangent is~$1/2^k$. @= @!spec_atan:array[1..26] of angle; {$\arctan2^{-k}$ times $2^{20}\cdot180/\pi$} @ @= spec_atan[1]:=27855475; spec_atan[2]:=14718068; spec_atan[3]:=7471121; spec_atan[4]:=3750058; spec_atan[5]:=1876857; spec_atan[6]:=938658; spec_atan[7]:=469357; spec_atan[8]:=234682; spec_atan[9]:=117342; spec_atan[10]:=58671; spec_atan[11]:=29335; spec_atan[12]:=14668; spec_atan[13]:=7334; spec_atan[14]:=3667; spec_atan[15]:=1833; spec_atan[16]:=917; spec_atan[17]:=458; spec_atan[18]:=229; spec_atan[19]:=115; spec_atan[20]:=57; spec_atan[21]:=29; spec_atan[22]:=14; spec_atan[23]:=7; spec_atan[24]:=4; spec_atan[25]:=2; spec_atan[26]:=1; @ Given integers |x| and |y|, not both zero, the |n_arg| function returns the |angle| whose tangent points in the direction $(x,y)$. This subroutine first determines the correct octant, then solves the problem for |0<=y<=x|, then converts the result appropriately to return an answer in the range |-one_eighty_deg<=@t$\theta$@><=one_eighty_deg|. (The answer is |+one_eighty_deg| if |y=0| and |x<0|, but an answer of |-one_eighty_deg| is possible if, for example, |y=-1| and $x=-2^{30}$.) The octants are represented in a ``Gray code,'' since that turns out to be computationally simplest. @d negate_x=1 @d negate_y=2 @d switch_x_and_y=4 @d first_octant=1 @d second_octant=first_octant+switch_x_and_y @d third_octant=first_octant+switch_x_and_y+negate_x @d fourth_octant=first_octant+negate_x @d fifth_octant=first_octant+negate_x+negate_y @d sixth_octant=first_octant+switch_x_and_y+negate_x+negate_y @d seventh_octant=first_octant+switch_x_and_y+negate_y @d eighth_octant=first_octant+negate_y @p function n_arg(@!x,@!y:integer):angle; var @!z:angle; {auxiliary register} @!t:integer; {temporary storage} @!k:small_number; {loop counter} @!octant:first_octant..sixth_octant; {octant code} begin if x>=0 then octant:=first_octant else begin negate(x); octant:=first_octant+negate_x; end; if y<0 then begin negate(y); octant:=octant+negate_y; end; if x else begin @; @; end; end; @ @= begin print_err("angle(0,0) is taken as zero"); @.angle(0,0)...zero@> help2("The `angle' between two identical points is undefined.")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); error; n_arg:=0; end @ @= case octant of first_octant:n_arg:=z; second_octant:n_arg:=ninety_deg-z; third_octant:n_arg:=ninety_deg+z; fourth_octant:n_arg:=one_eighty_deg-z; fifth_octant:n_arg:=z-one_eighty_deg; sixth_octant:n_arg:=-z-ninety_deg; seventh_octant:n_arg:=z-ninety_deg; eighth_octant:n_arg:=-z; end {there are no other cases} @ At this point we have |x>=y>=0|, and |x>0|. The numbers are scaled up or down until $2^{28}\L x<2^{29}$, so that accurate fixed-point calculations will be made. @= while x>=fraction_two do begin x:=half(x); y:=half(y); end; z:=0; if y>0 then begin while x; end @ During the calculations of this section, variables |x| and~|y| represent actual coordinates $(x,2^{-k}y)$. We will maintain the condition |x>=y|, so that the tangent will be at most $2^{-k}$. If $x<2y$, the tangent is greater than $2^{-k-1}$. The transformation $(a,b)\mapsto(a+b\tan\phi,b-a\tan\phi)$ replaces $(a,b)$ by coordinates whose angle has decreased by~$\phi$; in the special case $a=x$, $b=2^{-k}y$, and $\tan\phi=2^{-k-1}$, this operation reduces to the particularly simple iteration shown here. [Cf.~John E. Meggitt, @^Meggitt, John E.@> {\sl IBM Journal of Research and Development\/ \bf6} (1962), 210--226.] The initial value of |x| will be multiplied by at most $(1+{1\over2})(1+{1\over8})(1+{1\over32})\cdots\approx 1.7584$; hence there is no chance of integer overflow. @= k:=0; repeat double(y); incr(k); if y>x then begin z:=z+spec_atan[k]; t:=x; x:=x+(y div two_to_the[k+k]); y:=y-t; end; until k=15; repeat double(y); incr(k); if y>x then begin z:=z+spec_atan[k]; y:=y-x; end; until k=26 @ Conversely, the |n_sin_cos| routine takes an |angle| and produces the sine and cosine of that angle. The results of this routine are stored in global integer variables |n_sin| and |n_cos|. @= @!n_sin,@!n_cos:fraction; {results computed by |n_sin_cos|} @ Given an integer |z| that is $2^{20}$ times an angle $\theta$ in degrees, the purpose of |n_sin_cos(z)| is to set |x=@t$r\cos\theta$@>| and |y=@t$r\sin\theta$@>| (approximately), for some rather large number~|r|. The maximum of |x| and |y| will be between $2^{28}$ and $2^{30}$, so that there will be hardly any loss of accuracy. Then |x| and~|y| are divided by~|r|. @p procedure n_sin_cos(@!z:angle); {computes a multiple of the sine and cosine} var @!k:small_number; {loop control variable} @!q:0..7; {specifies the quadrant} @!r:fraction; {magnitude of |(x,y)|} @!x,@!y,@!t:integer; {temporary registers} begin while z<0 do z:=z+three_sixty_deg; z:=z mod three_sixty_deg; {now |0<=z; @; r:=pyth_add(x,y); n_cos:=make_fraction(x,r); n_sin:=make_fraction(y,r); end; @ In this case the octants are numbered sequentially. @= case q of 0:do_nothing; 1:begin t:=x; x:=y; y:=t; end; 2:begin t:=x; x:=-y; y:=t; end; 3:negate(x); 4:begin negate(x); negate(y); end; 5:begin t:=x; x:=-y; y:=-t; end; 6:begin t:=x; x:=y; y:=-t; end; 7:negate(y); end {there are no other cases} @ The main iteration of |n_sin_cos| is similar to that of |n_arg| but applied in reverse. The values of |spec_atan[k]| decrease slowly enough that this loop is guaranteed to terminate before the (nonexistent) value |spec_atan[27]| would be required. @= k:=1; while z>0 do begin if z>=spec_atan[k] then begin z:=z-spec_atan[k]; t:=x;@/ x:=t+y div two_to_the[k]; y:=y-t div two_to_the[k]; end; incr(k); end; if y<0 then y:=0 {this precaution may never be needed} @ And now let's complete our collection of numeric utility routines by considering random number generation. \MF\ generates pseudo-random numbers with the additive scheme recommended in Section 3.6 of {\sl The Art of Computer Programming}; however, the results are random fractions between 0 and |fraction_one-1|, inclusive. There's an auxiliary array |randoms| that contains 55 pseudo-random fractions. Using the recurrence $x_n=(x_{n-55}-x_{n-31})\bmod 2^{28}$, we generate batches of 55 new $x_n$'s at a time by calling |new_randoms|. The global variable |j_random| tells which element has most recently been consumed. @= @!randoms:array[0..54] of fraction; {the last 55 random values generated} @!j_random:0..54; {the number of unused |randoms|} @ To consume a random fraction, the program below will say `|next_random|' and then it will fetch |randoms[j_random]|. @d next_random==if j_random=0 then new_randoms else decr(j_random) @p procedure new_randoms; var @!k:0..54; {index into |randoms|} @!x:fraction; {accumulator} begin for k:=0 to 23 do begin x:=randoms[k]-randoms[k+31]; if x<0 then x:=x+fraction_one; randoms[k]:=x; end; for k:=24 to 54 do begin x:=randoms[k]-randoms[k-24]; if x<0 then x:=x+fraction_one; randoms[k]:=x; end; j_random:=54; end; @ To initialize the |randoms| table, we call the following routine. @p procedure init_randoms(@!seed:scaled); var @!j,@!jj,@!k:fraction; {more or less random integers} @!i:0..54; {index into |randoms|} begin j:=abs(seed); while j>=fraction_one do j:=half(j); k:=1; for i:=0 to 54 do begin jj:=k; k:=j-k; j:=jj; if k<0 then k:=k+fraction_one; randoms[(i*21)mod 55]:=j; end; new_randoms; new_randoms; new_randoms; {``warm up'' the array} end; @ To produce a uniform random number in the range |0<=u=u>x| or |0=u=x|, given a |scaled| value~|x|, we proceed as shown here. Note that the call of |take_fraction| will produce the values 0 and~|x| with about half the probability that it will produce any other particular values between 0 and~|x|, because it rounds its answers. @p function unif_rand(@!x:scaled):scaled; var @!y:scaled; {trial value} begin next_random; y:=take_fraction(abs(x),randoms[j_random]); if y=abs(x) then unif_rand:=0 else if x>0 then unif_rand:=y else unif_rand:=-y; end; @ Finally, a normal deviate with mean zero and unit standard deviation can readily be obtained with the ratio method (Algorithm 3.4.1R in {\sl The Art of Computer Programming\/}). @p function norm_rand:scaled; var @!x,@!u,@!l:integer; {what the book would call $2^{16}X$, $2^{28}U$, and $-2^{24}\ln U$} begin repeat repeat next_random; x:=take_fraction(112429,randoms[j_random]-fraction_half); {$2^{16}\sqrt{8/e}\approx 112428.82793$} next_random; u:=randoms[j_random]; until abs(x)=0; norm_rand:=x; end; @* \[9] Packed data. In order to make efficient use of storage space, \MF\ bases its major data structures on a |memory_word|, which contains either a (signed) integer, possibly scaled, or a small number of fields that are one half or one quarter of the size used for storing integers. If |x| is a variable of type |memory_word|, it contains up to four fields that can be referred to as follows: $$\vbox{\halign{\hfil#&#\hfil&#\hfil\cr |x|&.|int|&(an |integer|)\cr |x|&.|sc|\qquad&(a |scaled| integer)\cr |x.hh.lh|, |x.hh|&.|rh|&(two halfword fields)\cr |x.hh.b0|, |x.hh.b1|, |x.hh|&.|rh|&(two quarterword fields, one halfword field)\cr |x.qqqq.b0|, |x.qqqq.b1|, |x.qqqq|&.|b2|, |x.qqqq.b3|\hskip-100pt &\qquad\qquad\qquad(four quarterword fields)\cr}}$$ This is somewhat cumbersome to write, and not very readable either, but macros will be used to make the notation shorter and more transparent. The \PASCAL\ code below gives a formal definition of |memory_word| and its subsidiary types, using packed variant records. \MF\ makes no assumptions about the relative positions of the fields within a word. Since we are assuming 32-bit integers, a halfword must contain at least 16 bits, and a quarterword must contain at least 8 bits. @^system dependencies@> But it doesn't hurt to have more bits; for example, with enough 36-bit words you might be able to have |mem_max| as large as 262142. N.B.: Valuable memory space will be dreadfully wasted unless \MF\ is compiled by a \PASCAL\ that packs all of the |memory_word| variants into the space of a single integer. Some \PASCAL\ compilers will pack an integer whose subrange is `|0..255|' into an eight-bit field, but others insist on allocating space for an additional sign bit; on such systems you can get 256 values into a quarterword only if the subrange is `|-128..127|'. The present implementation tries to accommodate as many variations as possible, so it makes rather general assumptions. If integers having the subrange `|min_quarterword..max_quarterword|' can be packed into a quarterword, and if integers having the subrange `|min_halfword..max_halfword|' can be packed into a halfword, everything should work satisfactorily. It is usually most efficient to have |min_quarterword=min_halfword=0|, so one should try to achieve this unless it causes a severe problem. The values defined here are recommended for most 32-bit computers. @d min_quarterword=0 {smallest allowable value in a |quarterword|} @d max_quarterword=255 {largest allowable value in a |quarterword|} @d min_halfword==0 {smallest allowable value in a |halfword|} @d max_halfword==65535 {largest allowable value in a |halfword|} @ Here are the inequalities that the quarterword and halfword values must satisfy (or rather, the inequalities that they mustn't satisfy): @= init if mem_max<>mem_top then bad:=10;@+tini@;@/ if mem_max0)or(max_quarterword<127) then bad:=11; if (min_halfword>0)or(max_halfword<32767) then bad:=12; if (min_quarterwordmax_halfword) then bad:=13; if (mem_min=max_halfword) then bad:=14; if max_strings>max_halfword then bad:=15; if buf_size>max_halfword then bad:=16; if (max_quarterword-min_quarterword<255)or@| (max_halfword-min_halfword<65535) then bad:=17; @ The operation of subtracting |min_halfword| occurs rather frequently in \MF, so it is convenient to abbreviate this operation by using the macro |ho| defined here. \MF\ will run faster with respect to compilers that don't optimize the expression `|x-0|', if this macro is simplified in the obvious way when |min_halfword=0|. Similarly, |qi| and |qo| are used for input to and output from quarterwords. @^system dependencies@> @d ho(#)==#-min_halfword {to take a sixteen-bit item from a halfword} @d qo(#)==#-min_quarterword {to read eight bits from a quarterword} @d qi(#)==#+min_quarterword {to store eight bits in a quarterword} @ The reader should study the following definitions closely: @^system dependencies@> @d sc==int {|scaled| data is equivalent to |integer|} @= @!quarterword = min_quarterword..max_quarterword; {1/4 of a word} @!halfword=min_halfword..max_halfword; {1/2 of a word} @!two_choices = 1..2; {used when there are two variants in a record} @!three_choices = 1..3; {used when there are three variants in a record} @!two_halves = packed record@;@/ @!rh:halfword; case two_choices of 1: (@!lh:halfword); 2: (@!b0:quarterword; @!b1:quarterword); end; @!four_quarters = packed record@;@/ @!b0:quarterword; @!b1:quarterword; @!b2:quarterword; @!b3:quarterword; end; @!memory_word = record@;@/ case three_choices of 1: (@!int:integer); 2: (@!hh:two_halves); 3: (@!qqqq:four_quarters); end; @!word_file = file of memory_word; @ When debugging, we may want to print a |memory_word| without knowing what type it is; so we print it in all modes. @^dirty \PASCAL@>@^debugging@> @p @!debug procedure print_word(@!w:memory_word); {prints |w| in all ways} begin print_int(w.int); print_char(" ");@/ print_scaled(w.sc); print_char(" "); print_scaled(w.sc div @'10000); print_ln;@/ print_int(w.hh.lh); print_char("="); print_int(w.hh.b0); print_char(":"); print_int(w.hh.b1); print_char(";"); print_int(w.hh.rh); print_char(" ");@/ print_int(w.qqqq.b0); print_char(":"); print_int(w.qqqq.b1); print_char(":"); print_int(w.qqqq.b2); print_char(":"); print_int(w.qqqq.b3); end; gubed @* \[10] Dynamic memory allocation. The \MF\ system does nearly all of its own memory allocation, so that it can readily be transported into environments that do not have automatic facilities for strings, garbage collection, etc., and so that it can be in control of what error messages the user receives. The dynamic storage requirements of \MF\ are handled by providing a large array |mem| in which consecutive blocks of words are used as nodes by the \MF\ routines. Pointer variables are indices into this array, or into another array called |eqtb| that will be explained later. A pointer variable might also be a special flag that lies outside the bounds of |mem|, so we allow pointers to assume any |halfword| value. The minimum halfword value represents a null pointer. @d pointer==halfword {a flag or a location in |mem| or |eqtb|} @d null==mem_min {the null pointer} @ The |mem| array is divided into two regions that are allocated separately, but the dividing line between these two regions is not fixed; they grow together until finding their ``natural'' size in a particular job. Locations less than or equal to |lo_mem_max| are used for storing variable-length records consisting of two or more words each. This region is maintained using an algorithm similar to the one described in exercise 2.5--19 of {\sl The Art of Computer Programming}. However, no size field appears in the allocated nodes; the program is responsible for knowing the relevant size when a node is freed. Locations greater than or equal to |hi_mem_min| are used for storing one-word records; a conventional \.{AVAIL} stack is used for allocation in this region. Locations of |mem| between |mem_min| and |mem_top| may be dumped as part of preloaded format files, by the \.{INIMF} preprocessor. @.INIMF@> Production versions of \MF\ may extend the memory at the top end in order to provide more space; these locations, between |mem_top| and |mem_max|, are always used for single-word nodes. The key pointers that govern |mem| allocation have a prescribed order: $$\hbox{|null<=mem_min= @!mem : array[mem_min..mem_max] of memory_word; {the big dynamic storage area} @!lo_mem_max : pointer; {the largest location of variable-size memory in use} @!hi_mem_min : pointer; {the smallest location of one-word memory in use} @ In order to study the memory requirements of particular applications, it is possible to prepare a version of \MF\ that keeps track of current and maximum memory usage. When code between the delimiters |@!stat| $\ldots$ |tats| is not ``commented out,'' \MF\ will run a bit slower but it will report these statistics when |tracing_stats| is positive. @= @!var_used, @!dyn_used : integer; {how much memory is in use} @ Let's consider the one-word memory region first, since it's the simplest. The pointer variable |mem_end| holds the highest-numbered location of |mem| that has ever been used. The free locations of |mem| that occur between |hi_mem_min| and |mem_end|, inclusive, are of type |two_halves|, and we write |info(p)| and |link(p)| for the |lh| and |rh| fields of |mem[p]| when it is of this type. The single-word free locations form a linked list $$|avail|,\;\hbox{|link(avail)|},\;\hbox{|link(link(avail))|},\;\ldots$$ terminated by |null|. @d link(#) == mem[#].hh.rh {the |link| field of a memory word} @d info(#) == mem[#].hh.lh {the |info| field of a memory word} @= @!avail : pointer; {head of the list of available one-word nodes} @!mem_end : pointer; {the last one-word node used in |mem|} @ If one-word memory is exhausted, it might mean that the user has forgotten a token like `\&{enddef}' or `\&{endfor}'. We will define some procedures later that try to help pinpoint the trouble. @p @t\4@>@@; @t\4@>@ @ The function |get_avail| returns a pointer to a new one-word node whose |link| field is null. However, \MF\ will halt if there is no more room left. @^inner loop@> @p function get_avail : pointer; {single-word node allocation} var @!p:pointer; {the new node being got} begin p:=avail; {get top location in the |avail| stack} if p<>null then avail:=link(avail) {and pop it off} else if mem_end end; end; link(p):=null; {provide an oft-desired initialization of the new node} @!stat incr(dyn_used);@+tats@;{maintain statistics} get_avail:=p; end; @ Conversely, a one-word node is recycled by calling |free_avail|. @d free_avail(#)== {single-word node liberation} begin link(#):=avail; avail:=#; @!stat decr(dyn_used);@+tats@/ end @ There's also a |fast_get_avail| routine, which saves the procedure-call overhead at the expense of extra programming. This macro is used in the places that would otherwise account for the most calls of |get_avail|. @^inner loop@> @d fast_get_avail(#)==@t@>@;@/ begin #:=avail; {avoid |get_avail| if possible, to save time} if #=null then #:=get_avail else begin avail:=link(#); link(#):=null; @!stat incr(dyn_used);@+tats@/ end; end @ The available-space list that keeps track of the variable-size portion of |mem| is a nonempty, doubly-linked circular list of empty nodes, pointed to by the roving pointer |rover|. Each empty node has size 2 or more; the first word contains the special value |max_halfword| in its |link| field and the size in its |info| field; the second word contains the two pointers for double linking. Each nonempty node also has size 2 or more. Its first word is of type |two_halves|\kern-1pt, and its |link| field is never equal to |max_halfword|. Otherwise there is complete flexibility with respect to the contents of its other fields and its other words. (We require |mem_max= @!rover : pointer; {points to some node in the list of empties} @ A call to |get_node| with argument |s| returns a pointer to a new node of size~|s|, which must be 2~or more. The |link| field of the first word of this new node is set to null. An overflow stop occurs if no suitable space exists. If |get_node| is called with $s=2^{30}$, it simply merges adjacent free areas and returns the value |max_halfword|. @p function get_node(@!s:integer):pointer; {variable-size node allocation} label found,exit,restart; var @!p:pointer; {the node currently under inspection} @!q:pointer; {the node physically after node |p|} @!r:integer; {the newly allocated node, or a candidate for this honor} @!t,@!tt:integer; {temporary registers} @^inner loop@> begin restart: p:=rover; {start at some free node in the ring} repeat @; p:=rlink(p); {move to the next node in the ring} until p=rover; {repeat until the whole list has been traversed} if s=@'10000000000 then begin get_node:=max_halfword; return; end; if lo_mem_max+2; overflow("main memory size",mem_max+1-mem_min); {sorry, nothing satisfactory is left} @:METAFONT capacity exceeded main memory size}{\quad main memory size@> found: link(r):=null; {this node is now nonempty} @!stat var_used:=var_used+s; {maintain usage statistics} tats@;@/ get_node:=r; exit:end; @ The lower part of |mem| grows by 1000 words at a time, unless we are very close to going under. When it grows, we simply link a new node into the available-space list. This method of controlled growth helps to keep the |mem| usage consecutive when \MF\ is implemented on ``virtual memory'' systems. @^virtual memory@> @= begin if hi_mem_min-lo_mem_max>=1998 then t:=lo_mem_max+1000 else t:=lo_mem_max+1+(hi_mem_min-lo_mem_max) div 2; {|lo_mem_max+2<=tmem_min+max_halfword then t:=mem_min+max_halfword; p:=llink(rover); q:=lo_mem_max; rlink(p):=q; llink(rover):=q;@/ rlink(q):=rover; llink(q):=p; link(q):=empty_flag; node_size(q):=t-lo_mem_max;@/ lo_mem_max:=t; link(lo_mem_max):=null; info(lo_mem_max):=null; rover:=q; goto restart; end @ @= q:=p+node_size(p); {find the physical successor} while is_empty(q) do {merge node |p| with node |q|} begin t:=rlink(q); tt:=llink(q); @^inner loop@> if q=rover then rover:=t; llink(t):=tt; rlink(tt):=t;@/ q:=q+node_size(q); end; r:=q-s; if r>p+1 then @; if r=p then if rlink(p)<>p then @; node_size(p):=q-p {reset the size in case it grew} @ @= begin node_size(p):=r-p; {store the remaining size} rover:=p; {start searching here next time} goto found; end @ Here we delete node |p| from the ring, and let |rover| rove around. @= begin rover:=rlink(p); t:=llink(p); llink(rover):=t; rlink(t):=rover; goto found; end @ Conversely, when some variable-size node |p| of size |s| is no longer needed, the operation |free_node(p,s)| will make its words available, by inserting |p| as a new empty node just before where |rover| now points. @p procedure free_node(@!p:pointer; @!s:halfword); {variable-size node liberation} var @!q:pointer; {|llink(rover)|} begin node_size(p):=s; link(p):=empty_flag; @^inner loop@> q:=llink(rover); llink(p):=q; rlink(p):=rover; {set both links} llink(rover):=p; rlink(q):=p; {insert |p| into the ring} @!stat var_used:=var_used-s;@+tats@;{maintain statistics} end; @ Just before \.{INIMF} writes out the memory, it sorts the doubly linked available space list. The list is probably very short at such times, so a simple insertion sort is used. The smallest available location will be pointed to by |rover|, the next-smallest by |rlink(rover)|, etc. @p @!init procedure sort_avail; {sorts the available variable-size nodes by location} var @!p,@!q,@!r: pointer; {indices into |mem|} @!old_rover:pointer; {initial |rover| setting} begin p:=get_node(@'10000000000); {merge adjacent free areas} p:=rlink(rover); rlink(rover):=max_halfword; old_rover:=rover; while p<>old_rover do @; p:=rover; while rlink(p)<>max_halfword do begin llink(rlink(p)):=p; p:=rlink(p); end; rlink(p):=rover; llink(rover):=p; end; tini @ The following |while| loop terminates, since the list that starts at |rover| ends with |max_halfword| during the sorting procedure. @= if p= @^data structure assumptions@> rover:=lo_mem_stat_max+1; {initialize the dynamic memory} link(rover):=empty_flag; node_size(rover):=1000; {which is a 1000-word available node} llink(rover):=rover; rlink(rover):=rover;@/ lo_mem_max:=rover+1000; link(lo_mem_max):=null; info(lo_mem_max):=null;@/ for k:=hi_mem_stat_min to mem_top do mem[k]:=mem[lo_mem_max]; {clear list heads} avail:=null; mem_end:=mem_top; hi_mem_min:=hi_mem_stat_min; {initialize the one-word memory} var_used:=lo_mem_stat_max+1-mem_min; dyn_used:=mem_top+1-hi_mem_stat_min; {initialize statistics} @ The procedure |flush_list(p)| frees an entire linked list of one-word nodes that starts at a given position, until coming to |sentinel| or a pointer that is not in the one-word region. Another procedure, |flush_node_list|, frees an entire linked list of one-word and two-word nodes, until coming to a |null| pointer. @^inner loop@> @p procedure flush_list(@!p:pointer); {makes list of single-word nodes available} label done; var @!q,@!r:pointer; {list traversers} begin if p>=hi_mem_min then if p<>sentinel then begin r:=p; repeat q:=r; r:=link(r); @!stat decr(dyn_used);@+tats@/ if rnull do begin q:=p; p:=link(p); if q are debugging.) @= @!debug @!free: packed array [mem_min..mem_max] of boolean; {free cells} @t\hskip1em@>@!was_free: packed array [mem_min..mem_max] of boolean; {previously free cells} @t\hskip1em@>@!was_mem_end,@!was_lo_max,@!was_hi_min: pointer; {previous |mem_end|, |lo_mem_max|,and |hi_mem_min|} @t\hskip1em@>@!panicking:boolean; {do we want to check memory constantly?} gubed @ @= @!debug was_mem_end:=mem_min; {indicate that everything was previously free} was_lo_max:=mem_min; was_hi_min:=mem_max; panicking:=false; gubed @ Procedure |check_mem| makes sure that the available space lists of |mem| are well formed, and it optionally prints out all locations that are reserved now but were free the last time this procedure was called. @p @!debug procedure check_mem(@!print_locs : boolean); label done1,done2; {loop exits} var @!p,@!q,@!r:pointer; {current locations of interest in |mem|} @!clobbered:boolean; {is something amiss?} begin for p:=mem_min to lo_mem_max do free[p]:=false; {you can probably do this faster} for p:=hi_mem_min to mem_end do free[p]:=false; {ditto} @; @; @; @; if print_locs then @; for p:=mem_min to lo_mem_max do was_free[p]:=free[p]; for p:=hi_mem_min to mem_end do was_free[p]:=free[p]; {|was_free:=free| might be faster} was_mem_end:=mem_end; was_lo_max:=lo_mem_max; was_hi_min:=hi_mem_min; end; gubed @ @= p:=avail; q:=null; clobbered:=false; while p<>null do begin if (p>mem_end)or(p print_int(q); goto done1; end; free[p]:=true; q:=p; p:=link(q); end; done1: @ @= p:=rover; q:=null; clobbered:=false; repeat if (p>=lo_mem_max)or(p=lo_mem_max)or(rlink(p)lo_mem_max)or@| (llink(rlink(p))<>p) then clobbered:=true; if clobbered then begin print_nl("Double-AVAIL list clobbered at "); @.Double-AVAIL list clobbered...@> print_int(q); goto done2; end; for q:=p to p+node_size(p)-1 do {mark all locations free} begin if free[q] then begin print_nl("Doubly free location at "); @.Doubly free location...@> print_int(q); goto done2; end; free[q]:=true; end; q:=p; p:=rlink(p); until p=rover; done2: @ @= p:=mem_min; while p<=lo_mem_max do {node |p| should not be empty} begin if is_empty(p) then begin print_nl("Bad flag at "); print_int(p); @.Bad flag...@> end; while (p<=lo_mem_max) and not free[p] do incr(p); while (p<=lo_mem_max) and free[p] do incr(p); end @ @= begin print_nl("New busy locs:"); @.New busy locs@> for p:=mem_min to lo_mem_max do if not free[p] and ((p>was_lo_max) or was_free[p]) then begin print_char(" "); print_int(p); end; for p:=hi_mem_min to mem_end do if not free[p] and ((pwas_mem_end) or was_free[p]) then begin print_char(" "); print_int(p); end; end @ The |search_mem| procedure attempts to answer the question ``Who points to node~|p|?'' In doing so, it fetches |link| and |info| fields of |mem| that might not be of type |two_halves|. Strictly speaking, this is @^dirty \PASCAL@> undefined in \PASCAL, and it can lead to ``false drops'' (words that seem to point to |p| purely by coincidence). But for debugging purposes, we want to rule out the places that do {\sl not\/} point to |p|, so a few false drops are tolerable. @p @!debug procedure search_mem(@!p:pointer); {look for pointers to |p|} var @!q:integer; {current position being searched} begin for q:=mem_min to lo_mem_max do begin if link(q)=p then begin print_nl("LINK("); print_int(q); print_char(")"); end; if info(q)=p then begin print_nl("INFO("); print_int(q); print_char(")"); end; end; for q:=hi_mem_min to mem_end do begin if link(q)=p then begin print_nl("LINK("); print_int(q); print_char(")"); end; if info(q)=p then begin print_nl("INFO("); print_int(q); print_char(")"); end; end; @; end; gubed @* \[12] The command codes. Before we can go much further, we need to define symbolic names for the internal code numbers that represent the various commands obeyed by \MF. These codes are somewhat arbitrary, but not completely so. For example, some codes have been made adjacent so that |case| statements in the program need not consider cases that are widely spaced, or so that |case| statements can be replaced by |if| statements. A command can begin an expression if and only if its code lies between |min_primary_command| and |max_primary_command|, inclusive. The first token of a statement that doesn't begin with an expression has a command code between |min_command| and |max_statement_command|, inclusive. The ordering of the highest-numbered commands (|commacomma @d semicolon=80 {the operator `\.;', must be |comma+1|} @d end_group=81 {end a group (\&{endgroup}), must be |semicolon+1|} @d stop=82 {end a job (\&{end}, \&{dump}), must be |end_group+1|} @d max_command_code=stop @d outer_tag=max_command_code+1 {protection code added to command code} @= @!command_code=1..max_command_code; @ Variables and capsules in \MF\ have a variety of ``types,'' distinguished by the following code numbers: @d undefined=0 {no type has been declared} @d unknown_tag=1 {this constant is added to certain type codes below} @d vacuous=1 {no expression was present} @d boolean_type=2 {\&{boolean} with a known value} @d unknown_boolean=boolean_type+unknown_tag @d string_type=4 {\&{string} with a known value} @d unknown_string=string_type+unknown_tag @d pen_type=6 {\&{pen} with a known value} @d unknown_pen=pen_type+unknown_tag @d future_pen=8 {subexpression that will become a \&{pen} at a higher level} @d path_type=9 {\&{path} with a known value} @d unknown_path=path_type+unknown_tag @d picture_type=11 {\&{picture} with a known value} @d unknown_picture=picture_type+unknown_tag @d transform_type=13 {\&{transform} variable or capsule} @d pair_type=14 {\&{pair} variable or capsule} @d numeric_type=15 {variable that has been declared \&{numeric} but not used} @d known=16 {\&{numeric} with a known value} @d dependent=17 {a linear combination with |fraction| coefficients} @d proto_dependent=18 {a linear combination with |scaled| coefficients} @d independent=19 {\&{numeric} with unknown value} @d token_list=20 {variable name or suffix argument or text argument} @d structured=21 {variable with subscripts and attributes} @d unsuffixed_macro=22 {variable defined with \&{vardef} but no \.{\AT!\#}} @d suffixed_macro=23 {variable defined with \&{vardef} and \.{\AT!\#}} @# @d unknown_types==unknown_boolean,unknown_string, unknown_pen,unknown_picture,unknown_path @= procedure print_type(@!t:small_number); begin case t of vacuous:print("vacuous"); boolean_type:print("boolean"); unknown_boolean:print("unknown boolean"); string_type:print("string"); unknown_string:print("unknown string"); pen_type:print("pen"); unknown_pen:print("unknown pen"); future_pen:print("future pen"); path_type:print("path"); unknown_path:print("unknown path"); picture_type:print("picture"); unknown_picture:print("unknown picture"); transform_type:print("transform"); pair_type:print("pair"); known:print("known numeric"); dependent:print("dependent"); proto_dependent:print("proto-dependent"); numeric_type:print("numeric"); independent:print("independent"); token_list:print("token list"); structured:print("structured"); unsuffixed_macro:print("unsuffixed macro"); suffixed_macro:print("suffixed macro"); othercases print("undefined") endcases; end; @ Values inside \MF\ are stored in two-word nodes that have a |name_type| as well as a |type|. The possibilities for |name_type| are defined here; they will be explained in more detail later. @d root=0 {|name_type| at the top level of a variable} @d saved_root=1 {same, when the variable has been saved} @d structured_root=2 {|name_type| where a |structured| branch occurs} @d subscr=3 {|name_type| in a subscript node} @d attr=4 {|name_type| in an attribute node} @d x_part_sector=5 {|name_type| in the \&{xpart} of a node} @d y_part_sector=6 {|name_type| in the \&{ypart} of a node} @d xx_part_sector=7 {|name_type| in the \&{xxpart} of a node} @d xy_part_sector=8 {|name_type| in the \&{xypart} of a node} @d yx_part_sector=9 {|name_type| in the \&{yxpart} of a node} @d yy_part_sector=10 {|name_type| in the \&{yypart} of a node} @d capsule=11 {|name_type| in stashed-away subexpressions} @d token=12 {|name_type| in a numeric token or string token} @ Primitive operations that produce values have a secondary identification code in addition to their command code; it's something like genera and species. For example, `\.*' has the command code |primary_binary|, and its secondary identification is |times|. The secondary codes start at 30 so that they don't overlap with the type codes; some type codes (e.g., |string_type|) are used as operators as well as type identifications. @d true_code=30 {operation code for \.{true}} @d false_code=31 {operation code for \.{false}} @d null_picture_code=32 {operation code for \.{nullpicture}} @d null_pen_code=33 {operation code for \.{nullpen}} @d job_name_op=34 {operation code for \.{jobname}} @d read_string_op=35 {operation code for \.{readstring}} @d pen_circle=36 {operation code for \.{pencircle}} @d normal_deviate=37 {operation code for \.{normaldeviate}} @d odd_op=38 {operation code for \.{odd}} @d known_op=39 {operation code for \.{known}} @d unknown_op=40 {operation code for \.{unknown}} @d not_op=41 {operation code for \.{not}} @d decimal=42 {operation code for \.{decimal}} @d reverse=43 {operation code for \.{reverse}} @d make_path_op=44 {operation code for \.{makepath}} @d make_pen_op=45 {operation code for \.{makepen}} @d total_weight_op=46 {operation code for \.{totalweight}} @d oct_op=47 {operation code for \.{oct}} @d hex_op=48 {operation code for \.{hex}} @d ASCII_op=49 {operation code for \.{ASCII}} @d char_op=50 {operation code for \.{char}} @d length_op=51 {operation code for \.{length}} @d turning_op=52 {operation code for \.{turningnumber}} @d x_part=53 {operation code for \.{xpart}} @d y_part=54 {operation code for \.{ypart}} @d xx_part=55 {operation code for \.{xxpart}} @d xy_part=56 {operation code for \.{xypart}} @d yx_part=57 {operation code for \.{yxpart}} @d yy_part=58 {operation code for \.{yypart}} @d sqrt_op=59 {operation code for \.{sqrt}} @d m_exp_op=60 {operation code for \.{mexp}} @d m_log_op=61 {operation code for \.{mlog}} @d sin_d_op=62 {operation code for \.{sind}} @d cos_d_op=63 {operation code for \.{cosd}} @d floor_op=64 {operation code for \.{floor}} @d uniform_deviate=65 {operation code for \.{uniformdeviate}} @d char_exists_op=66 {operation code for \.{charexists}} @d angle_op=67 {operation code for \.{angle}} @d cycle_op=68 {operation code for \.{cycle}} @d plus=69 {operation code for \.+} @d minus=70 {operation code for \.-} @d times=71 {operation code for \.*} @d over=72 {operation code for \./} @d pythag_add=73 {operation code for \.{++}} @d pythag_sub=74 {operation code for \.{+-+}} @d or_op=75 {operation code for \.{or}} @d and_op=76 {operation code for \.{and}} @d less_than=77 {operation code for \.<} @d less_or_equal=78 {operation code for \.{<=}} @d greater_than=79 {operation code for \.>} @d greater_or_equal=80 {operation code for \.{>=}} @d equal_to=81 {operation code for \.=} @d unequal_to=82 {operation code for \.{<>}} @d concatenate=83 {operation code for \.\&} @d rotated_by=84 {operation code for \.{rotated}} @d slanted_by=85 {operation code for \.{slanted}} @d scaled_by=86 {operation code for \.{scaled}} @d shifted_by=87 {operation code for \.{shifted}} @d transformed_by=88 {operation code for \.{transformed}} @d x_scaled=89 {operation code for \.{xscaled}} @d y_scaled=90 {operation code for \.{yscaled}} @d z_scaled=91 {operation code for \.{zscaled}} @d intersect=92 {operation code for \.{intersectiontimes}} @d double_dot=93 {operation code for improper \.{..}} @d substring_of=94 {operation code for \.{substring}} @d min_of=substring_of @d subpath_of=95 {operation code for \.{subpath}} @d direction_time_of=96 {operation code for \.{directiontime}} @d point_of=97 {operation code for \.{point}} @d precontrol_of=98 {operation code for \.{precontrol}} @d postcontrol_of=99 {operation code for \.{postcontrol}} @d pen_offset_of=100 {operation code for \.{penoffset}} @p procedure print_op(@!c:quarterword); begin if c<=numeric_type then print_type(c) else case c of true_code:print("true"); false_code:print("false"); null_picture_code:print("nullpicture"); null_pen_code:print("nullpen"); job_name_op:print("jobname"); read_string_op:print("readstring"); pen_circle:print("pencircle"); normal_deviate:print("normaldeviate"); odd_op:print("odd"); known_op:print("known"); unknown_op:print("unknown"); not_op:print("not"); decimal:print("decimal"); reverse:print("reverse"); make_path_op:print("makepath"); make_pen_op:print("makepen"); total_weight_op:print("totalweight"); oct_op:print("oct"); hex_op:print("hex"); ASCII_op:print("ASCII"); char_op:print("char"); length_op:print("length"); turning_op:print("turningnumber"); x_part:print("xpart"); y_part:print("ypart"); xx_part:print("xxpart"); xy_part:print("xypart"); yx_part:print("yxpart"); yy_part:print("yypart"); sqrt_op:print("sqrt"); m_exp_op:print("mexp"); m_log_op:print("mlog"); sin_d_op:print("sind"); cos_d_op:print("cosd"); floor_op:print("floor"); uniform_deviate:print("uniformdeviate"); char_exists_op:print("charexists"); angle_op:print("angle"); cycle_op:print("cycle"); plus:print_char("+"); minus:print_char("-"); times:print_char("*"); over:print_char("/"); pythag_add:print("++"); pythag_sub:print("+-+"); or_op:print("or"); and_op:print("and"); less_than:print_char("<"); less_or_equal:print("<="); greater_than:print_char(">"); greater_or_equal:print(">="); equal_to:print_char("="); unequal_to:print("<>"); concatenate:print("&"); rotated_by:print("rotated"); slanted_by:print("slanted"); scaled_by:print("scaled"); shifted_by:print("shifted"); transformed_by:print("transformed"); x_scaled:print("xscaled"); y_scaled:print("yscaled"); z_scaled:print("zscaled"); intersect:print("intersectiontimes"); substring_of:print("substring"); subpath_of:print("subpath"); direction_time_of:print("directiontime"); point_of:print("point"); precontrol_of:print("precontrol"); postcontrol_of:print("postcontrol"); pen_offset_of:print("penoffset"); othercases print("..") endcases; end; @ \MF\ also has a bunch of internal parameters that a user might want to fuss with. Every such parameter has an identifying code number, defined here. @d tracing_titles=1 {show titles online when they appear} @d tracing_equations=2 {show each variable when it becomes known} @d tracing_capsules=3 {show capsules too} @d tracing_choices=4 {show the control points chosen for paths} @d tracing_specs=5 {show subdivision of paths into octants before digitizing} @d tracing_pens=6 {show details of pens that are made} @d tracing_commands=7 {show commands and operations before they are performed} @d tracing_restores=8 {show when a variable or internal is restored} @d tracing_macros=9 {show macros before they are expanded} @d tracing_edges=10 {show digitized edges as they are computed} @d tracing_output=11 {show digitized edges as they are output} @d tracing_stats=12 {show memory usage at end of job} @d tracing_online=13 {show long diagnostics on terminal and in the log file} @d year=14 {the current year (e.g., 1984)} @d month=15 {the current month (e.g, 3 $\equiv$ March)} @d day=16 {the current day of the month} @d time=17 {the number of minutes past midnight when this job started} @d char_code=18 {the number of the next character to be output} @d char_ext=19 {the extension code of the next character to be output} @d char_wd=20 {the width of the next character to be output} @d char_ht=21 {the height of the next character to be output} @d char_dp=22 {the depth of the next character to be output} @d char_ic=23 {the italic correction of the next character to be output} @d char_dx=24 {the device's $x$ movement for the next character, in pixels} @d char_dy=25 {the device's $y$ movement for the next character, in pixels} @d design_size=26 {the unit of measure used for |char_wd..char_ic|, in points} @d hppp=27 {the number of horizontal pixels per point} @d vppp=28 {the number of vertical pixels per point} @d x_offset=29 {horizontal displacement of shipped-out characters} @d y_offset=30 {vertical displacement of shipped-out characters} @d pausing=31 {positive to display lines on the terminal before they are read} @d showstopping=32 {positive to stop after each \&{show} command} @d fontmaking=33 {positive if font metric output is to be produced} @d proofing=34 {positive for proof mode, negative to suppress output} @d smoothing=35 {positive if moves are to be ``smoothed''} @d autorounding=36 {controls path modification to ``good'' points} @d granularity=37 {autorounding uses this pixel size} @d fillin=38 {extra darkness of diagonal lines} @d turning_check=39 {controls reorientation of clockwise paths} @d warning_check=40 {controls error message when variable value is large} @d max_given_internal=warning_check @= @!internal:array[1..max_internal] of scaled; {the values of internal quantities} @!int_name:array[1..max_internal] of str_number; {their names} @!int_ptr:max_given_internal..max_internal; {the maximum internal quantity defined so far} @ @= for k:=1 to max_given_internal do internal[k]:=0; int_ptr:=max_given_internal; @ The symbolic names for internal quantities are put into \MF's hash table by using a routine called |primitive|, which will be defined later. Let us enter them now, so that we don't have to list all those names again anywhere else. @= primitive("tracingtitles",internal_quantity,tracing_titles);@/ @!@:tracingtitles_}{\&{tracingtitles} primitive@> primitive("tracingequations",internal_quantity,tracing_equations);@/ @!@:tracing_equations_}{\&{tracingequations} primitive@> primitive("tracingcapsules",internal_quantity,tracing_capsules);@/ @!@:tracing_capsules_}{\&{tracingcapsules} primitive@> primitive("tracingchoices",internal_quantity,tracing_choices);@/ @!@:tracing_choices_}{\&{tracingchoices} primitive@> primitive("tracingspecs",internal_quantity,tracing_specs);@/ @!@:tracing_specs_}{\&{tracingspecs} primitive@> primitive("tracingpens",internal_quantity,tracing_pens);@/ @!@:tracing_pens_}{\&{tracingpens} primitive@> primitive("tracingcommands",internal_quantity,tracing_commands);@/ @!@:tracing_commands_}{\&{tracingcommands} primitive@> primitive("tracingrestores",internal_quantity,tracing_restores);@/ @!@:tracing_restores_}{\&{tracingrestores} primitive@> primitive("tracingmacros",internal_quantity,tracing_macros);@/ @!@:tracing_macros_}{\&{tracingmacros} primitive@> primitive("tracingedges",internal_quantity,tracing_edges);@/ @!@:tracing_edges_}{\&{tracingedges} primitive@> primitive("tracingoutput",internal_quantity,tracing_output);@/ @!@:tracing_output_}{\&{tracingoutput} primitive@> primitive("tracingstats",internal_quantity,tracing_stats);@/ @!@:tracing_stats_}{\&{tracingstats} primitive@> primitive("tracingonline",internal_quantity,tracing_online);@/ @!@:tracing_online_}{\&{tracingonline} primitive@> primitive("year",internal_quantity,year);@/ @!@:year_}{\&{year} primitive@> primitive("month",internal_quantity,month);@/ @!@:month_}{\&{month} primitive@> primitive("day",internal_quantity,day);@/ @!@:day_}{\&{day} primitive@> primitive("time",internal_quantity,time);@/ @!@:time_}{\&{time} primitive@> primitive("charcode",internal_quantity,char_code);@/ @!@:char_code_}{\&{charcode} primitive@> primitive("charext",internal_quantity,char_ext);@/ @!@:char_ext_}{\&{charext} primitive@> primitive("charwd",internal_quantity,char_wd);@/ @!@:char_wd_}{\&{charwd} primitive@> primitive("charht",internal_quantity,char_ht);@/ @!@:char_ht_}{\&{charht} primitive@> primitive("chardp",internal_quantity,char_dp);@/ @!@:char_dp_}{\&{chardp} primitive@> primitive("charic",internal_quantity,char_ic);@/ @!@:char_ic_}{\&{charic} primitive@> primitive("chardx",internal_quantity,char_dx);@/ @!@:char_dx_}{\&{chardx} primitive@> primitive("chardy",internal_quantity,char_dy);@/ @!@:char_dy_}{\&{chardy} primitive@> primitive("designsize",internal_quantity,design_size);@/ @!@:design_size_}{\&{designsize} primitive@> primitive("hppp",internal_quantity,hppp);@/ @!@:hppp_}{\&{hppp} primitive@> primitive("vppp",internal_quantity,vppp);@/ @!@:vppp_}{\&{vppp} primitive@> primitive("xoffset",internal_quantity,x_offset);@/ @!@:x_offset_}{\&{xoffset} primitive@> primitive("yoffset",internal_quantity,y_offset);@/ @!@:y_offset_}{\&{yoffset} primitive@> primitive("pausing",internal_quantity,pausing);@/ @!@:pausing_}{\&{pausing} primitive@> primitive("showstopping",internal_quantity,showstopping);@/ @!@:showstopping_}{\&{showstopping} primitive@> primitive("fontmaking",internal_quantity,fontmaking);@/ @!@:fontmaking_}{\&{fontmaking} primitive@> primitive("proofing",internal_quantity,proofing);@/ @!@:proofing_}{\&{proofing} primitive@> primitive("smoothing",internal_quantity,smoothing);@/ @!@:smoothing_}{\&{smoothing} primitive@> primitive("autorounding",internal_quantity,autorounding);@/ @!@:autorounding_}{\&{autorounding} primitive@> primitive("granularity",internal_quantity,granularity);@/ @!@:granularity_}{\&{granularity} primitive@> primitive("fillin",internal_quantity,fillin);@/ @!@:fillin_}{\&{fillin} primitive@> primitive("turningcheck",internal_quantity,turning_check);@/ @!@:turning_check_}{\&{turningcheck} primitive@> primitive("warningcheck",internal_quantity,warning_check);@/ @!@:warning_check_}{\&{warningcheck} primitive@> @ Well, we do have to list the names one more time, for use in symbolic printouts. @= int_name[tracing_titles]:="tracingtitles"; int_name[tracing_equations]:="tracingequations"; int_name[tracing_capsules]:="tracingcapsules"; int_name[tracing_choices]:="tracingchoices"; int_name[tracing_specs]:="tracingspecs"; int_name[tracing_pens]:="tracingpens"; int_name[tracing_commands]:="tracingcommands"; int_name[tracing_restores]:="tracingrestores"; int_name[tracing_macros]:="tracingmacros"; int_name[tracing_edges]:="tracingedges"; int_name[tracing_output]:="tracingoutput"; int_name[tracing_stats]:="tracingstats"; int_name[tracing_online]:="tracingonline"; int_name[year]:="year"; int_name[month]:="month"; int_name[day]:="day"; int_name[time]:="time"; int_name[char_code]:="charcode"; int_name[char_ext]:="charext"; int_name[char_wd]:="charwd"; int_name[char_ht]:="charht"; int_name[char_dp]:="chardp"; int_name[char_ic]:="charic"; int_name[char_dx]:="chardx"; int_name[char_dy]:="chardy"; int_name[design_size]:="designsize"; int_name[hppp]:="hppp"; int_name[vppp]:="vppp"; int_name[x_offset]:="xoffset"; int_name[y_offset]:="yoffset"; int_name[pausing]:="pausing"; int_name[showstopping]:="showstopping"; int_name[fontmaking]:="fontmaking"; int_name[proofing]:="proofing"; int_name[smoothing]:="smoothing"; int_name[autorounding]:="autorounding"; int_name[granularity]:="granularity"; int_name[fillin]:="fillin"; int_name[turning_check]:="turningcheck"; int_name[warning_check]:="warningcheck"; @ The following procedure, which is called just before \MF\ initializes its input and output, establishes the initial values of the date and time. @^system dependencies@> Since standard \PASCAL\ cannot provide such information, something special is needed. The program here simply specifies July 4, 1776, at noon; but users probably want a better approximation to the truth. Note that the values are |scaled| integers. Hence \MF\ can no longer be used after the year 32767. @p procedure fix_date_and_time; begin internal[time]:=12*60*unity; {minutes since midnight} internal[day]:=4*unity; {fourth day of the month} internal[month]:=7*unity; {seventh month of the year} internal[year]:=1776*unity; {Anno Domini} end; @ \MF\ is occasionally supposed to print diagnostic information that goes only into the transcript file, unless |tracing_online| is positive. Now that we have defined |tracing_online| we can define two routines that adjust the destination of print commands: @= procedure begin_diagnostic; {prepare to do some tracing} begin old_setting:=selector; if(internal[tracing_online]<=0)and(selector=term_and_log) then begin decr(selector); if history=spotless then history:=warning_issued; end; end; @# procedure end_diagnostic(@!blank_line:boolean); {restore proper conditions after tracing} begin print_nl(""); if blank_line then print_ln; selector:=old_setting; end; @ Of course we had better declare another global variable, if the previous routines are going to work. @= @!old_setting:0..max_selector; @ We will occasionally use |begin_diagnostic| in connection with line-number printing, as follows. (The parameter |s| is typically |"Path"| or |"Cycle spec"|, etc.) @= procedure print_diagnostic(@!s,@!t:str_number;@!nuline:boolean); begin begin_diagnostic; if nuline then print_nl(s)@+else print(s); print(" at line "); print_int(line); print(t); print_char(":"); end; @ The 128 |ASCII_code| characters are grouped into classes by means of the |char_class| table. Individual class numbers have no semantic or syntactic significance, except in a few instances defined here. There's also |max_class|, which can be used as a basis for additional class numbers in nonstandard extensions of \MF. @d digit_class=0 {the class number of \.{0123456789}} @d period_class=1 {the class number of `\..'} @d space_class=2 {the class number of spaces and nonstandard characters} @d percent_class=3 {the class number of `\.\%'} @d string_class=4 {the class number of `\."'} @d right_paren_class=8 {the class number of `\.)'} @d isolated_classes==5,6,7,8 {characters that make length-one tokens only} @d letter_class=9 {letters and the underline character} @d left_bracket_class=17 {`\.['} @d right_bracket_class=18 {`\.]'} @d invalid_class=20 {bad character in the input} @d max_class=20 {the largest class number} @= @!char_class:array[ASCII_code] of 0..max_class; {the class numbers} @ If changes are made to accommodate non-ASCII character sets, they should follow the guidelines in Appendix~C of {\sl The {\logos METAFONT\/}book}. @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> @^system dependencies@> @= for k:="0" to "9" do char_class[k]:=digit_class; char_class["."]:=period_class; char_class[" "]:=space_class; char_class["%"]:=percent_class; char_class[""""]:=string_class;@/ char_class[","]:=5; char_class[";"]:=6; char_class["("]:=7; char_class[")"]:=right_paren_class; for k:="A" to "Z" do char_class[k]:=letter_class; for k:="a" to "z" do char_class[k]:=letter_class; char_class["_"]:=letter_class;@/ char_class["<"]:=10; char_class["="]:=10; char_class[">"]:=10; char_class[":"]:=10; char_class["|"]:=10;@/ char_class["`"]:=11; char_class["'"]:=11;@/ char_class["+"]:=12; char_class["-"]:=12;@/ char_class["/"]:=13; char_class["*"]:=13; char_class["\"]:=13;@/ char_class["!"]:=14; char_class["?"]:=14;@/ char_class["#"]:=15; char_class["&"]:=15; char_class["@@"]:=15; char_class["$"]:=15;@/ char_class["^"]:=16; char_class["~"]:=16;@/ char_class["["]:=left_bracket_class; char_class["]"]:=right_bracket_class;@/ char_class["{"]:=19; char_class["}"]:=19;@/ for k:=0 to " "-1 do char_class[k]:=invalid_class; char_class[127]:=invalid_class; @* \[13] The hash table. Symbolic tokens are stored and retrieved by means of a fairly standard hash table algorithm called the method of ``coalescing lists'' (cf.\ Algorithm 6.4C in {\sl The Art of Computer Programming\/}). Once a symbolic token enters the table, it is never removed. The actual sequence of characters forming a symbolic token is stored in the |str_pool| array together with all the other strings. An auxiliary array |hash| consists of items with two halfword fields per word. The first of these, called |next(p)|, points to the next identifier belonging to the same coalesced list as the identifier corresponding to~|p|; and the other, called |text(p)|, points to the |str_start| entry for |p|'s identifier. If position~|p| of the hash table is empty, we have |text(p)=0|; if position |p| is either empty or the end of a coalesced hash list, we have |next(p)=0|. An auxiliary pointer variable called |hash_used| is maintained in such a way that all locations |p>=hash_used| are nonempty. The global variable |st_count| tells how many symbolic tokens have been defined, if statistics are being kept. The first 128 locations of |hash| are reserved for symbols of length one. There's a parallel array called |eqtb| that contains the current equivalent values of each symbolic token. The entries of this array consist of two halfwords called |eq_type| (a command code) and |equiv| (a secondary piece of information that qualifies the |eq_type|). @d next(#) == hash[#].lh {link for coalesced lists} @d text(#) == hash[#].rh {string number for symbolic token name} @d hash_is_full == (hash_used=1) {test if all positions are occupied} @d eq_type(#) == eqtb[#].lh {the current ``meaning'' of a symbolic token} @d equiv(#) == eqtb[#].rh {parametric part of a token's meaning} @d hash_base=129 {hashing actually starts here} @= @!hash_used:pointer; {allocation pointer for |hash|} @!st_count:integer; {total number of known identifiers} @ Certain entries in the hash table are ``frozen'' and not redefinable, since they are used in error recovery. @d hash_top==hash_base+hash_size {the first location of the frozen area} @d frozen_inaccessible==hash_top {|hash| location to protect the frozen area} @d frozen_repeat_loop==hash_top+1 {|hash| location of a loop-repeat token} @d frozen_right_delimiter==hash_top+2 {|hash| location of a permanent `\.)'} @d frozen_left_bracket==hash_top+3 {|hash| location of a permanent `\.['} @d frozen_slash==hash_top+4 {|hash| location of a permanent `\./'} @d frozen_colon==hash_top+5 {|hash| location of a permanent `\.:'} @d frozen_semicolon==hash_top+6 {|hash| location of a permanent `\.;'} @d frozen_end_for==hash_top+7 {|hash| location of a permanent \&{endfor}} @d frozen_end_def==hash_top+8 {|hash| location of a permanent \&{enddef}} @d frozen_fi==hash_top+9 {|hash| location of a permanent \&{fi}} @d frozen_end_group==hash_top+10 {|hash| location of a permanent `\.{endgroup}'} @d frozen_bad_vardef==hash_top+11 {|hash| location of `\.{a bad variable}'} @d frozen_undefined==hash_top+12 {|hash| location that never gets defined} @d hash_end==hash_top+12 {the actual size of the |hash| and |eqtb| arrays} @= @!hash: array[1..hash_end] of two_halves; {the hash table} @!eqtb: array[1..hash_end] of two_halves; {the equivalents} @ @= next(1):=0; text(1):=0; eq_type(1):=tag_token; equiv(1):=null; for k:=2 to hash_end do begin hash[k]:=hash[1]; eqtb[k]:=eqtb[1]; end; @ @= hash_used:=frozen_inaccessible; {nothing is used} st_count:=0;@/ text(frozen_bad_vardef):="a bad variable"; text(frozen_fi):="fi"; text(frozen_end_group):="endgroup"; text(frozen_end_def):="enddef"; text(frozen_end_for):="endfor";@/ text(frozen_semicolon):=";"; text(frozen_colon):=":"; text(frozen_slash):="/"; text(frozen_left_bracket):="["; text(frozen_right_delimiter):=")";@/ text(frozen_inaccessible):=" INACCESSIBLE";@/ eq_type(frozen_right_delimiter):=right_delimiter; @ @= if hash_end+max_internal>max_halfword then bad:=21; @ Here is the subroutine that searches the hash table for an identifier that matches a given string of length~|l| appearing in |buffer[j.. (j+l-1)]|. If the identifier is not found, it is inserted; hence it will always be found, and the corresponding hash table address will be returned. @p function id_lookup(@!j,@!l:integer):pointer; {search the hash table} label found; {go here when you've found it} var @!h:integer; {hash code} @!p:pointer; {index in |hash| array} @!k:pointer; {index in |buffer| array} begin if l=1 then @; @; p:=h+hash_base; {we start searching here; note that |0<=h0 then if length(text(p))=l then if str_eq_buf(text(p),j) then goto found; if next(p)=0 then @; p:=next(p); end; found: id_lookup:=p; end; @ @= begin p:=buffer[j]+1; text(p):=p-1; goto found; end @ @= begin if text(p)>0 then begin repeat if hash_is_full then overflow("hash size",hash_size); @:METAFONT capacity exceeded hash size}{\quad hash size@> decr(hash_used); until text(hash_used)=0; {search for an empty location in |hash|} next(p):=hash_used; p:=hash_used; end; str_room(l); for k:=j to j+l-1 do append_char(buffer[k]); text(p):=make_string; str_ref[text(p)]:=max_str_ref; @!stat incr(st_count);@+tats@;@/ goto found; end @ The value of |hash_prime| should be roughly 85\% of |hash_size|, and it should be a prime number. The theory of hashing tells us to expect fewer than two table probes, on the average, when the search is successful. [See J.~S. Vitter, {\sl Journal of the ACM\/ \bf30} (1983), 231--258.] @^Vitter, Jeffrey Scott@> @= h:=buffer[j]; for k:=j+1 to j+l-1 do begin h:=h+h+buffer[k]; while h>=hash_prime do h:=h-hash_prime; end @ @= for q:=1 to hash_end do begin if equiv(q)=p then begin print_nl("EQUIV("); print_int(q); print_char(")"); end; end @ We need to put \MF's ``primitive'' symbolic tokens into the hash table, together with their command code (which will be the |eq_type|) and an operand (which will be the |equiv|). The |primitive| procedure does this, in a way that no \MF\ user can. The global value |cur_sym| contains the new |eqtb| pointer after |primitive| has acted. @p @!init procedure primitive(@!s:str_number;@!c:halfword;@!o:halfword); var @!k:pool_pointer; {index into |str_pool|} @!j:small_number; {index into |buffer|} @!l:small_number; {length of the string} begin k:=str_start[s]; l:=str_start[s+1]-k; {we will move |s| into the (empty) |buffer|} for j:=0 to l-1 do buffer[j]:=str_pool[k+j]; cur_sym:=id_lookup(0,l);@/ if s>=128 then {we don't want to have the string twice} begin flush_string(str_ptr-1); text(cur_sym):=s; end; eq_type(cur_sym):=c; equiv(cur_sym):=o; end; tini @ Many of \MF's primitives need no |equiv|, since they are identifiable by their |eq_type| alone. These primitives are loaded into the hash table as follows: @= primitive("..",path_join,0);@/ @!@:.._}{\.{..} primitive@> primitive("[",left_bracket,0); eqtb[frozen_left_bracket]:=eqtb[cur_sym];@/ @!@:[ }{\.{[} primitive@> primitive("]",right_bracket,0);@/ @!@:] }{\.{]} primitive@> primitive("}",right_brace,0);@/ @!@:]]}{\.{\char`\}} primitive@> primitive("{",left_brace,0);@/ @!@:][}{\.{\char`\{} primitive@> primitive(":",colon,0); eqtb[frozen_colon]:=eqtb[cur_sym];@/ @!@:: }{\.{:} primitive@> primitive(":=",assignment,0);@/ @!@::=_}{\.{:=} primitive@> primitive(",",comma,0);@/ @!@:, }{\., primitive@> primitive(";",semicolon,0); eqtb[frozen_semicolon]:=eqtb[cur_sym];@/ @!@:; }{\.; primitive@> primitive("\",relax,0);@/ @!@:]]\\}{\.{\char`\\} primitive@> @# primitive("addto",add_to_command,0);@/ @!@:add_to_}{\&{addto} primitive@> primitive("at",at_token,0);@/ @!@:at_}{\&{at} primitive@> primitive("atleast",at_least,0);@/ @!@:at_least_}{\&{atleast} primitive@> primitive("begingroup",begin_group,0); bg_loc:=cur_sym;@/ @!@:begin_group_}{\&{begingroup} primitive@> primitive("controls",controls,0);@/ @!@:controls_}{\&{controls} primitive@> primitive("cull",cull_command,0);@/ @!@:cull_}{\&{cull} primitive@> primitive("curl",curl_command,0);@/ @!@:curl_}{\&{curl} primitive@> primitive("delimiters",delimiters,0);@/ @!@:delimiters_}{\&{delimiters} primitive@> primitive("display",display_command,0);@/ @!@:display_}{\&{display} primitive@> primitive("endgroup",end_group,0); eqtb[frozen_end_group]:=eqtb[cur_sym]; eg_loc:=cur_sym;@/ @!@:endgroup_}{\&{endgroup} primitive@> primitive("everyjob",every_job_command,0);@/ @!@:every_job_}{\&{everyjob} primitive@> primitive("exitif",exit_test,0);@/ @!@:exit_if_}{\&{exitif} primitive@> primitive("expandafter",expand_after,0);@/ @!@:expand_after_}{\&{expandafter} primitive@> primitive("from",from_token,0);@/ @!@:from_}{\&{from} primitive@> primitive("inwindow",in_window,0);@/ @!@:in_window_}{\&{inwindow} primitive@> primitive("interim",interim_command,0);@/ @!@:interim_}{\&{interim} primitive@> primitive("let",let_command,0);@/ @!@:let_}{\&{let} primitive@> primitive("newinternal",new_internal,0);@/ @!@:new_internal_}{\&{newinternal} primitive@> primitive("of",of_token,0);@/ @!@:of_}{\&{of} primitive@> primitive("openwindow",open_window,0);@/ @!@:open_window_}{\&{openwindow} primitive@> primitive("randomseed",random_seed,0);@/ @!@:random_seed_}{\&{randomseed} primitive@> primitive("save",save_command,0);@/ @!@:save_}{\&{save} primitive@> primitive("scantokens",scan_tokens,0);@/ @!@:scan_tokens_}{\&{scantokens} primitive@> primitive("shipout",ship_out_command,0);@/ @!@:ship_out_}{\&{shipout} primitive@> primitive("step",step_token,0);@/ @!@:step_}{\&{step} primitive@> primitive("str",str_op,0);@/ @!@:str_}{\&{str} primitive@> primitive("tension",tension,0);@/ @!@:tension_}{\&{tension} primitive@> primitive("to",to_token,0);@/ @!@:to_}{\&{to} primitive@> primitive("until",until_token,0);@/ @!@:until_}{\&{until} primitive@> @ Each primitive has a corresponding inverse, so that it is possible to display the cryptic numeric contents of |eqtb| in symbolic form. Every call of |primitive| in this program is therefore accompanied by some straightforward code that forms part of the |print_cmd_mod| routine explained below. @= add_to_command:print("addto"); assignment:print(":="); at_least:print("atleast"); at_token:print("at"); begin_group:print("begingroup"); colon:print(":"); comma:print(","); controls:print("controls"); cull_command:print("cull"); curl_command:print("curl"); delimiters:print("delimiters"); display_command:print("display"); end_group:print("endgroup"); every_job_command:print("everyjob"); exit_test:print("exitif"); expand_after:print("expandafter"); from_token:print("from"); in_window:print("inwindow"); interim_command:print("interim"); left_brace:print("{"); left_bracket:print("["); let_command:print("let"); new_internal:print("newinternal"); of_token:print("of"); open_window:print("openwindow"); path_join:print(".."); random_seed:print("randomseed"); relax:print_char("\"); right_brace:print("}"); right_bracket:print("]"); save_command:print("save"); scan_tokens:print("scantokens"); semicolon:print(";"); ship_out_command:print("shipout"); step_token:print("step"); str_op:print("str"); tension:print("tension"); to_token:print("to"); until_token:print("until"); @ We will deal with the other primitives later, at some point in the program where their |eq_type| and |equiv| values are more meaningful. For example, the primitives for macro definitions will be loaded when we consider the routines that define macros. It is easy to find where each particular primitive was treated by looking in the index at the end; for example, the section where |"def"| entered |eqtb| is listed under `\&{def} primitive'. @* \[14] Token lists. A \MF\ token is either symbolic or numeric or a string, or it denotes a macro parameter or capsule; so there are five corresponding ways to encode it @^token@> internally: (1)~A symbolic token whose hash code is~|p| is represented by the number |p|, in the |info| field of a single-word node in~|mem|. (2)~A numeric token whose |scaled| value is~|v| is represented in a two-word node of~|mem|; the |type| field is |known|, the |name_type| field is |token|, and the |value| field holds~|v|. The fact that this token appears in a two-word node rather than a one-word node is, of course, clear from the node address. (3)~A string token is also represented in a two-word node; the |type| field is |string_type|, the |name_type| field is |token|, and the |value| field holds the corresponding |str_number|. (4)~Capsules have |name_type=capsule|, and their |type| and |value| fields represent arbitrary values (in ways to be explained later). (5)~Macro parameters are like symbolic tokens in that they appear in |info| fields of one-word nodes. The $k$th parameter is represented by |expr_base+k| if it is of type \&{expr}, or by |suffix_base+k| if it is of type \&{suffix}, or by |text_base+k| if it is of type \&{text}. (Here |0<=k= if text_base+param_size>max_halfword then bad:=22; @ A numeric token is created by the following trivial routine. @p function new_num_tok(@!v:scaled):pointer; var @!p:pointer; {the new node} begin p:=get_node(token_node_size); value(p):=v; type(p):=known; name_type(p):=token; new_num_tok:=p; end; @ A token list is a singly linked list of nodes in |mem|, where each node contains a token and a link. Here's a subroutine that gets rid of a token list when it is no longer needed. @p procedure@?token_recycle; forward;@t\2@>@;@/ procedure flush_token_list(@!p:pointer); var @!q:pointer; {the node being recycled} begin while p<>null do begin q:=p; p:=link(p); if q>=hi_mem_min then free_avail(q) else begin case type(q) of vacuous,boolean_type,known:do_nothing; string_type:delete_str_ref(value(q)); unknown_types,pen_type,path_type,future_pen,picture_type, pair_type,transform_type,dependent,proto_dependent,independent: begin g_pointer:=q; token_recycle; end; othercases confusion("token") @:this can't happen token}{\quad token@> endcases;@/ free_node(q,token_node_size); end; end; end; @ The procedure |show_token_list|, which prints a symbolic form of the token list that starts at a given node |p|, illustrates these conventions. The token list being displayed should not begin with a reference count. However, the procedure is intended to be fairly robust, so that if the memory links are awry or if |p| is not really a pointer to a token list, almost nothing catastrophic can happen. An additional parameter |q| is also given; this parameter is either null or it points to a node in the token list where a certain magic computation takes place that will be explained later. (Basically, |q| is non-null when we are printing the two-line context information at the time of an error message; |q| marks the place corresponding to where the second line should begin.) The generation will stop, and `\.{\char`\ ETC.}' will be printed, if the length of printing exceeds a given limit~|l|; the length of printing upon entry is assumed to be a given amount called |null_tally|. (Note that |show_token_list| sometimes uses itself recursively to print the name of a variable within a capsule.) @^recursion@> Unusual entries are printed in the form of all-caps symbolic tokens preceded by a space, e.g., `\.{\char`\ BAD}'. @= procedure@?print_capsule; forward; @t\2@>@;@/ procedure show_token_list(@!p,@!q:integer;@!l,@!null_tally:integer); label exit; var @!class,@!c:small_number; {the |char_class| of previous and new tokens} @!r,@!v:integer; {temporary registers} begin class:=percent_class; tally:=null_tally; while (p<>null) and (tally; @; class:=c; p:=link(p); end; if p<>null then print(" ETC."); @.ETC@> exit: end; @ @= c:=letter_class; {the default} if (pmem_end) then begin print(" CLOBBERED"); return; @.CLOBBERED@> end; if p else begin r:=info(p); if r>=expr_base then @ else if r<1 then if r=0 then @ else print(" IMPOSSIBLE") @.IMPOSSIBLE@> else begin r:=text(r); if (r<0)or(r>=str_ptr) then print(" NONEXISTENT") @.NONEXISTENT@> else @; end; end @ @= if name_type(p)=token then if type(p)=known then @ else if type(p)<>string_type then print(" BAD") @.BAD@> else begin print_char(""""); slow_print(value(p)); print_char(""""); c:=string_class; end else if (name_type(p)<>capsule)or(type(p)independent) then print(" BAD") else begin g_pointer:=p; print_capsule; c:=right_paren_class; end @ @= begin if class=digit_class then print_char(" "); v:=value(p); if v<0 then begin if class=left_bracket_class then print_char(" "); print_char("["); print_scaled(v); print_char("]"); c:=right_bracket_class; end else begin print_scaled(v); c:=digit_class; end; end @ Strictly speaking, a genuine token will never have |info(p)=0|. But we will see later (in the |print_variable_name| routine) that it is convenient to let |info(p)=0| stand for `\.{[]}'. @= begin if class=left_bracket_class then print_char(" "); print("[]"); c:=right_bracket_class; end @ @= begin if r end else if r end else begin print("(TEXT"); r:=r-(text_base); @.TEXT@> end; print_int(r); print_char(")"); c:=right_paren_class; end @ @= begin c:=char_class[str_pool[str_start[r]]]; if c=class then case c of letter_class:print_char("."); isolated_classes:do_nothing; othercases print_char(" ") endcases; print(r); end @ The following procedures have been declared |forward| with no parameters, because the author dislikes \PASCAL's convention about |forward| procedures with parameters. It was necessary to do something, because |show_token_list| is recursive (although the recursion is limited to one level), and because |flush_token_list| is syntactically (but not semantically) recursive. @^recursion@> @= procedure print_capsule; begin print_char("("); print_exp(g_pointer,0); print_char(")"); end; @# procedure token_recycle; begin recycle_value(g_pointer); end; @ @= @!g_pointer:pointer; {(global) parameter to the |forward| procedures} @ Macro definitions are kept in \MF's memory in the form of token lists that have a few extra one-word nodes at the beginning. The first node contains a reference count that is used to tell when the list is no longer needed. To emphasize the fact that a reference count is present, we shall refer to the |info| field of this special node as the |ref_count| field. @^reference counts@> The next node or nodes after the reference count serve to describe the formal parameters. They either contain a code word that specifies all of the parameters, or they contain zero or more parameter tokens followed by the code `|general_macro|'. @d ref_count==info {reference count preceding a macro definition or pen header} @d add_mac_ref(#)==incr(ref_count(#)) {make a new reference to a macro list} @d general_macro=0 {preface to a macro defined with a parameter list} @d primary_macro=1 {preface to a macro with a \&{primary} parameter} @d secondary_macro=2 {preface to a macro with a \&{secondary} parameter} @d tertiary_macro=3 {preface to a macro with a \&{tertiary} parameter} @d expr_macro=4 {preface to a macro with an undelimited \&{expr} parameter} @d of_macro=5 {preface to a macro with undelimited `\&{expr} |x| \&{of}~|y|' parameters} @d suffix_macro=6 {preface to a macro with an undelimited \&{suffix} parameter} @d text_macro=7 {preface to a macro with an undelimited \&{text} parameter} @p procedure delete_mac_ref(@!p:pointer); {|p| points to the reference count of a macro list that is losing one reference} begin if ref_count(p)=null then flush_token_list(p) else decr(ref_count(p)); end; @ The following subroutine displays a macro, given a pointer to its reference count. @p @t\4@>@@; procedure show_macro(@!p:pointer;@!q,@!l:integer); label exit; var @!r:pointer; {temporary storage} begin p:=link(p); {bypass the reference count} while info(p)>text_macro do begin r:=link(p); link(p):=null; show_token_list(p,null,l,0); link(p):=r; p:=r; if l>0 then l:=l-tally@+else return; end; {control printing of `\.{ETC.}'} @.ETC@> tally:=0; case info(p) of general_macro:print("->"); @.->@> primary_macro,secondary_macro,tertiary_macro:begin print_char("<"); print_cmd_mod(param_type,info(p)); print(">->"); end; expr_macro:print("->"); of_macro:print("of->"); suffix_macro:print("->"); text_macro:print("->"); end; {there are no other cases} show_token_list(link(p),q,l-tally,0); exit:end; @* \[15] Data structures for variables. The variables of \MF\ programs can be simple, like `\.x', or they can combine the structural properties of arrays and records, like `\.{x20a.b}'. A \MF\ user assigns a type to a variable like \.{x20a.b} by saying, for example, `\.{boolean} \.{x20a.b}'. It's time for us to study how such things are represented inside of the computer. Each variable value occupies two consecutive words, either in a two-word node called a value node, or as a two-word subfield of a larger node. One of those two words is called the |value| field; it is an integer, containing either a |scaled| numeric value or the representation of some other type of quantity. (It might also be subdivided into halfwords, in which case it is referred to by other names instead of |value|.) The other word is broken into subfields called |type|, |name_type|, and |link|. The |type| field is a quarterword that specifies the variable's type, and |name_type| is a quarterword from which \MF\ can reconstruct the variable's name (sometimes by using the |link| field as well). Thus, only 1.25 words are actually devoted to the value itself; the other three-quarters of a word are overhead, but they aren't wasted because they allow \MF\ to deal with sparse arrays and to provide meaningful diagnostics. In this section we shall be concerned only with the structural aspects of variables, not their values. Later parts of the program will change the |type| and |value| fields, but we shall treat those fields as black boxes whose contents should not be touched. However, if the |type| field is |structured|, there is no |value| field, and the second word is broken into two pointer fields called |attr_head| and |subscr_head|. Those fields point to additional nodes that contain structural information, as we shall see. @d subscr_head_loc(#) == #+1 {where |value|, |subscr_head| and |attr_head| are} @d attr_head(#) == info(subscr_head_loc(#)) {pointer to attribute info} @d subscr_head(#) == link(subscr_head_loc(#)) {pointer to subscript info} @d value_node_size=2 {the number of words in a value node} @ An attribute node is three words long. Two of these words contain |type| and |value| fields as described above, and the third word contains additional information: There is an |attr_loc| field, which contains the hash address of the token that names this attribute; and there's also a |parent| field, which points to the value node of |structured| type at the next higher level (i.e., at the level to which this attribute is subsidiary). The |name_type| in an attribute node is `|attr|'. The |link| field points to the next attribute with the same parent; these are arranged in increasing order, so that |attr_loc(link(p))>attr_loc(p)|. The final attribute node links to the constant |end_attr|, whose |attr_loc| field is greater than any legal hash address. The |attr_head| in the parent points to a node whose |name_type| is |structured_root|; this node represents the null attribute, i.e., the variable that is relevant when no attributes are attached to the parent. The |attr_head| node is either a value node, a subscript node, or an attribute node, depending on what the parent would be if it were not structured; but the subscript and attribute fields are ignored, so it effectively contains only the data of a value node. The |link| field in this special node points to an attribute node whose |attr_loc| field is zero; the latter node represents a collective subscript `\.{[]}' attached to the parent, and its |link| field points to the first non-special attribute node (or to |end_attr| if there are none). A subscript node likewise occupies three words, with |type| and |value| fields plus extra information; its |name_type| is |subscr|. In this case the third word is called the |subscript| field, which is a |scaled| integer. The |link| field points to the subscript node with the next larger subscript, if any; otherwise the |link| points to the attribute node for collective subscripts at this level. We have seen that the latter node contains an upward pointer, so that the parent can be deduced. The |name_type| in a parent-less value node is |root|, and the |link| is the hash address of the token that names this value. In other words, variables have a hierarchical structure that includes enough threads running around so that the program is able to move easily between siblings, parents, and children. An example should be helpful: (The reader is advised to draw a picture while reading the following description, since that will help to firm up the ideas.) Suppose that `\.x' and `\.{x.a}' and `\.{x[]b}' and `\.{x5}' and `\.{x20b}' have been mentioned in a user's program, where \.{x[]b} has been declared to be of \&{boolean} type. Let |h(x)|, |h(a)|, and |h(b)| be the hash addresses of \.x, \.a, and \.b. Then |eq_type(h(x))=name| and |equiv(h(x))=p|, where |p|~is a two-word value node with |name_type(p)=root| and |link(p)=h(x)|. We have |type(p)=structured|, |attr_head(p)=q|, and |subscr_head(p)=r|, where |q| points to a value node and |r| to a subscript node. (Are you still following this? Use a pencil to draw a diagram.) The lone variable `\.x' is represented by |type(q)| and |value(q)|; furthermore |name_type(q)=structured_root| and |link(q)=q1|, where |q1| points to an attribute node representing `\.{x[]}'. Thus |name_type(q1)=attr|, |attr_loc(q1)=collective_subscript=0|, |parent(q1)=p|, |type(q1)=structured|, |attr_head(q1)=qq|, and |subscr_head(q1)=qq1|; |qq| is a value node with |type(qq)=numeric_type| (assuming that \.{x5} is numeric, because |qq| represents `\.{x[]}' with no further attributes), |name_type(qq)=structured_root|, and |link(qq)=qq1|. (Now pay attention to the next part.) Node |qq1| is an attribute node representing `\.{x[][]}', which has never yet occurred; its |type| field is |undefined|, and its |value| field is undefined. We have |name_type(qq1)=attr|, |attr_loc(qq1)=collective_subscript|, |parent(qq1)=q1|, and |link(qq1)=qq2|. Since |qq2| represents `\.{x[]b}', |type(qq2)=unknown_boolean|; also |attr_loc(qq2)=h(b)|, |parent(qq2)=q1|, |name_type(qq2)=attr|, |link(qq2)=end_attr|. (Maybe colored lines will help untangle your picture.) Node |r| is a subscript node with |type| and |value| representing `\.{x5}'; |name_type(r)=subscr|, |subscript(r)=5.0|, and |link(r)=r1| is another subscript node. To complete the picture, see if you can guess what |link(r1)| is; give up? It's~|q1|. Furthermore |subscript(r1)=20.0|, |name_type(r1)=subscr|, |type(r1)=structured|, |attr_head(r1)=qqq|, |subscr_head(r1)=qqq1|, and we finish things off with three more nodes |qqq|, |qqq1|, and |qqq2| hung onto~|r1|. (Perhaps you should start again with a larger sheet of paper.) The value of variable \.{x20b} appears in node~|qqq2|, as you can well imagine. If the example in the previous paragraph doesn't make things crystal clear, a glance at some of the simpler subroutines below will reveal how things work out in practice. The only really unusual thing about these conventions is the use of collective subscript attributes. The idea is to avoid repeating a lot of type information when many elements of an array are identical macros (for which distinct values need not be stored) or when they don't have all of the possible attributes. Branches of the structure below collective subscript attributes do not carry actual values except for macro identifiers; branches of the structure below subscript nodes do not carry significant information in their collective subscript attributes. @d attr_loc_loc(#)==#+2 {where the |attr_loc| and |parent| fields are} @d attr_loc(#)==info(attr_loc_loc(#)) {hash address of this attribute} @d parent(#)==link(attr_loc_loc(#)) {pointer to |structured| variable} @d subscript_loc(#)==#+2 {where the |subscript| field lives} @d subscript(#)==mem[subscript_loc(#)].sc {subscript of this variable} @d attr_node_size=3 {the number of words in an attribute node} @d subscr_node_size=3 {the number of words in a subscript node} @d collective_subscript=0 {code for the attribute `\.{[]}'} @= attr_loc(end_attr):=hash_end+1; parent(end_attr):=null; @ Variables of type \&{pair} will have values that point to four-word nodes containing two numeric values. The first of these values has |name_type=x_part_sector| and the second has |name_type=y_part_sector|; the |link| in the first points back to the node whose |value| points to this four-word node. Variables of type \&{transform} are similar, but in this case their |value| points to a 12-word node containing six values, identified by |x_part_sector|, |y_part_sector|, |xx_part_sector|, |xy_part_sector|, |yx_part_sector|, and |yy_part_sector|. When an entire structured variable is saved, the |root| indication is temporarily replaced by |saved_root|. Some variables have no name; they just are used for temporary storage while expressions are being evaluated. We call them {\sl capsules}. @d x_part_loc(#)==# {where the \&{xpart} is found in a pair or transform node} @d y_part_loc(#)==#+2 {where the \&{ypart} is found in a pair or transform node} @d xx_part_loc(#)==#+4 {where the \&{xxpart} is found in a transform node} @d xy_part_loc(#)==#+6 {where the \&{xypart} is found in a transform node} @d yx_part_loc(#)==#+8 {where the \&{yxpart} is found in a transform node} @d yy_part_loc(#)==#+10 {where the \&{yypart} is found in a transform node} @# @d pair_node_size=4 {the number of words in a pair node} @d transform_node_size=12 {the number of words in a transform node} @= @!big_node_size:array[transform_type..pair_type] of small_number; @ The |big_node_size| array simply contains two constants that \MF\ occasionally needs to know. @= big_node_size[transform_type]:=transform_node_size; big_node_size[pair_type]:=pair_node_size; @ If |type(p)=pair_type| or |transform_type| and if |value(p)=null|, the procedure call |init_big_node(p)| will allocate a pair or transform node for~|p|. The individual parts of such nodes are initially of type |independent|. @p procedure init_big_node(@!p:pointer); var @!q:pointer; {the new node} @!s:small_number; {its size} begin s:=big_node_size[type(p)]; q:=get_node(s); repeat s:=s-2; @; name_type(q+s):=half(s)+x_part_sector; link(q+s):=null; until s=0; link(q):=p; value(p):=q; end; @ The |id_transform| function creates a capsule for the identity transformation. @p function id_transform:pointer; var @!p,@!q,@!r:pointer; {list manipulation registers} begin p:=get_node(value_node_size); type(p):=transform_type; name_type(p):=capsule; value(p):=null; init_big_node(p); q:=value(p); r:=q+transform_node_size; repeat r:=r-2; type(r):=known; value(r):=0; until r=q; value(xx_part_loc(q)):=unity; value(yy_part_loc(q)):=unity; id_transform:=p; end; @ Tokens are of type |tag_token| when they first appear, but they point to |null| until they are first used as the root of a variable. The following subroutine establishes the root node on such grand occasions. @p procedure new_root(@!x:pointer); var @!p:pointer; {the new node} begin p:=get_node(value_node_size); type(p):=undefined; name_type(p):=root; link(p):=x; equiv(x):=p; end; @ These conventions for variable representation are illustrated by the |print_variable_name| routine, which displays the full name of a variable given only a pointer to its two-word value packet. @p procedure print_variable_name(@!p:pointer); label found,exit; var @!q:pointer; {a token list that will name the variable's suffix} @!r:pointer; {temporary for token list creation} begin while name_type(p)>=x_part_sector do @; q:=null; while name_type(p)>saved_root do @; r:=get_avail; info(r):=link(p); link(r):=q; if name_type(p)=saved_root then print("(SAVED)"); @.SAVED@> show_token_list(r,null,el_gordo,tally); flush_token_list(r); exit:end; @ @= begin if name_type(p)=subscr then begin r:=new_num_tok(subscript(p)); repeat p:=link(p); until name_type(p)=attr; end else if name_type(p)=structured_root then begin p:=link(p); goto found; end else begin if name_type(p)<>attr then confusion("var"); @:this can't happen var}{\quad var@> r:=get_avail; info(r):=attr_loc(p); end; link(r):=q; q:=r; found: p:=parent(p); end @ @= begin case name_type(p) of x_part_sector: print_char("x"); y_part_sector: print_char("y"); xx_part_sector: print("xx"); xy_part_sector: print("xy"); yx_part_sector: print("yx"); yy_part_sector: print("yy"); capsule: begin print("%CAPSULE"); print_int(p-null); return; @.CAPSULE@> end; end; {there are no other cases} print("part "); p:=link(p-2*(name_type(p)-x_part_sector)); end @ The |interesting| function returns |true| if a given variable is not in a capsule, or if the user wants to trace capsules. @p function interesting(@!p:pointer):boolean; var @!t:small_number; {a |name_type|} begin if internal[tracing_capsules]>0 then interesting:=true else begin t:=name_type(p); if t>=x_part_sector then if t<>capsule then t:=name_type(link(p-2*(t-x_part_sector))); interesting:=(t<>capsule); end; end; @ Now here is a subroutine that converts an unstructured type into an equivalent structured type, by inserting a |structured| node that is capable of growing. This operation is done only when |name_type(p)=root|, |subscr|, or |attr|. The procedure returns a pointer to the new node that has taken node~|p|'s place in the structure. Node~|p| itself does not move, nor are its |value| or |type| fields changed in any way. @p function new_structure(@!p:pointer):pointer; var @!q,@!r:pointer; {list manipulation registers} begin case name_type(p) of root: begin q:=link(p); r:=get_node(value_node_size); equiv(q):=r; end; subscr: @; attr: @; othercases confusion("struct") @:this can't happen struct}{\quad struct@> endcases;@/ link(r):=link(p); type(r):=structured; name_type(r):=name_type(p); attr_head(r):=p; name_type(p):=structured_root;@/ q:=get_node(attr_node_size); link(p):=q; subscr_head(r):=q; parent(q):=r; type(q):=undefined; name_type(q):=attr; link(q):=end_attr; attr_loc(q):=collective_subscript; new_structure:=r; end; @ @= begin q:=p; repeat q:=link(q); until name_type(q)=attr; q:=parent(q); r:=subscr_head_loc(q); {|link(r)=subscr_head(q)|} repeat q:=r; r:=link(r); until r=p; r:=get_node(subscr_node_size); link(q):=r; subscript(r):=subscript(p); end @ If the attribute is |collective_subscript|, there are two pointers to node~|p|, so we must change both of them. @= begin q:=parent(p); r:=attr_head(q); repeat q:=r; r:=link(r); until r=p; r:=get_node(attr_node_size); link(q):=r;@/ mem[attr_loc_loc(r)]:=mem[attr_loc_loc(p)]; {copy |attr_loc| and |parent|} if attr_loc(p)=collective_subscript then begin q:=subscr_head_loc(parent(p)); while link(q)<>p do q:=link(q); link(q):=r; end; end @ The |find_variable| routine is given a pointer~|t| to a nonempty token list of suffixes; it returns a pointer to the corresponding two-word value. For example, if |t| points to token \.x followed by a numeric token containing the value~7, |find_variable| finds where the value of \.{x7} is stored in memory. This may seem a simple task, and it usually is, except when \.{x7} has never been referenced before. Indeed, \.x may never have even been subscripted before; complexities arise with respect to updating the collective subscript information. If a macro type is detected anywhere along path~|t|, or if the first item on |t| isn't a |tag_token|, the value |null| is returned. Otherwise |p| will be a non-null pointer to a node such that |undefined begin p:=info(t); t:=link(t); if eq_type(p) mod outer_tag<>tag_token then abort_find; if equiv(p)=null then new_root(p); p:=equiv(p); pp:=p; while t<>null do begin @; if t else @; t:=link(t); end; if type(pp)>=structured then if type(pp)=structured then pp:=attr_head(pp)@+else abort_find; if type(p)=structured then p:=attr_head(p); if type(p)=undefined then begin if type(pp)=undefined then begin type(pp):=numeric_type; value(pp):=null; end; type(p):=type(pp); value(p):=null; end; find_variable:=p; exit:end; @ Although |pp| and |p| begin together, they diverge when a subscript occurs; |pp|~stays in the collective line while |p|~goes through actual subscript values. @= if type(pp)<>structured then begin if type(pp)>structured then abort_find; ss:=new_structure(pp); if p=pp then p:=ss; pp:=ss; end; {now |type(pp)=structured|} if type(p)<>structured then {it cannot be |>structured|} p:=new_structure(p) {now |type(p)=structured|} @ We want this part of the program to be reasonably fast, in case there are @^inner loop@> lots of subscripts at the same level of the data structure. Therefore we store an ``infinite'' value in the word that appears at the end of the subscript list, even though that word isn't part of a subscript node. @= begin n:=value(t); pp:=link(attr_head(pp)); {now |attr_loc(pp)=collective_subscript|} q:=link(attr_head(p)); save_word:=mem[subscript_loc(q)]; subscript(q):=el_gordo; s:=subscr_head_loc(p); {|link(s)=subscr_head(p)|} repeat r:=s; s:=link(s); until n<=subscript(s); if n=subscript(s) then p:=s else begin p:=get_node(subscr_node_size); link(r):=p; link(p):=s; subscript(p):=n; name_type(p):=subscr; type(p):=undefined; end; mem[subscript_loc(q)]:=save_word; end @ @= begin n:=info(t); ss:=attr_head(pp); repeat rr:=ss; ss:=link(ss); until n<=attr_loc(ss); if n Parameter |p| points to the root information of the variable; parameter |t| points to a list of one-word nodes that represent suffixes, with |info=collective_subscript| for subscripts. @p @t\4@>@@;@/ @t\4@>@@; @t\4@>@@; @t\4@>@@; @t\4@>@@; procedure flush_variable(@!p,@!t:pointer;@!discard_suffixes:boolean); label exit; var @!q,@!r:pointer; {list manipulation} @!n:halfword; {attribute to match} begin while t<>null do begin if type(p)<>structured then return; n:=info(t); t:=link(t); if n=collective_subscript then begin r:=subscr_head_loc(p); q:=link(r); {|q=subscr_head(p)|} while name_type(q)=subscr do begin flush_variable(q,t,discard_suffixes); if t=null then if type(q)=structured then r:=q else begin link(r):=link(q); free_node(q,subscr_node_size); end else r:=q; q:=link(r); end; end; p:=attr_head(p); repeat r:=p; p:=link(p); until attr_loc(p)>=n; if attr_loc(p)<>n then return; end; if discard_suffixes then flush_below_variable(p) else begin if type(p)=structured then p:=attr_head(p); recycle_value(p); end; exit:end; @ The next procedure is simpler; it wipes out everything but |p| itself, which becomes undefined. @= procedure flush_below_variable(@!p:pointer); var @!q,@!r:pointer; {list manipulation registers} begin if type(p)<>structured then recycle_value(p) {this sets |type(p)=undefined|} else begin q:=subscr_head(p); while name_type(q)=subscr do begin flush_below_variable(q); r:=q; q:=link(q); free_node(r,subscr_node_size); end; r:=attr_head(p); q:=link(r); recycle_value(r); if name_type(p)<=saved_root then free_node(r,value_node_size) else free_node(r,subscr_node_size); {we assume that |subscr_node_size=attr_node_size|} repeat flush_below_variable(q); r:=q; q:=link(q); free_node(r,attr_node_size); until q=end_attr; type(p):=undefined; end; end; @ Just before assigning a new value to a variable, we will recycle the old value and make the old value undefined. The |und_type| routine determines what type of undefined value should be given, based on the current type before recycling. @p function und_type(@!p:pointer):small_number; begin case type(p) of undefined,vacuous:und_type:=undefined; boolean_type,unknown_boolean:und_type:=unknown_boolean; string_type,unknown_string:und_type:=unknown_string; pen_type,unknown_pen,future_pen:und_type:=unknown_pen; path_type,unknown_path:und_type:=unknown_path; picture_type,unknown_picture:und_type:=unknown_picture; transform_type,pair_type,numeric_type:und_type:=type(p); known,dependent,proto_dependent,independent:und_type:=numeric_type; end; {there are no other cases} end; @ The |clear_symbol| routine is used when we want to redefine the equivalent of a symbolic token. It must remove any variable structure or macro definition that is currently attached to that symbol. If the |saving| parameter is true, a subsidiary structure is saved instead of destroyed. @p procedure clear_symbol(@!p:pointer;@!saving:boolean); var @!q:pointer; {|equiv(p)|} begin q:=equiv(p); case eq_type(p) mod outer_tag of defined_macro,secondary_primary_macro,tertiary_secondary_macro, expression_tertiary_macro: if not saving then delete_mac_ref(q); tag_token:if q<>null then if saving then name_type(q):=saved_root else begin flush_below_variable(q); free_node(q,value_node_size); end; othercases do_nothing endcases;@/ eqtb[p]:=eqtb[frozen_undefined]; end; @* \[16] Saving and restoring equivalents. The nested structure that is provided by \&{begingroup} and \&{endgroup} allows |eqtb| entries to be saved and restored, so that temporary changes can be made without difficulty. When the user requests a current value to be saved, \MF\ puts that value into its ``save stack.'' An appearance of \&{endgroup} ultimately causes the old values to be removed from the save stack and put back in their former places. The save stack is a linked list containing three kinds of entries, distinguished by their |info| fields. If |p| points to a saved item, then \smallskip\hang |info(p)=0| stands for a group boundary; each \&{begingroup} contributes such an item to the save stack and each \&{endgroup} cuts back the stack until the most recent such entry has been removed. \smallskip\hang |info(p)=q|, where |1<=q<=hash_end|, means that |mem[p+1]| holds the former contents of |eqtb[q]|. Such save stack entries are generated by \&{save} commands or suitable \&{interim} commands. \smallskip\hang |info(p)=hash_end+q|, where |q>0|, means that |value(p)| is a |scaled| integer to be restored to internal parameter number~|q|. Such entries are generated by \&{interim} commands. \smallskip\noindent The global variable |save_ptr| points to the top item on the save stack. @d save_node_size=2 {number of words per non-boundary save-stack node} @d saved_equiv(#)==mem[#+1].hh {where an |eqtb| entry gets saved} @d save_boundary_item(#)==begin #:=get_avail; info(#):=0; link(#):=save_ptr; save_ptr:=#; end @=@!save_ptr:pointer; {the most recently saved item} @ @=save_ptr:=null; @ The |save_variable| routine is given a hash address |q|; it salts this address in the save stack, together with its current equivalent, then makes token~|q| behave as though it were brand new. Nothing is stacked when |save_ptr=null|, however; there's no way to remove things from the stack when the program is not inside a group, so there's no point in wasting the space. @p procedure save_variable(@!q:pointer); var @!p:pointer; {temporary register} begin if save_ptr<>null then begin p:=get_node(save_node_size); info(p):=q; link(p):=save_ptr; saved_equiv(p):=eqtb[q]; save_ptr:=p; end; clear_symbol(q,(save_ptr<>null)); end; @ Similarly, |save_internal| is given the location |q| of an internal quantity like |tracing_commands|. It creates a save stack entry of the third kind. @p procedure save_internal(@!q:halfword); var @!p:pointer; {new item for the save stack} begin if save_ptr<>null then begin p:=get_node(save_node_size); info(p):=hash_end+q; link(p):=save_ptr; value(p):=internal[q]; save_ptr:=p; end; end; @ At the end of a group, the |unsave| routine restores all of the saved equivalents in reverse order. This routine will be called only when there is at least one boundary item on the save stack. @p procedure unsave; var @!q:pointer; {index to saved item} @!p:pointer; {temporary register} begin while info(save_ptr)<>0 do begin q:=info(save_ptr); if q>hash_end then begin if internal[tracing_restores]>0 then begin begin_diagnostic; print_nl("{restoring "); print(int_name[q-(hash_end)]); print_char("="); print_scaled(value(save_ptr)); print_char("}"); end_diagnostic(false); end; internal[q-(hash_end)]:=value(save_ptr); end else begin if internal[tracing_restores]>0 then begin begin_diagnostic; print_nl("{restoring "); print(text(q)); print_char("}"); end_diagnostic(false); end; clear_symbol(q,false); eqtb[q]:=saved_equiv(save_ptr); if eq_type(q) mod outer_tag=tag_token then begin p:=equiv(q); if p<>null then name_type(p):=root; end; end; p:=link(save_ptr); free_node(save_ptr,save_node_size); save_ptr:=p; end; p:=link(save_ptr); free_avail(save_ptr); save_ptr:=p; end; @* \[17] Data structures for paths. When a \MF\ user specifies a path, \MF\ will create a list of knots and control points for the associated cubic spline curves. If the knots are $z_0$, $z_1$, \dots, $z_n$, there are control points $z_k^+$ and $z_{k+1}^-$ such that the cubic splines between knots $z_k$ and $z_{k+1}$ are defined by B\'ezier's formula @:Bezier}{B\'ezier, Pierre Etienne@> $$\eqalign{z(t)&=B(z_k,z_k^+,z_{k+1}^-,z_{k+1};t)\cr &=(1-t)^3z_k+3(1-t)^2tz_k^++3(1-t)t^2z_{k+1}^-+t^3z_{k+1}\cr}$$ for |0<=t<=1|. There is a 7-word node for each knot $z_k$, containing one word of control information and six words for the |x| and |y| coordinates of $z_k^-$ and $z_k$ and~$z_k^+$. The control information appears in the |left_type| and |right_type| fields, which each occupy a quarter of the first word in the node; they specify properties of the curve as it enters and leaves the knot. There's also a halfword |link| field, which points to the following knot. If the path is a closed contour, knots 0 and |n| are identical; i.e., the |link| in knot |n-1| points to knot~0. But if the path is not closed, the |left_type| of knot~0 and the |right_type| of knot~|n| are equal to |endpoint|. In the latter case the |link| in knot~|n| points to knot~0, and the control points $z_0^-$ and $z_n^+$ are not used. @d left_type(#) == mem[#].hh.b0 {characterizes the path entering this knot} @d right_type(#) == mem[#].hh.b1 {characterizes the path leaving this knot} @d endpoint=0 {|left_type| at path beginning and |right_type| at path end} @d x_coord(#) == mem[#+1].sc {the |x| coordinate of this knot} @d y_coord(#) == mem[#+2].sc {the |y| coordinate of this knot} @d left_x(#) == mem[#+3].sc {the |x| coordinate of previous control point} @d left_y(#) == mem[#+4].sc {the |y| coordinate of previous control point} @d right_x(#) == mem[#+5].sc {the |x| coordinate of next control point} @d right_y(#) == mem[#+6].sc {the |y| coordinate of next control point} @d knot_node_size=7 {number of words in a knot node} @ Before the B\'ezier control points have been calculated, the memory space they will ultimately occupy is taken up by information that can be used to compute them. There are four cases: \yskip \textindent{$\bullet$} If |right_type=open|, the curve should leave the knot in the same direction it entered; \MF\ will figure out a suitable direction. \yskip \textindent{$\bullet$} If |right_type=curl|, the curve should leave the knot in a direction depending on the angle at which it enters the next knot and on the curl parameter stored in |right_curl|. \yskip \textindent{$\bullet$} If |right_type=given|, the curve should leave the knot in a nonzero direction stored as an |angle| in |right_given|. \yskip \textindent{$\bullet$} If |right_type=explicit|, the B\'ezier control point for leaving this knot has already been computed; it is in the |right_x| and |right_y| fields. \yskip\noindent The rules for |left_type| are similar, but they refer to the curve entering the knot, and to \\{left} fields instead of \\{right} fields. Non-|explicit| control points will be chosen based on ``tension'' parameters stored in the |left_tension| and |right_tension| fields. The `\&{atleast}' option is represented by negative tension values. @!@:at_least_}{\&{atleast} primitive@> For example, the \MF\ path specification $$\.{z0..z1..tension atleast 1..\{curl 2\}z2..z3\{-1,-2\}..tension 3 and 4..p},$$ where \.p is the path `\.{z4..controls z45 and z54..z5}', will be represented by the six knots \def\lodash{\hbox to 1.1em{\thinspace\hrulefill\thinspace}} $$\vbox{\halign{#\hfil&&\qquad#\hfil\cr |left_type|&\\{left} info&|x_coord,y_coord|&|right_type|&\\{right} info\cr \noalign{\yskip} |endpoint|&\lodash$,\,$\lodash&$x_0,y_0$&|curl|&$1.0,1.0$\cr |open|&\lodash$,1.0$&$x_1,y_1$&|open|&\lodash$,-1.0$\cr |curl|&$2.0,-1.0$&$x_2,y_2$&|curl|&$2.0,1.0$\cr |given|&$d,1.0$&$x_3,y_3$&|given|&$d,3.0$\cr |open|&\lodash$,4.0$&$x_4,y_4$&|explicit|&$x_{45},y_{45}$\cr |explicit|&$x_{54},y_{54}$&$x_5,y_5$&|endpoint|&\lodash$,\,$\lodash\cr}}$$ Here |d| is the |angle| obtained by calling |n_arg(-unity,-two)|. Of course, this example is more complicated than anything a normal user would ever write. These types must satisfy certain restrictions because of the form of \MF's path syntax: (i)~|open| type never appears in the same node together with |endpoint|, |given|, or |curl|. (ii)~The |right_type| of a node is |explicit| if and only if the |left_type| of the following node is |explicit|. (iii)~|endpoint| types occur only at the ends, as mentioned above. @d left_curl==left_x {curl information when entering this knot} @d left_given==left_x {given direction when entering this knot} @d left_tension==left_y {tension information when entering this knot} @d right_curl==right_x {curl information when leaving this knot} @d right_given==right_x {given direction when leaving this knot} @d right_tension==right_y {tension information when leaving this knot} @d explicit=1 {|left_type| or |right_type| when control points are known} @d given=2 {|left_type| or |right_type| when a direction is given} @d curl=3 {|left_type| or |right_type| when a curl is desired} @d open=4 {|left_type| or |right_type| when \MF\ should choose the direction} @ Here is a diagnostic routine that prints a given knot list in symbolic form. It illustrates the conventions discussed above, and checks for anomalies that might arise while \MF\ is being debugged. @= procedure print_path(@!h:pointer;@!s:str_number;@!nuline:boolean); label done,done1; var @!p,@!q:pointer; {for list traversal} begin print_diagnostic("Path",s,nuline); print_ln; @.Path at line...@> p:=h; repeat q:=link(p); if (p=null)or(q=null) then begin print_nl("???"); goto done; {this won't happen} @.???@> end; @; p:=q; if (p<>h)or(left_type(h)<>endpoint) then @; until p=h; if left_type(h)<>endpoint then print("cycle"); done:end_diagnostic(true); end; @ @= print_two(x_coord(p),y_coord(p)); case right_type(p) of endpoint: begin if left_type(p)=open then print("{open?}"); {can't happen} @.open?@> if (left_type(q)<>endpoint)or(q<>h) then q:=null; {force an error} goto done1; end; explicit: @; open: @; curl,given: @; othercases print("???") {can't happen} @.???@> endcases;@/ if left_type(q)<=explicit then print("..control?") {can't happen} @.control?@> else if (right_tension(p)<>unity)or(left_tension(q)<>unity) then @; done1: @ Since |n_sin_cos| produces |fraction| results, which we will print as if they were |scaled|, the magnitude of a |given| direction vector will be~4096. @= begin print_nl(" .."); if left_type(p)=given then begin n_sin_cos(left_given(p)); print_char("{"); print_scaled(n_cos); print_char(","); print_scaled(n_sin); print_char("}"); end else if left_type(p)=curl then begin print("{curl "); print_scaled(left_curl(p)); print_char("}"); end; end @ @= begin print("..tension "); if right_tension(p)<0 then print("atleast"); print_scaled(abs(right_tension(p))); if right_tension(p)<>left_tension(q) then begin print(" and "); if left_tension(q)<0 then print("atleast"); print_scaled(abs(left_tension(q))); end; end @ @= begin print("..controls "); print_two(right_x(p),right_y(p)); print(" and "); if left_type(q)<>explicit then print("??") {can't happen} @.??@> else print_two(left_x(q),left_y(q)); goto done1; end @ @= if (left_type(p)<>explicit)and(left_type(p)<>open) then print("{open?}") {can't happen} @.open?@> @ A curl of 1 is shown explicitly, so that the user sees clearly that \MF's default curl is present. The code here uses the fact that |left_curl==left_given| and |right_curl==right_given|. @= begin if left_type(p)=open then print("??"); {can't happen} @.??@> if right_type(p)=curl then begin print("{curl "); print_scaled(right_curl(p)); end else begin n_sin_cos(right_given(p)); print_char("{"); print_scaled(n_cos); print_char(","); print_scaled(n_sin); end; print_char("}"); end @ If we want to duplicate a knot node, we can say |copy_knot|: @p function copy_knot(@!p:pointer):pointer; var @!q:pointer; {the copy} @!k:0..knot_node_size-1; {runs through the words of a knot node} begin q:=get_node(knot_node_size); for k:=0 to knot_node_size-1 do mem[q+k]:=mem[p+k]; copy_knot:=q; end; @ The |copy_path| routine makes a clone of a given path. @p function copy_path(@!p:pointer):pointer; label exit; var @!q,@!pp,@!qq:pointer; {for list manipulation} begin q:=get_node(knot_node_size); {this will correspond to |p|} qq:=q; pp:=p; loop@+ begin left_type(qq):=left_type(pp); right_type(qq):=right_type(pp);@/ x_coord(qq):=x_coord(pp); y_coord(qq):=y_coord(pp);@/ left_x(qq):=left_x(pp); left_y(qq):=left_y(pp);@/ right_x(qq):=right_x(pp); right_y(qq):=right_y(pp);@/ if link(pp)=p then begin link(qq):=q; copy_path:=q; return; end; link(qq):=get_node(knot_node_size); qq:=link(qq); pp:=link(pp); end; exit:end; @ Similarly, there's a way to copy the {\sl reverse\/} of a path. This procedure returns a pointer to the first node of the copy, if the path is a cycle, but to the final node of a non-cyclic copy. The global variable |path_tail| will point to the final node of the original path; this trick makes it easier to implement `\&{doublepath}'. All node types are assumed to be |endpoint| or |explicit| only. @p function htap_ypoc(@!p:pointer):pointer; label exit; var @!q,@!pp,@!qq,@!rr:pointer; {for list manipulation} begin q:=get_node(knot_node_size); {this will correspond to |p|} qq:=q; pp:=p; loop@+ begin right_type(qq):=left_type(pp); left_type(qq):=right_type(pp);@/ x_coord(qq):=x_coord(pp); y_coord(qq):=y_coord(pp);@/ right_x(qq):=left_x(pp); right_y(qq):=left_y(pp);@/ left_x(qq):=right_x(pp); left_y(qq):=right_y(pp);@/ if link(pp)=p then begin link(q):=qq; path_tail:=pp; htap_ypoc:=q; return; end; rr:=get_node(knot_node_size); link(rr):=qq; qq:=rr; pp:=link(pp); end; exit:end; @ @= @!path_tail:pointer; {the node that links to the beginning of a path} @ When a cyclic list of knot nodes is no longer needed, it can be recycled by calling the following subroutine. @= procedure toss_knot_list(@!p:pointer); var @!q:pointer; {the node being freed} @!r:pointer; {the next node} begin q:=p; repeat r:=link(q); free_node(q,knot_node_size); q:=r; until q=p; end; @* \[18] Choosing control points. Now we must actually delve into one of \MF's more difficult routines, the |make_choices| procedure that chooses angles and control points for the splines of a curve when the user has not specified them explicitly. The parameter to |make_choices| points to a list of knots and path information, as described above. A path decomposes into independent segments at ``breakpoint'' knots, which are knots whose left and right angles are both prespecified in some way (i.e., their |left_type| and |right_type| aren't both open). @p @t\4@>@@; procedure make_choices(@!knots:pointer); label done; var @!h:pointer; {the first breakpoint} @!p,@!q:pointer; {consecutive breakpoints being processed} @@; begin check_arith; {make sure that |arith_error=false|} if internal[tracing_choices]>0 then print_path(knots,", before choices",true); @; @; p:=h; repeat @; until p=h; if internal[tracing_choices]>0 then print_path(knots,", after choices",true); if arith_error then @; end; @ @= begin print_err("Some number got too big"); @.Some number got too big@> help2("The path that I just computed is out of range.")@/ ("So it will probably look funny. Proceed, for a laugh."); put_get_error; arith_error:=false; end @ Two knots in a row with the same coordinates will always be joined by an explicit ``curve'' whose control points are identical with the knots. @= p:=knots; repeat q:=link(p); if x_coord(p)=x_coord(q) then if y_coord(p)=y_coord(q) then if right_type(p)>explicit then begin right_type(p):=explicit; if left_type(p)=open then begin left_type(p):=curl; left_curl(p):=unity; end; left_type(q):=explicit; if right_type(q)=open then begin right_type(q):=curl; right_curl(q):=unity; end; right_x(p):=x_coord(p); left_x(q):=x_coord(p);@/ right_y(p):=y_coord(p); left_y(q):=y_coord(p); end; p:=q; until p=knots @ If there are no breakpoints, it is necessary to compute the direction angles around an entire cycle. In this case the |left_type| of the first node is temporarily changed to |end_cycle|. @d end_cycle=open+1 @= h:=knots; loop@+ begin if left_type(h)<>open then goto done; if right_type(h)<>open then goto done; h:=link(h); if h=knots then begin left_type(h):=end_cycle; goto done; end; end; done: @ If |right_type(p)= q:=link(p); if right_type(p)>=given then begin while (left_type(q)=open)and(right_type(q)=open) do q:=link(q); @; end; p:=q @ Before we can go further into the way choices are made, we need to consider the underlying theory. The basic ideas implemented in |make_choices| are due to John Hobby, who introduced the notion of ``mock curvature'' @^Hobby, John Douglas@> at a knot. Angles are chosen so that they preserve mock curvature when a knot is passed, and this has been found to produce excellent results. It is convenient to introduce some notations that simplify the necessary formulas. Let $d_{k,k+1}=\vert z\k-z_k\vert$ be the (nonzero) distance between knots |k| and |k+1|; and let $${z\k-z_k\over z_k-z_{k-1}}={d_{k,k+1}\over d_{k-1,k}}e^{i\psi_k}$$ so that a polygonal line from $z_{k-1}$ to $z_k$ to $z\k$ turns left through an angle of~$\psi_k$. We assume that $\vert\psi_k\vert\L180^\circ$. The control points for the spline from $z_k$ to $z\k$ will be denoted by $$\eqalign{z_k^+&=z_k+ \textstyle{1\over3}\rho_k e^{i\theta_k}(z\k-z_k),\cr z\k^-&=z\k- \textstyle{1\over3}\sigma\k e^{-i\phi\k}(z\k-z_k),\cr}$$ where $\rho_k$ and $\sigma\k$ are nonnegative ``velocity ratios'' at the beginning and end of the curve, while $\theta_k$ and $\phi\k$ are the corresponding ``offset angles.'' These angles satisfy the condition $$\theta_k+\phi_k+\psi_k=0,\eqno(*)$$ whenever the curve leaves an intermediate knot~|k| in the direction that it enters. @ Let $\alpha_k$ and $\beta\k$ be the reciprocals of the ``tension'' of the curve at its beginning and ending points. This means that $\rho_k=\alpha_k f(\theta_k,\phi\k)$ and $\sigma\k=\beta\k f(\phi\k,\theta_k)$, where $f(\theta,\phi)$ is \MF's standard velocity function defined in the |velocity| subroutine. The cubic spline $B(z_k^{\phantom+},z_k^+, z\k^-,z\k^{\phantom+};t)$ has curvature @^curvature@> $${2\sigma\k\sin(\theta_k+\phi\k)-6\sin\theta_k\over\rho_k^2d_{k,k+1}} \qquad{\rm and}\qquad {2\rho_k\sin(\theta_k+\phi\k)-6\sin\phi\k\over\sigma\k^2d_{k,k+1}}$$ at |t=0| and |t=1|, respectively. The mock curvature is the linear @^mock curvature@> approximation to this true curvature that arises in the limit for small $\theta_k$ and~$\phi\k$, if second-order terms are discarded. The standard velocity function satisfies $$f(\theta,\phi)=1+O(\theta^2+\theta\phi+\phi^2);$$ hence the mock curvatures are respectively $${2\beta\k(\theta_k+\phi\k)-6\theta_k\over\alpha_k^2d_{k,k+1}} \qquad{\rm and}\qquad {2\alpha_k(\theta_k+\phi\k)-6\phi\k\over\beta\k^2d_{k,k+1}}.\eqno(**)$$ @ The turning angles $\psi_k$ are given, and equation $(*)$ above determines $\phi_k$ when $\theta_k$ is known, so the task of angle selection is essentially to choose appropriate values for each $\theta_k$. When equation~$(*)$ is used to eliminate $\phi$~variables from $(**)$, we obtain a system of linear equations of the form $$A_k\theta_{k-1}+(B_k+C_k)\theta_k+D_k\theta\k=-B_k\psi_k-D_k\psi\k,$$ where $$A_k={\alpha_{k-1}\over\beta_k^2d_{k-1,k}}, \qquad B_k={3-\alpha_{k-1}\over\beta_k^2d_{k-1,k}}, \qquad C_k={3-\beta\k\over\alpha_k^2d_{k,k+1}}, \qquad D_k={\beta\k\over\alpha_k^2d_{k,k+1}}.$$ The tensions are always $3\over4$ or more, hence each $\alpha$ and~$\beta$ will be at most $4\over3$. It follows that $B_k\G{5\over4}A_k$ and $C_k\G{5\over4}D_k$; hence the equations are diagonally dominant; hence they have a unique solution. Moreover, in most cases the tensions are equal to~1, so that $B_k=2A_k$ and $C_k=2D_k$. This makes the solution numerically stable, and there is an exponential damping effect: The data at knot $k\pm j$ affects the angle at knot~$k$ by a factor of~$O(2^{-j})$. @ However, we still must consider the angles at the starting and ending knots of a non-cyclic path. These angles might be given explicitly, or they might be specified implicitly in terms of an amount of ``curl.'' Let's assume that angles need to be determined for a non-cyclic path starting at $z_0$ and ending at~$z_n$. Then equations of the form $$A_k\theta_{k-1}+(B_k+C_k)\theta_k+D_k\theta_{k+1}=R_k$$ have been given for $00$ and $C_0B_1-A_1D_0>0$ when $\gamma_0\G0$, hence the linear equations remain nonsingular. Similar considerations apply at the right end, when the final angle $\phi_n$ may or may not need to be determined. It is convenient to let $\psi_n=0$, hence $\theta_n=-\phi_n$. We either have an explicit equation $\theta_n=E_n$, or we have $$\bigl((3-\beta_n)\chi_n+\alpha_{n-1}\bigr)\theta_{n-1}+ (\beta_n\chi_n+3-\alpha_{n-1})\theta_n=0,\qquad \chi_n={\beta_n^2\gamma_n\over\alpha_{n-1}^2}.$$ When |make_choices| chooses angles, it must compute the coefficients of these linear equations, then solve the equations. To compute the coefficients, it is necessary to compute arctangents of the given turning angles~$\psi_k$. When the equations are solved, the chosen directions $\theta_k$ are put back into the form of control points by essentially computing sines and cosines. @ OK, we are ready to make the hard choices of |make_choices|. Most of the work is relegated to an auxiliary procedure called |solve_choices|, which has been introduced to keep |make_choices| from being extremely long. @= @; @; solve_choices(p,q,n) @ It's convenient to precompute quantities that will be needed several times later. The values of |delta_x[k]| and |delta_y[k]| will be the coordinates of $z\k-z_k$, and the magnitude of this vector will be |delta[k]=@t$d_{k,k+1}$@>|. The path angle $\psi_k$ between $z_k-z_{k-1}$ and $z\k-z_k$ will be stored in |psi[k]|. @= @!delta_x,@!delta_y,@!delta:array[0..path_size] of scaled; {knot differences} @!psi:array[1..path_size] of angle; {turning angles} @ @= @!k,@!n:0..path_size; {current and final knot numbers} @!s,@!t:pointer; {registers for list traversal} @!delx,@!dely:scaled; {directions where |open| meets |explicit|} @!sine,@!cosine:fraction; {trig functions of various angles} @ @= k:=0; s:=p; n:=path_size; repeat t:=link(s); delta_x[k]:=x_coord(t)-x_coord(s); delta_y[k]:=y_coord(t)-y_coord(s); delta[k]:=pyth_add(delta_x[k],delta_y[k]); if k>0 then begin sine:=make_fraction(delta_y[k-1],delta[k-1]); cosine:=make_fraction(delta_x[k-1],delta[k-1]); psi[k]:=n_arg(take_fraction(delta_x[k],cosine)+ take_fraction(delta_y[k],sine), take_fraction(delta_y[k],cosine)- take_fraction(delta_x[k],sine)); end; @:METAFONT capacity exceeded path size}{\quad path size@> incr(k); s:=t; if k=path_size then overflow("path size",path_size); if s=q then n:=k; until (k>=n)and(left_type(s)<>end_cycle); if k=n then psi[n]:=0@+else psi[k]:=psi[1] @ When we get to this point of the code, |right_type(p)| is either |given| or |curl| or |open|. If it is |open|, we must have |left_type(p)=end_cycle| or |left_type(p)=explicit|. In the latter case, the |open| type is converted to |given|; however, if the velocity coming into this knot is zero, the |open| type is converted to a |curl|, since we don't know the incoming direction. Similarly, |left_type(q)| is either |given| or |curl| or |open| or |end_cycle|. The |open| possibility is reduced either to |given| or to |curl|. @= if left_type(q)=open then begin delx:=right_x(q)-x_coord(q); dely:=right_y(q)-y_coord(q); if (delx=0)and(dely=0) then begin left_type(q):=curl; left_curl(q):=unity; end else begin left_type(q):=given; left_given(q):=n_arg(delx,dely); end; end; if (right_type(p)=open)and(left_type(p)=explicit) then begin delx:=x_coord(p)-left_x(p); dely:=y_coord(p)-left_y(p); if (delx=0)and(dely=0) then begin right_type(p):=curl; right_curl(p):=unity; end else begin right_type(p):=given; right_given(p):=n_arg(delx,dely); end; end @ Linear equations need to be solved whenever |n>1|; and also when |n=1| and exactly one of the breakpoints involves a curl. The simplest case occurs when |n=1| and there is a curl at both breakpoints; then we simply draw a straight line. But before coding up the simple cases, we might as well face the general case, since we must deal with it sooner or later, and since the general case is likely to give some insight into the way simple cases can be handled best. When there is no cycle, the linear equations to be solved form a tri-diagonal system, and we can apply the standard technique of Gaussian elimination to convert that system to a sequence of equations of the form $$\theta_0+u_0\theta_1=v_0,\quad \theta_1+u_1\theta_2=v_1,\quad\ldots,\quad \theta_{n-1}+u_{n-1}\theta_n=v_{n-1},\quad \theta_n=v_n.$$ It is possible to do this diagonalization while generating the equations. Once $\theta_n$ is known, it is easy to determine $\theta_{n-1}$, \dots, $\theta_1$, $\theta_0$; thus, the equations will be solved. The procedure is slightly more complex when there is a cycle, but the basic idea will be nearly the same. In the cyclic case the right-hand sides will be $v_k+w_k\theta_0$ instead of simply $v_k$, and we will start the process off with $u_0=v_0=0$, $w_0=1$. The final equation will be not $\theta_n=v_n$ but $\theta_n+u_n\theta_1=v_n+w_n\theta_0$; an appropriate ending routine will take account of the fact that $\theta_n=\theta_0$ and eliminate the $w$'s from the system, after which the solution can be obtained as before. When $u_k$, $v_k$, and $w_k$ are being computed, the three pointer variables |r|, |s|,~|t| will point respectively to knots |k-1|, |k|, and~|k+1|. The $u$'s and $w$'s are scaled by $2^{28}$, i.e., they are of type |fraction|; the $\theta$'s and $v$'s are of type |angle|. @= @!theta:array[0..path_size] of angle; {values of $\theta_k$} @!uu:array[0..path_size] of fraction; {values of $u_k$} @!vv:array[0..path_size] of angle; {values of $v_k$} @!ww:array[0..path_size] of fraction; {values of $w_k$} @ Our immediate problem is to get the ball rolling by setting up the first equation or by realizing that no equations are needed, and to fit this initialization into a framework suitable for the overall computation. @= @t\4@>@@; procedure solve_choices(@!p,@!q:pointer;@!n:halfword); label found,exit; var @!k:0..path_size; {current knot number} @!r,@!s,@!t:pointer; {registers for list traversal} @@; begin k:=0; s:=p; loop@+ begin t:=link(s); if k=0 then @ else case left_type(s) of end_cycle,open:@; curl:@; given:@; end; {there are no other cases} r:=s; s:=t; incr(k); end; found:@; exit:end; @ On the first time through the loop, we have |k=0| and |r| is not yet defined. The first linear equation, if any, will have $A_0=B_0=0$. @= case right_type(s) of given: if left_type(t)=given then @ else @; curl: if left_type(t)=curl then @ else @; open: begin uu[0]:=0; vv[0]:=0; ww[0]:=fraction_one; end; {this begins a cycle} end {there are no other cases} @ The general equation that specifies equality of mock curvature at $z_k$ is $$A_k\theta_{k-1}+(B_k+C_k)\theta_k+D_k\theta\k=-B_k\psi_k-D_k\psi\k,$$ as derived above. We want to combine this with the already-derived equation $\theta_{k-1}+u_{k-1}\theta_k=v_{k-1}+w_{k-1}\theta_0$ in order to obtain a new equation $\theta_k+u_k\theta\k=v_k+w_k\theta_0$. This can be done by dividing the equation $$(B_k-u_{k-1}A_k+C_k)\theta_k+D_k\theta\k=-B_k\psi_k-D_k\psi\k-A_kv_{k-1} -A_kw_{k-1}\theta_0$$ by $B_k-u_{k-1}A_k+C_k$. The trick is to do this carefully with fixed-point arithmetic, avoiding the chance of overflow while retaining suitable precision. The calculations will be performed in several registers that provide temporary storage for intermediate quantities. @= @!aa,@!bb,@!cc,@!ff,@!acc:fraction; {temporary registers} @!dd,@!ee:scaled; {likewise, but |scaled|} @!lt,@!rt:scaled; {tension values} @ @= begin @; @; uu[k]:=take_fraction(ff,bb); @; if left_type(s)=end_cycle then @; end @ Since tension values are never less than 3/4, the values |aa| and |bb| computed here are never more than 4/5. @= if abs(right_tension(r))=unity then begin aa:=fraction_half; dd:=2*delta[k]; end else begin aa:=make_fraction(unity,3*abs(right_tension(r))-unity); dd:=take_fraction(delta[k], fraction_three-make_fraction(unity,abs(right_tension(r)))); end; if abs(left_tension(t))=unity then begin bb:=fraction_half; ee:=2*delta[k-1]; end else begin bb:=make_fraction(unity,3*abs(left_tension(t))-unity); ee:=take_fraction(delta[k-1], fraction_three-make_fraction(unity,abs(left_tension(t)))); end; cc:=fraction_one-take_fraction(uu[k-1],aa) @ The ratio to be calculated in this step can be written in the form $$\beta_k^2\cdot\\{ee}\over\beta_k^2\cdot\\{ee}+\alpha_k^2\cdot \\{cc}\cdot\\{dd},$$ because of the quantities just calculated. The values of |dd| and |ee| will not be needed after this step has been performed. @= dd:=take_fraction(dd,cc); lt:=abs(left_tension(s)); rt:=abs(right_tension(s)); if lt<>rt then {$\beta_k^{-1}<>\alpha_k^{-1}$} if lt= acc:=-take_fraction(psi[k+1],uu[k]); if right_type(r)=curl then begin ww[k]:=0; vv[k]:=acc-take_fraction(psi[1],fraction_one-ff); end else begin ff:=make_fraction(fraction_one-ff,cc); {this is $B_k/(C_k+B_k-u_{k-1}A_k)<5$} acc:=acc-take_fraction(psi[k],ff); ff:=take_fraction(ff,aa); {this is $A_k/(C_k+B_k-u_{k-1}A_k)$} vv[k]:=acc-take_fraction(vv[k-1],ff); if ww[k-1]=0 then ww[k]:=0 else ww[k]:=-take_fraction(ww[k-1],ff); end @ When a complete cycle has been traversed, we have $\theta_k+u_k\theta\k= v_k+w_k\theta_0$, for |1<=k<=n|. We would like to determine the value of $\theta_n$ and reduce the system to the form $\theta_k+u_k\theta\k=v_k$ for |0<=k= begin aa:=0; bb:=fraction_one; {we have |k=n|} repeat decr(k); if k=0 then k:=n; aa:=vv[k]-take_fraction(aa,uu[k]); bb:=ww[k]-take_fraction(bb,uu[k]); until k=n; {now $\theta_n=\\{aa}+\\{bb}\cdot\theta_n$} aa:=make_fraction(aa,fraction_one-bb); theta[n]:=aa; vv[0]:=aa; for k:=1 to n-1 do vv[k]:=vv[k]+take_fraction(aa,ww[k]); goto found; end @ @d reduce_angle(#)==if abs(#)>one_eighty_deg then if #>0 then #:=#-three_sixty_deg@+else #:=#+three_sixty_deg @= begin theta[n]:=left_given(s)-n_arg(delta_x[n-1],delta_y[n-1]); reduce_angle(theta[n]); goto found; end @ @= begin vv[0]:=right_given(s)-n_arg(delta_x[0],delta_y[0]); reduce_angle(vv[0]); uu[0]:=0; ww[0]:=0; end @ @= begin cc:=right_curl(s); lt:=abs(left_tension(t)); rt:=abs(right_tension(s)); if (rt=unity)and(lt=unity) then uu[0]:=make_fraction(cc+cc+unity,cc+two) else uu[0]:=curl_ratio(cc,rt,lt); vv[0]:=-take_fraction(psi[1],uu[0]); ww[0]:=0; end @ @= begin cc:=left_curl(s); lt:=abs(left_tension(s)); rt:=abs(right_tension(r)); if (rt=unity)and(lt=unity) then ff:=make_fraction(cc+cc+unity,cc+two) else ff:=curl_ratio(cc,lt,rt); theta[n]:=-make_fraction(take_fraction(vv[n-1],ff), fraction_one-take_fraction(ff,uu[n-1])); goto found; end @ The |curl_ratio| subroutine has three arguments, which our previous notation encourages us to call $\gamma$, $\alpha^{-1}$, and $\beta^{-1}$. It is a somewhat tedious program to calculate $${(3-\alpha)\alpha^2\gamma+\beta^3\over \alpha^3\gamma+(3-\beta)\beta^2},$$ with the result reduced to 4 if it exceeds 4. (This reduction of curl is necessary only if the curl and tension are both large.) The values of $\alpha$ and $\beta$ will be at most~4/3. @= function curl_ratio(@!gamma,@!a_tension,@!b_tension:scaled):fraction; var @!alpha,@!beta,@!num,@!denom,@!ff:fraction; {registers} begin alpha:=make_fraction(unity,a_tension); beta:=make_fraction(unity,b_tension);@/ if alpha<=beta then begin ff:=make_fraction(alpha,beta); ff:=take_fraction(ff,ff); gamma:=take_fraction(gamma,ff);@/ beta:=beta div @'10000; {convert |fraction| to |scaled|} denom:=take_fraction(gamma,alpha)+three-beta; num:=take_fraction(gamma,fraction_three-alpha)+beta; end else begin ff:=make_fraction(beta,alpha); ff:=take_fraction(ff,ff); beta:=take_fraction(beta,ff) div @'10000; {convert |fraction| to |scaled|} denom:=take_fraction(gamma,alpha)+(ff div 1365)-beta; {$1365\approx 2^{12}/3$} num:=take_fraction(gamma,fraction_three-alpha)+beta; end; if num>=denom+denom+denom+denom then curl_ratio:=fraction_four else curl_ratio:=make_fraction(num,denom); end; @ We're in the home stretch now. @= for k:=n-1 downto 0 do theta[k]:=vv[k]-take_fraction(theta[k+1],uu[k]); s:=p; k:=0; repeat t:=link(s);@/ n_sin_cos(theta[k]); st:=n_sin; ct:=n_cos;@/ n_sin_cos(-psi[k+1]-theta[k+1]); sf:=n_sin; cf:=n_cos;@/ set_controls(s,t,k);@/ incr(k); s:=t; until k=n @ The |set_controls| routine actually puts the control points into a pair of consecutive nodes |p| and~|q|. Global variables are used to record the values of $\sin\theta$, $\cos\theta$, $\sin\phi$, and $\cos\phi$ needed in this calculation. @= @!st,@!ct,@!sf,@!cf:fraction; {sines and cosines} @ @= procedure set_controls(@!p,@!q:pointer;@!k:integer); var @!rr,@!ss:fraction; {velocities, divided by thrice the tension} @!lt,@!rt:scaled; {tensions} @!sine:fraction; {$\sin(\theta+\phi)$} begin lt:=abs(left_tension(q)); rt:=abs(right_tension(p)); rr:=velocity(st,ct,sf,cf,rt); ss:=velocity(sf,cf,st,ct,lt); if (right_tension(p)<0)or(left_tension(q)<0) then @; right_x(p):=x_coord(p)+take_fraction( take_fraction(delta_x[k],ct)-take_fraction(delta_y[k],st),rr); right_y(p):=y_coord(p)+take_fraction( take_fraction(delta_y[k],ct)+take_fraction(delta_x[k],st),rr); left_x(q):=x_coord(q)-take_fraction( take_fraction(delta_x[k],cf)+take_fraction(delta_y[k],sf),ss); left_y(q):=y_coord(q)-take_fraction( take_fraction(delta_y[k],cf)-take_fraction(delta_x[k],sf),ss); right_type(p):=explicit; left_type(q):=explicit; end; @ The boundedness conditions $\\{rr}\L\sin\phi\,/\sin(\theta+\phi)$ and $\\{ss}\L\sin\theta\,/\sin(\theta+\phi)$ are to be enforced if $\sin\theta$, $\sin\phi$, and $\sin(\theta+\phi)$ all have the same sign. Otherwise there is no ``bounding triangle.'' @!@:at_least_}{\&{atleast} primitive@> @= if((st>=0)and(sf>=0))or((st<=0)and(sf<=0)) then begin sine:=take_fraction(abs(st),cf)+take_fraction(abs(sf),ct); if sine>0 then begin sine:=take_fraction(sine,fraction_one+unity); {safety factor} if right_tension(p)<0 then if ab_vs_cd(abs(sf),fraction_one,rr,sine)<0 then rr:=make_fraction(abs(sf),sine); if left_tension(q)<0 then if ab_vs_cd(abs(st),fraction_one,ss,sine)<0 then ss:=make_fraction(abs(st),sine); end; end @ Only the simple cases remain to be handled. @= begin aa:=n_arg(delta_x[0],delta_y[0]);@/ n_sin_cos(right_given(p)-aa); ct:=n_cos; st:=n_sin;@/ n_sin_cos(left_given(q)-aa); cf:=n_cos; sf:=-n_sin;@/ set_controls(p,q,0); return; end @ @= begin right_type(p):=explicit; left_type(q):=explicit; lt:=abs(left_tension(q)); rt:=abs(right_tension(p)); if rt=unity then begin if delta_x[0]>=0 then right_x(p):=x_coord(p)+((delta_x[0]+1) div 3) else right_x(p):=x_coord(p)+((delta_x[0]-1) div 3); if delta_y[0]>=0 then right_y(p):=y_coord(p)+((delta_y[0]+1) div 3) else right_y(p):=y_coord(p)+((delta_y[0]-1) div 3); end else begin ff:=make_fraction(unity,3*rt); {$\alpha/3$} right_x(p):=x_coord(p)+take_fraction(delta_x[0],ff); right_y(p):=y_coord(p)+take_fraction(delta_y[0],ff); end; if lt=unity then begin if delta_x[0]>=0 then left_x(q):=x_coord(q)-((delta_x[0]+1) div 3) else left_x(q):=x_coord(q)-((delta_x[0]-1) div 3); if delta_y[0]>=0 then left_y(q):=y_coord(q)-((delta_y[0]+1) div 3) else left_y(q):=y_coord(q)-((delta_y[0]-1) div 3); end else begin ff:=make_fraction(unity,3*lt); {$\beta/3$} left_x(q):=x_coord(q)-take_fraction(delta_x[0],ff); left_y(q):=y_coord(q)-take_fraction(delta_y[0],ff); end; return; end @* \[19] Generating discrete moves. The purpose of the next part of \MF\ is to compute discrete approximations to curves described as parametric polynomial functions $z(t)$. We shall start with the low level first, because an efficient ``engine'' is needed to support the high-level constructions. Most of the subroutines are based on variations of a single theme, namely the idea of {\sl bisection}. Given a Bernshte{\u\i}n polynomial @^Bernshte{\u\i}n, Serge{\u\i} Natanovich@> $$B(z_0,z_1,\ldots,z_n;t)=\sum_k{n\choose k}t^k(1-t)^{n-k}z_k,$$ we can conveniently bisect its range as follows: \smallskip \textindent{1)} Let $z_k^{(0)}=z_k$, for |0<=k<=n|. \smallskip \textindent{2)} Let $z_k^{(j+1)}={1\over2}(z_k^{(j)}+z\k^{(j)})$, for |0<=k @^Riesenfeld, Richard Franklin@> and Machine Intelligence\/ \bf PAMI-2} (1980), 35--46]: Assuming that $\vert Z\k-Z_k\vert\L M$ before bisection, we want to prove that $\vert Z\k-Z_k\vert\L\lceil M/2\rceil$ afterward. First we show that $\vert Z\k^{(j)}-Z_k^{(j)}\vert\L M$ for all $j$ and~$k$, by induction on~$j$; this follows from the fact that $$\bigl\vert\\{half}(a+b)-\\{half}(b+c)\bigr\vert\L \max\bigl(\vert a-b\vert,\vert b-c\vert\bigr)$$ holds for both of the rounding rules $\\{half}(x)=\lfloor x/2\rfloor$ and $\\{half}(x)={\rm sign}(x)\lfloor\vert x/2\vert\rfloor$. (If $\vert a-b\vert$ and $\vert b-c\vert$ are equal, then $a+b$ and $b+c$ are both even or both odd. The rounding errors either cancel or round the numbers toward each other; hence $$\eqalign{\bigl\vert\\{half}(a+b)-\\{half}(b+c)\bigr\vert &\L\textstyle\bigl\vert{1\over2}(a+b)-{1\over2}(b+c)\bigr\vert\cr &=\textstyle\bigl\vert{1\over2}(a-b)+{1\over2}(b-c)\bigr\vert \L\max\bigl(\vert a-b\vert,\vert b-c\vert\bigr),\cr}$$ as required. A simpler argument applies if $\vert a-b\vert$ and $\vert b-c\vert$ are unequal.) Now it is easy to see that $\vert Z_1^{(j+1)}-Z_1^{(j)}\vert\L\bigl\lfloor{1\over2} \vert Z_2^{(j)}-Z_1^{(j)}\vert+{1\over2}\bigr\rfloor \L\bigl\lfloor{1\over2}(M+1)\bigr\rfloor=\lceil M/2\rceil$. Another interesting fact about bisection is the identity $$Z_1'+\cdots+Z_n'+Z_1''+\cdots+Z_n''=2(Z_1+\cdots+Z_n+E),$$ where $E$ is the sum of the rounding errors in all of the halving operations ($\vert E\vert\L n(n-1)/4$). @ We will later reduce the problem of digitizing a complex cubic $z(t)=B(z_0,z_1,z_2,z_3;t)$ to the following simpler problem: Given two real cubics $x(t)=B(x_0,x_1,x_2,x_3;t)$ and $y(t)=B(y_0,y_1,y_2,y_3;t)$ that are monotone nondecreasing, determine the set of integer points $$P=\bigl\{\bigl(\lfloor x(t)\rfloor,\lfloor y(t)\rfloor\bigr) \bigm\vert 0\L t\L 1\bigr\}.$$ Well, the problem isn't actually quite so clean as this; when the path goes very near an integer point $(a,b)$, computational errors may make us think that $P$ contains $(a-1,b)$ while in reality it should contain $(a,b-1)$. Furthermore, if the path goes {\sl exactly\/} through the integer points $(a-1,b-1)$ and $(a,b)$, we will want $P$ to contain one of the two points $(a-1,b)$ or $(a,b-1)$, so that $P$ can be described entirely by ``rook moves'' upwards or to the right; no diagonal moves from $(a-1,b-1)$ to~$(a,b)$ will be allowed. Thus, the set $P$ we wish to compute will merely be an approximation to the set described in the formula above. It will consist of $\lfloor x(1)\rfloor-\lfloor x(0)\rfloor$ rightward moves and $\lfloor y(1)\rfloor-\lfloor y(0)\rfloor$ upward moves, intermixed in some order. Our job will be to figure out a suitable order. The following recursive strategy suggests itself, when we recall that $x(0)=x_0$, $x(1)=x_3$, $y(0)=y_0$, and $y(1)=y_3$: \smallskip If $\lfloor x_0\rfloor=\lfloor x_3\rfloor$ then take $\lfloor y_3\rfloor-\lfloor y_0\rfloor$ steps up. Otherwise if $\lfloor y_0\rfloor=\lfloor y_3\rfloor$ then take $\lfloor x_3\rfloor-\lfloor x_0\rfloor$ steps to the right. Otherwise bisect the current cubics and repeat the process on both halves. \yskip\noindent This intuitively appealing formulation does not quite solve the problem, because it may never terminate. For example, it's not hard to see that no steps will {\sl ever\/} be taken if $(x_0,x_1,x_2,x_3)=(y_0,y_1,y_2,y_3)$! However, we can surmount this difficulty with a bit of care; so let's proceed to flesh out the algorithm as stated, before worrying about such details. The bisect-and-double strategy discussed above suggests that we represent $(x_0,x_1,x_2,x_3)$ by $(X_1,X_2,X_3)$, where $X_k=2^l(x_k-x_{k-1})$ for some~$l$. Initially $l=16$, since the $x$'s are |scaled|. In order to deal with other aspects of the algorithm we will want to maintain also the quantities $m=\lfloor x_3\rfloor-\lfloor x_0\rfloor$ and $R=2^l(x_0\bmod 1)$. Similarly, $(y_0,y_1,y_2,y_3)$ will be represented by $(Y_1,Y_2,Y_3)$, $n=\lfloor y_3\rfloor-\lfloor y_0\rfloor$, and $S=2^l(y_0\bmod 1)$. The algorithm now takes the following form: \smallskip If $m=0$ then take $n$ steps up. Otherwise if $n=0$ then take $m$ steps to the right. Otherwise bisect the current cubics and repeat the process on both halves. \smallskip\noindent The bisection process for $(X_1,X_2,X_3,m,R,l)$ reduces, in essence, to the following formulas: $$\vbox{\halign{$#\hfil$\cr X_2'=\\{half}(X_1+X_2),\quad X_2''=\\{half}(X_2+X_3),\quad X_3'=\\{half}(X_2'+X_2''),\cr X_1'=X_1,\quad X_1''=X_3',\quad X_3''=X_3,\cr R'=2R,\quad T=X_1'+X_2'+X_3'+R',\quad R''=T\bmod 2^{l+1},\cr m'=\lfloor T/2^{l+1}\rfloor,\quad m''=m-m'.\cr}}$$ @ When $m=n=1$, the computation can be speeded up because we simply need to decide between two alternatives, (up,\thinspace right) versus (right,\thinspace up). There appears to be no simple, direct way to make the correct decision by looking at the values of $(X_1,X_2,X_3,R)$ and $(Y_1,Y_2,Y_3,S)$; but we can streamline the bisection process, and we can use the fact that only one of the two descendants needs to be examined after each bisection. Furthermore, we observed earlier that after several levels of bisection the $X$'s and $Y$'s will be nearly equal; so we will be justified in assuming that the curve is essentially a straight line. (This, incidentally, solves the problem of infinite recursion mentioned earlier.) It is possible to show that $$m=\bigl\lfloor(X_1+X_2+X_3+R+E)\,/\,2^l\bigr\rfloor,$$ where $E$ is an accumulated rounding error that is at most $3\cdot(2^{l-16}-1)$ in absolute value. We will make sure that the $X$'s are less than $2^{28}$; hence when $l=30$ we must have |m<=1|. This proves that the special case $m=n=1$ is bound to be reached by the time $l=30$. Furthermore $l=30$ is a suitable time to make the straight line approximation, if the recursion hasn't already died out, because the maximum difference between $X$'s will then be $<2^{14}$; this corresponds to an error of $<1$ with respect to the original scaling. (Stating this another way, each bisection makes the curve two bits closer to a straight line, hence 14 bisections are sufficient for 28-bit accuracy.) In the case of a straight line, the curve goes first right, then up, if and only if $(T-2^l)(2^l-S)>(U-2^l)(2^l-R)$, where $T=X_1+X_2+X_3+R$ and $U=Y_1+Y_2+Y_3+S$. For the actual curve essentially runs from $(R/2^l,S/2^l)$ to $(T/2^l,U/2^l)$, and we are testing whether or not $(1,1)$ is above the straight line connecting these two points. (This formula assumes that $(1,1)$ is not exactly on the line.) @ We have glossed over the problem of tie-breaking in ambiguous cases when the cubic curve passes exactly through integer points. \MF\ finesses this problem by assuming that coordinates $(x,y)$ actually stand for slightly perturbed values $(x+\xi,y+\eta)$, where $\xi$ and~$\eta$ are infinitesimals whose signs will determine what to do when $x$ and/or~$y$ are exact integers. The quantities $\lfloor x\rfloor$ and~$\lfloor y\rfloor$ in the formulas above should actually read $\lfloor x+\xi\rfloor$ and $\lfloor y+\eta\rfloor$. If $x$ is a |scaled| value, we have $\lfloor x+\xi\rfloor=\lfloor x\rfloor$ if $\xi>0$, and $\lfloor x+\xi\rfloor=\lfloor x-2^{-16}\rfloor$ if $\xi<0$. It is convenient to represent $\xi$ by the integer |xi_corr|, defined to be 0~if $\xi>0$ and 1~if $\xi<0$; then, for example, the integer $\lfloor x+\xi\rfloor$ can be computed as |floor_unscaled(x-xi_corr)|. Similarly, $\eta$ is conveniently represented by~|eta_corr|. In our applications the sign of $\xi-\eta$ will always be the same as the sign of $\xi$. Therefore it turns out that the rule for straight lines, as stated above, should be modified as follows in the case of ties: The line goes first right, then up, if and only if $(T-2^l)(2^l-S)+\xi>(U-2^l)(2^l-R)$. And this relation holds iff $|ab_vs_cd|(T-2^l,2^l-S,U-2^l,2^l-R)-|xi_corr|>=0$. These conventions for rounding are symmetrical, in the sense that the digitized moves obtained from $(x_0,x_1,x_2,x_3,y_0,y_1,y_2,y_3,\xi,\eta)$ will be exactly complementary to the moves that would be obtained from $(-x_3,-x_2,-x_1,-x_0,-y_3,-y_2,-y_1,-y_0,-\xi,-\eta)$, if arithmetic is exact. However, truncation errors in the bisection process might upset the symmetry. We can restore much of the lost symmetry by adding |xi_corr| or |eta_corr| when halving the data. @ One further possibility needs to be mentioned: The algorithm will be applied only to cubic polynomials $B(x_0,x_1,x_2,x_3;t)$ that are nondecreasing as $t$~varies from 0 to~1; this condition turns out to hold if and only if $x_0\L x_1$, $x_2\L x_3$, and either $x_1\L x_2$ or $(x_1-x_2)^2\L(x_1-x_0)(x_3-x_2)$. If bisection were carried out with perfect accuracy, these relations would remain invariant. But rounding errors can creep in, hence the bisection algorithm can produce non-monotonic subproblems from monotonic initial conditions. This leads to the potential danger that $m$ or~$n$ could become negative in the algorithm described above. For example, if we start with $(x_1-x_0,x_2-x_1,x_3-x_2)= (X_1,X_2,X_3)=(7,-16,58)$, the corresponding polynomial is monotonic, because $16^2<7\cdot39$. But the bisection algorithm produces the left descendant $(7,-5,3)$, which is nonmonotonic; its right descendant is~$(0,-1,3)$. \def\xt{{\tilde x}} Fortunately we can prove that such rounding errors will never cause the algorithm to make a tragic mistake. At every stage we are working with numbers corresponding to a cubic polynomial $B(\xt_0, \xt_1,\xt_2,\xt_3)$ that approximates some monotonic polynomial $B(x_0,x_1,x_2,x_3)$. The accumulated errors are controlled so that $\vert x_k-\xt_k\vert<\epsilon=3\cdot2^{-16}$. If bisection is done at some stage of the recursion, we have $m=\lfloor\xt_3\rfloor-\lfloor\xt_0\rfloor>0$, and the algorithm computes a bisection value $\bar x$ such that $m'=\lfloor\bar x\rfloor- \lfloor\xt_0\rfloor$ and $m''=\lfloor\xt_3\rfloor-\lfloor\bar x\rfloor$. We want to prove that neither $m'$ nor $m''$ can be negative. Since $\bar x$ is an approximation to a value in the interval $[x_0,x_3]$, we have $\bar x>x_0-\epsilon$ and $\bar x \xt_0-2\epsilon$ and $\bar x<\xt_3+2\epsilon$. If $m'$ is negative we must have $\xt_0\bmod 1<2\epsilon$; if $m''$ is negative we must have $\xt_3\bmod 1>1-2\epsilon$. In either case the condition $\lfloor\xt_3\rfloor-\lfloor\xt_0\rfloor>0$ implies that $\xt_3-\xt_0>1-2\epsilon$, hence $x_3-x_0>1-4\epsilon$. But it can be shown that if $B(x_0,x_1,x_2,x_3;t)$ is a monotonic cubic, then $B(x_0,x_1,x_2,x_3;{1\over2})$ is always between $.14[x_0,x_3]$ and $.86[x_0,x_3]$; and it is impossible for $\bar x$ to be within~$\epsilon$ of such a number. Contradiction! (The constant .14 is actually $(7-\sqrt{28}\,)/12$; the worst case occurs for polynomials like $B(0,28-4\sqrt{28},14-5\sqrt{28},42;t)$.) @ OK, now that a long theoretical preamble has justified the bisection-and-doubling algorithm, we are ready to proceed with its actual coding. But we still haven't discussed the form of the output. For reasons to be discussed later, we shall find it convenient to record the output as follows: Moving one step up is represented by appending a `1' to a list; moving one step right is represented by adding unity to the element at the end of the list. Thus, for example, the net effect of ``(up, right, right, up, right)'' is to append $(3,2)$. The list is kept in a global array called |move|. Before starting the algorithm, \MF\ should check that $\\{move\_ptr}+\lfloor y_3\rfloor -\lfloor y_0\rfloor\L\\{move\_size}$, so that the list won't exceed the bounds of this array. @= @!move:array[0..move_size] of integer; {the recorded moves} @!move_ptr:0..move_size; {the number of items in the |move| list} @ When bisection occurs, we ``push'' the subproblem corresponding to the right-hand subinterval onto the |bisect_stack| while we continue to work on the left-hand subinterval. Thus, the |bisect_stack| will hold $(X_1,X_2,X_3,R,m,Y_1,Y_2,Y_3,S,n,l)$ values for subproblems yet to be tackled. At most 15 subproblems will be on the stack at once (namely, for $l=15$,~16, \dots,~29); but the stack is bigger than this, because it is used also for more complicated bisection algorithms. @d stack_x1==bisect_stack[bisect_ptr] {stacked value of $X_1$} @d stack_x2==bisect_stack[bisect_ptr+1] {stacked value of $X_2$} @d stack_x3==bisect_stack[bisect_ptr+2] {stacked value of $X_3$} @d stack_r==bisect_stack[bisect_ptr+3] {stacked value of $R$} @d stack_m==bisect_stack[bisect_ptr+4] {stacked value of $m$} @d stack_y1==bisect_stack[bisect_ptr+5] {stacked value of $Y_1$} @d stack_y2==bisect_stack[bisect_ptr+6] {stacked value of $Y_2$} @d stack_y3==bisect_stack[bisect_ptr+7] {stacked value of $Y_3$} @d stack_s==bisect_stack[bisect_ptr+8] {stacked value of $S$} @d stack_n==bisect_stack[bisect_ptr+9] {stacked value of $n$} @d stack_l==bisect_stack[bisect_ptr+10] {stacked value of $l$} @d move_increment=11 {number of items pushed by |make_moves|} @= @!bisect_stack:array[0..bistack_size] of integer; @!bisect_ptr:0..bistack_size; @ @= if 15*move_increment>bistack_size then bad:=31; @ The |make_moves| subroutine is given |scaled| values $(x_0,x_1,x_2,x_3)$ and $(y_0,y_1,y_2,y_3)$ that represent monotone-nondecreasing polynomials; it makes $\lfloor x_3+\xi\rfloor-\lfloor x_0+\xi\rfloor$ rightward moves and $\lfloor y_3+\eta\rfloor-\lfloor y_0+\eta\rfloor$ upward moves, as explained earlier. (Here $\lfloor x+\xi\rfloor$ actually stands for $\lfloor x/2^{16}-|xi_corr|\rfloor$, if $x$ is regarded as an integer without scaling.) The unscaled integers $x_k$ and~$y_k$ should be less than $2^{28}$ in magnitude. It is assumed that $|move_ptr| + \lfloor y_3+\eta\rfloor - \lfloor y_0+\eta\rfloor < |move_size|$ when this procedure is called, so that the capacity of the |move| array will not be exceeded. The variables |r| and |s| in this procedure stand respectively for $R-|xi_corr|$ and $S-|eta_corr|$ in the theory discussed above. @p procedure make_moves(@!xx0,@!xx1,@!xx2,@!xx3,@!yy0,@!yy1,@!yy2,@!yy3: scaled;@!xi_corr,@!eta_corr:small_number); label continue, done, exit; var @!x1,@!x2,@!x3,@!m,@!r,@!y1,@!y2,@!y3,@!n,@!s,@!l:integer; {bisection variables explained above} @!q,@!t,@!u,@!x2a,@!x3a,@!y2a,@!y3a:integer; {additional temporary registers} begin if (xx3 l:=16; bisect_ptr:=0;@/ x1:=xx1-xx0; x2:=xx2-xx1; x3:=xx3-xx2; if xx0>=xi_corr then r:=(xx0-xi_corr) mod unity else r:=unity-1-((-xx0+xi_corr-1) mod unity); m:=(xx3-xx0+r) div unity;@/ y1:=yy1-yy0; y2:=yy2-yy1; y3:=yy3-yy2; if yy0>=eta_corr then s:=(yy0-eta_corr) mod unity else s:=unity-1-((-yy0+eta_corr-1) mod unity); n:=(yy3-yy0+s) div unity;@/ if (xx3-xx0>=fraction_one)or(yy3-yy0>=fraction_one) then @; loop@+ begin continue:@; if bisect_ptr=0 then return; @; end; exit: end; @ @= bisect_ptr:=bisect_ptr-move_increment;@/ x1:=stack_x1; x2:=stack_x2; x3:=stack_x3; r:=stack_r; m:=stack_m;@/ y1:=stack_y1; y2:=stack_y2; y3:=stack_y3; s:=stack_s; n:=stack_n;@/ l:=stack_l @ Our variables |(x1,x2,x3)| correspond to $(X_1,X_2,X_3)$ in the notation of the theory developed above. We need to keep them less than $2^{28}$ in order to avoid integer overflow in weird circumstances. For example, data like $x_0=-2^{28}+2^{16}-1$ and $x_1=x_2=x_3=2^{28}-1$ would otherwise be problematical. Hence this part of the code is needed, if only to thwart malicious users. @= begin x1:=half(x1+xi_corr); x2:=half(x2+xi_corr); x3:=half(x3+xi_corr); r:=half(r+xi_corr);@/ y1:=half(y1+eta_corr); y2:=half(y2+eta_corr); y3:=half(y3+eta_corr); s:=half(s+eta_corr);@/ l:=15; end @ @= if m=0 then @ else if n=0 then @ else if m+n=2 then @ else begin incr(l); stack_l:=l;@/ stack_x3:=x3; stack_x2:=half(x2+x3+xi_corr); x2:=half(x1+x2+xi_corr); x3:=half(x2+stack_x2+xi_corr); stack_x1:=x3;@/ r:=r+r+xi_corr; t:=x1+x2+x3+r;@/ q:=t div two_to_the[l]; stack_r:=t mod two_to_the[l];@/ stack_m:=m-q; m:=q;@/ stack_y3:=y3; stack_y2:=half(y2+y3+eta_corr); y2:=half(y1+y2+eta_corr); y3:=half(y2+stack_y2+eta_corr); stack_y1:=y3;@/ s:=s+s+eta_corr; u:=y1+y2+y3+s;@/ q:=u div two_to_the[l]; stack_s:=u mod two_to_the[l];@/ stack_n:=n-q; n:=q;@/ bisect_ptr:=bisect_ptr+move_increment; goto continue; end @ @= while n>0 do begin incr(move_ptr); move[move_ptr]:=1; decr(n); end @ @= move[move_ptr]:=move[move_ptr]+m @ @= begin r:=two_to_the[l]-r; s:=two_to_the[l]-s;@/ while l<30 do begin x3a:=x3; x2a:=half(x2+x3+xi_corr); x2:=half(x1+x2+xi_corr); x3:=half(x2+x2a+xi_corr); t:=x1+x2+x3; r:=r+r-xi_corr;@/ y3a:=y3; y2a:=half(y2+y3+eta_corr); y2:=half(y1+y2+eta_corr); y3:=half(y2+y2a+eta_corr); u:=y1+y2+y3; s:=s+s-eta_corr;@/ if t else begin @; goto done; end else if u; goto done; end; incr(l); end; r:=r-xi_corr; s:=s-eta_corr; if ab_vs_cd(x1+x2+x3,s,y1+y2+y3,r)-xi_corr>=0 then @ else @; done: end @ @= begin x1:=x3; x2:=x2a; x3:=x3a; r:=r-t; y1:=y3; y2:=y2a; y3:=y3a; s:=s-u; end @ @= begin incr(move[move_ptr]); incr(move_ptr); move[move_ptr]:=1; end @ @= begin incr(move_ptr); move[move_ptr]:=2; end @ After |make_moves| has acted, possibly for several curves that move toward the same octant, a ``smoothing'' operation might be done on the |move| array. This removes optical glitches that can arise even when the curve has been digitized without rounding errors. The smoothing process replaces the integers $a_0\ldots a_n$ in |move[b..t]| by ``smoothed'' integers $a_0'\ldots a_n'$ defined as follows: $$a_k'=a_k+\delta\k-\delta_k;\qquad \delta_k=\cases{+1,&if $1=3 then begin k:=b+2; aa:=move[k-1]; aaa:=move[k-2]; repeat a:=move[k]; if abs(a-aa)>1 then @; incr(k); aaa:=aa; aa:=a; until k=t; end; end; @ @= if a>aa then begin if aaa>=aa then if a>=move[k+1] then begin incr(move[k-1]); move[k]:=a-1; end; end else begin if aaa<=aa then if a<=move[k+1] then begin decr(move[k-1]); move[k]:=a+1; end; end @* \[20] Edge structures. Now we come to \MF's internal scheme for representing what the user can actually ``see,'' the edges between pixels. Each pixel has an integer weight, obtained by summing the weights on all edges to its left. \MF\ represents only the nonzero edge weights, since most of the edges are weightless; in this way, the data storage requirements grow only linearly with respect to the number of pixels per point, even though two-dimensional data is being represented. (Well, the actual dependence on the underlying resolution is order $n\log n$, but the the $\log n$ factor is buried in our implicit restriction on the maximum raster size.) The sum of all edge weights in each row should be zero. The data structure for edge weights must be compact and flexible, yet it should support efficient updating and display operations. We want to be able to have many different edge structures in memory at once, and we want the computer to be able to translate them, reflect them, and/or merge them together with relative ease. \MF's solution to this problem requires one single-word node per nonzero edge weight, plus one two-word node for each row in a contiguous set of rows. There's also a header node that provides global information about the entire structure. @ Let's consider the edge-weight nodes first. The |info| field of such nodes contains both an $m$~value and a weight~$w$, in the form $8m+w+c$, where $c$ is a constant that depends on data found in the header. We shall consider $c$ in detail later; for now, it's best just to think of it as a way to compensate for the fact that $m$ and~$w$ can be negative, together with the fact that an |info| field must have a value between |min_halfword| and |max_halfword|. The $m$ value is an unscaled $x$~coordinate, so it satisfies $\vert m\vert< 4096$; the $w$ value is always in the range $1\L\vert w\vert\L3$. We can unpack the data in the |info| field by fetching |ho(info(p))= info(p)-min_halfword| and dividing this nonnegative number by~8; the constant~$c$ will be chosen so that the remainder of this division is $4+w$. Thus, for example, a remainder of~3 will correspond to the edge weight $w=-1$. Every row of an edge structure contains two lists of such edge-weight nodes, called the |sorted| and |unsorted| lists, linked together by their |link| fields in the normal way. The difference between them is that we always have |info(p)<=info(link(p))| in the |sorted| list, but there's no such restriction on the elements of the |unsorted| list. The reason for this distinction is that it would take unnecessarily long to maintain edge-weight lists in sorted order while they're being updated; but when we need to process an entire row from left to right in order of the $m$~values, it's fairly easy and quick to sort a short list of unsorted elements and to merge them into place among their sorted cohorts. Furthermore, the fact that the |unsorted| list is empty can sometimes be used to good advantage, because it allows us to conclude that a particular row has not changed since the last time we sorted it. The final |link| of the |sorted| list will be |sentinel|, which points to a special one-word node whose |info| field is essentially infinite; this facilitates the sorting and merging operations. The final |link| of the |unsorted| list will be either |null| or |void|, where |void=null+1| is used to avoid redisplaying data that has not changed: A |void| value is stored at the head of the unsorted list whenever the corresponding row has been displayed. @d zero_w=4 @d void==null+1 @= info(sentinel):=max_halfword; {|link(sentinel)=null|} @ The rows themselves are represented by row-header nodes that contain four link fields. Two of these four, |sorted| and |unsorted|, point to the first items of the edge-weight lists just mentioned. The other two, |link| and |knil|, point to the headers of the two adjacent rows. If |p| points to the header for row number~|n|, then |link(p)| points up to the header for row~|n+1|, and |knil(p)| points down to the header for row~|n-1|. This double linking makes it convenient to move through consecutive rows either upward or downward; as usual, we have |link(knil(p))=knil(link(p))=p| for all row headers~|p|. The row associated with a given value of |n| contains weights for edges that run between the lattice points |(m,n)| and |(m,n+1)|. @d knil==info {inverse of the |link| field, in a doubly linked list} @d sorted_loc(#)==#+1 {where the |sorted| link field resides} @d sorted(#)==link(sorted_loc(#)) {beginning of the list of sorted edge weights} @d unsorted(#)==info(#+1) {beginning of the list of unsorted edge weights} @d row_node_size=2 {number of words in a row header node} @ The main header node |h| for an edge structure has |link| and |knil| fields that link it above the topmost row and below the bottommost row. It also has fields called |m_min|, |m_max|, |n_min|, and |n_max| that bound the current extent of the edge data: All |m| values in edge-weight nodes should lie between |m_min(h)-4096| and |m_max(h)-4096|, inclusive. Furthermore the topmost row header, pointed to by |knil(h)|, is for row number |n_max(h)-4096|; the bottommost row header, pointed to by |link(h)|, is for row number |n_min(h)-4096|. The offset constant |c| that's used in all of the edge-weight data is represented implicitly in |m_offset(h)|; its actual value is $$\hbox{|c=min_halfword+zero_w+8*m_offset(h)|.}$$ Notice that it's possible to shift an entire edge structure by an amount $(\Delta m,\Delta n)$ by adding $\Delta n$ to |n_min(h)| and |n_max(h)|, adding $\Delta m$ to |m_min(h)| and |m_max(h)|, and subtracting $\Delta m$ from |m_offset(h)|; none of the other edge data needs to be modified. Initially the |m_offset| field is~4096, but it will change if the user requests such a shift. The contents of these five fields should always be positive and less than 8192; |n_max| should, in fact, be less than 8191. Furthermore |m_min+m_offset-4096| and |m_max+m_offset-4096| must also lie strictly between 0 and 8192, so that the |info| fields of edge-weight nodes will fit in a halfword. The header node of an edge structure also contains two somewhat unusual fields called |last_window(h)| and |last_window_time(h)|. When this structure is displayed in window~|k| of the user's screen, after that window has been updated |t| times, \MF\ sets |last_window(h):=k| and |last_window_time(h):=t|; it also sets |unsorted(p):=void| for all row headers~|p|, after merging any existing unsorted weights with the sorted ones. A subsequent display in the same window will be able to avoid redisplaying rows whose |unsorted| list is still |void|, if the window hasn't been used for something else in the meantime. A pointer to the row header of row |n_pos(h)-4096| is provided in |n_rover(h)|. Most of the algorithms that update an edge structure are able to get by without random row references; they usually access rows that are neighbors of each other or of the current |n_pos| row. Exception: If |link(h)=h| (so that the edge structure contains no rows), we have |n_rover(h)=h|, and |n_pos(h)| is irrelevant. @d zero_field=4096 {amount added to coordinates to make them positive} @d n_min(#)==info(#+1) {minimum row number present, plus |zero_field|} @d n_max(#)==link(#+1) {maximum row number present, plus |zero_field|} @d m_min(#)==info(#+2) {minimum column number present, plus |zero_field|} @d m_max(#)==link(#+2) {maximum column number present, plus |zero_field|} @d m_offset(#)==info(#+3) {translation of $m$ data in edge-weight nodes} @d last_window(#)==link(#+3) {the last display went into this window} @d last_window_time(#)==mem[#+4].int {after this many window updates} @d n_pos(#)==info(#+5) {the row currently in |n_rover|, plus |zero_field|} @d n_rover(#)==link(#+5) {a row recently referenced} @d edge_header_size=6 {number of words in an edge-structure header} @d valid_range(#)==(abs(#-4096)<4096) {is |#| strictly between 0 and 8192?} @d empty_edges(#)==link(#)=# {are there no rows in this edge header?} @p procedure init_edges(@!h:pointer); {initialize an edge header to null values} begin knil(h):=h; link(h):=h;@/ n_min(h):=zero_field+4095; n_max(h):=zero_field-4095; m_min(h):=zero_field+4095; m_max(h):=zero_field-4095; m_offset(h):=zero_field;@/ last_window(h):=0; last_window_time(h):=0;@/ n_rover(h):=h; n_pos(h):=0;@/ end; @ When a lot of work is being done on a particular edge structure, we plant a pointer to its main header in the global variable |cur_edges|. This saves us from having to pass this pointer as a parameter over and over again between subroutines. Similarly, |cur_wt| is a global weight that is being used by several procedures at once. @= @!cur_edges:pointer; {the edge structure of current interest} @!cur_wt:integer; {the edge weight of current interest} @ The |fix_offset| routine goes through all the edge-weight nodes of |cur_edges| and adds a constant to their |info| fields, so that |m_offset(cur_edges)| can be brought back to |zero_field|. (This is necessary only in unusual cases when the offset has gotten too large or too small.) @p procedure fix_offset; var @!p,@!q:pointer; {list traversers} @!delta:integer; {the amount of change} begin delta:=8*(m_offset(cur_edges)-zero_field); m_offset(cur_edges):=zero_field; q:=link(cur_edges); while q<>cur_edges do begin p:=sorted(q); while p<>sentinel do begin info(p):=info(p)-delta; p:=link(p); end; p:=unsorted(q); while p>void do begin info(p):=info(p)-delta; p:=link(p); end; q:=link(q); end; end; @ The |edge_prep| routine makes the |cur_edges| structure ready to accept new data whose coordinates satisfy |ml<=m<=mr| and |nl<=n<=nr-1|, assuming that |-4096m_max(cur_edges) then m_max(cur_edges):=mr; if not valid_range(m_min(cur_edges)+m_offset(cur_edges)-zero_field) or@| not valid_range(m_max(cur_edges)+m_offset(cur_edges)-zero_field) then fix_offset; if empty_edges(cur_edges) then {there are no rows} begin n_min(cur_edges):=nr+1; n_max(cur_edges):=nr; end; if nl; if nr>n_max(cur_edges) then @; end; @ @= begin delta:=n_min(cur_edges)-nl; n_min(cur_edges):=nl; p:=link(cur_edges); repeat q:=get_node(row_node_size); sorted(q):=sentinel; unsorted(q):=void; knil(p):=q; link(q):=p; p:=q; decr(delta); until delta=0; knil(p):=cur_edges; link(cur_edges):=p; if n_rover(cur_edges)=cur_edges then n_pos(cur_edges):=nl-1; end @ @= begin delta:=nr-n_max(cur_edges); n_max(cur_edges):=nr; p:=knil(cur_edges); repeat q:=get_node(row_node_size); sorted(q):=sentinel; unsorted(q):=void; link(p):=q; knil(q):=p; p:=q; decr(delta); until delta=0; link(p):=cur_edges; knil(cur_edges):=p; if n_rover(cur_edges)=cur_edges then n_pos(cur_edges):=nr+1; end @ The |print_edges| subroutine gives a symbolic rendition of an edge structure, for use in `\&{show}' commands. A rather terse output format has been chosen since edge structures can grow quite large. @= @t\4@>@@;@/ procedure print_edges(@!s:str_number;@!nuline:boolean;@!x_off,@!y_off:integer); var @!p,@!q,@!r:pointer; {for list traversal} @!n:integer; {row number} begin print_diagnostic("Edge structure",s,nuline); p:=knil(cur_edges); n:=n_max(cur_edges)-zero_field; while p<>cur_edges do begin q:=unsorted(p); r:=sorted(p); if(q>void)or(r<>sentinel) then begin print_nl("row "); print_int(n+y_off); print_char(":"); while q>void do begin print_weight(q,x_off); q:=link(q); end; print(" |"); while r<>sentinel do begin print_weight(r,x_off); r:=link(r); end; end; p:=knil(p); decr(n); end; end_diagnostic(true); end; @ @= procedure print_weight(@!q:pointer;@!x_off:integer); var @!w,@!m:integer; {unpacked weight and coordinate} @!d:integer; {temporary data register} begin d:=ho(info(q)); w:=d mod 8; m:=(d div 8)-m_offset(cur_edges); if file_offset>max_print_line-9 then print_nl(" ") else print_char(" "); print_int(m+x_off); while w>zero_w do begin print_char("+"); decr(w); end; while wh do begin pp:=get_node(row_node_size); link(qq):=pp; knil(pp):=qq; @; p:=link(p); qq:=pp; end; link(qq):=hh; knil(hh):=qq; copy_edges:=hh; end; @ @= r:=sorted(p); rr:=sorted_loc(pp); {|link(rr)=sorted(pp)|} while r<>sentinel do begin ss:=get_avail; link(rr):=ss; rr:=ss; info(rr):=info(r);@/ r:=link(r); end; link(rr):=sentinel;@/ r:=unsorted(p); rr:=temp_head; while r>void do begin ss:=get_avail; link(rr):=ss; rr:=ss; info(rr):=info(r);@/ r:=link(r); end; link(rr):=r; unsorted(pp):=link(temp_head) @ Another trivial routine flips |cur_edges| about the |x|-axis (i.e., negates all the |y| coordinates), assuming that at least one row is present. @p procedure y_reflect_edges; var @!p,@!q,@!r:pointer; {list manipulation registers} begin p:=n_min(cur_edges); n_min(cur_edges):=zero_field+zero_field-1-n_max(cur_edges); n_max(cur_edges):=zero_field+zero_field-1-p; n_pos(cur_edges):=zero_field+zero_field-1-n_pos(cur_edges);@/ p:=link(cur_edges); q:=cur_edges; {we assume that |p<>q|} repeat r:=link(p); link(p):=q; knil(q):=p; q:=p; p:=r; until q=cur_edges; last_window_time(cur_edges):=0; end; @ It's somewhat more difficult, yet not too hard, to reflect about the |y|-axis. @p procedure x_reflect_edges; var @!p,@!q,@!r,@!s:pointer; {list manipulation registers} @!m:integer; {|info| fields will be reflected with respect to this number} begin p:=m_min(cur_edges); m_min(cur_edges):=zero_field+zero_field-m_max(cur_edges); m_max(cur_edges):=zero_field+zero_field-p; m:=(zero_field+m_offset(cur_edges))*8+zero_w+min_halfword+zero_w+min_halfword; m_offset(cur_edges):=zero_field; p:=link(cur_edges); repeat @; @; p:=link(p); until p=cur_edges; last_window_time(cur_edges):=0; end; @ We want to change the sign of the weight as we change the sign of the |x|~coordinate. Fortunately, it's easier to do this than to negate one without the other. @= q:=unsorted(p); while q>void do begin info(q):=m-info(q); q:=link(q); end @ Reversing the order of a linked list is best thought of as the process of popping nodes off one stack and pushing them on another. In this case we pop from stack~|q| and push to stack~|r|. @= q:=sorted(p); r:=sentinel; while q<>sentinel do begin s:=link(q); link(q):=r; r:=q; info(r):=m-info(q); q:=s; end; sorted(p):=r @ Now let's multiply all the $y$~coordinates of a nonempty edge structure by a small integer $s>1$: @p procedure y_scale_edges(@!s:integer); var @!p,@!q,@!pp,@!r,@!rr,@!ss:pointer; {list manipulation registers} @!t:integer; {replication counter} begin if (s*(n_max(cur_edges)+1-zero_field)>=4096) or@| (s*(n_min(cur_edges)-zero_field)<=-4096) then begin print_err("Scaled picture would be too big"); @.Scaled picture...big@> help3("I can't yscale the picture as requested---it would")@/ ("make some coordinates too large or too small.")@/ ("Proceed, and I'll omit the transformation."); put_get_error; end else begin n_max(cur_edges):=s*(n_max(cur_edges)+1-zero_field)-1+zero_field; n_min(cur_edges):=s*(n_min(cur_edges)-zero_field)+zero_field; @; last_window_time(cur_edges):=0; end; end; @ @= p:=cur_edges; repeat q:=p; p:=link(p); for t:=2 to s do begin pp:=get_node(row_node_size); link(q):=pp; knil(p):=pp; link(pp):=p; knil(pp):=q; q:=pp; @; end; until link(p)=cur_edges @ Scaling the $x$~coordinates is, of course, our next task. @p procedure x_scale_edges(@!s:integer); var @!p,@!q:pointer; {list manipulation registers} @!t:0..65535; {unpacked |info| field} @!w:0..7; {unpacked weight} @!delta:integer; {amount added to scaled |info|} begin if (s*(m_max(cur_edges)-zero_field)>=4096) or@| (s*(m_min(cur_edges)-zero_field)<=-4096) then begin print_err("Scaled picture would be too big"); @.Scaled picture...big@> help3("I can't xscale the picture as requested---it would")@/ ("make some coordinates too large or too small.")@/ ("Proceed, and I'll omit the transformation."); put_get_error; end else if (m_max(cur_edges)<>zero_field)or(m_min(cur_edges)<>zero_field) then begin m_max(cur_edges):=s*(m_max(cur_edges)-zero_field)+zero_field; m_min(cur_edges):=s*(m_min(cur_edges)-zero_field)+zero_field; delta:=8*(zero_field-s*m_offset(cur_edges))+min_halfword; m_offset(cur_edges):=zero_field; @; last_window_time(cur_edges):=0; end; end; @ The multiplications cannot overflow because we know that |s<4096|. @= q:=link(cur_edges); repeat p:=sorted(q); while p<>sentinel do begin t:=ho(info(p)); w:=t mod 8; info(p):=(t-w)*s+w+delta; p:=link(p); end; p:=unsorted(q); while p>void do begin t:=ho(info(p)); w:=t mod 8; info(p):=(t-w)*s+w+delta; p:=link(p); end; q:=link(q); until q=cur_edges @ Here is a routine that changes the signs of all the weights, without changing anything else. @p procedure negate_edges(@!h:pointer); label done; var @!p,@!q,@!r,@!s,@!t,@!u:pointer; {structure traversers} begin p:=link(h); while p<>h do begin q:=unsorted(p); while q>void do begin info(q):=8-2*((ho(info(q))) mod 8)+info(q); q:=link(q); end; q:=sorted(p); if q<>sentinel then begin repeat info(q):=8-2*((ho(info(q))) mod 8)+info(q); q:=link(q); until q=sentinel; @; end; p:=link(p); end; last_window_time(h):=0; end; @ \MF\ would work even if the code in this section were omitted, because a list of edge-and-weight data that is sorted only by |m| but not~|w| turns out to be good enough for correct operation. However, the author decided not to make the program even trickier than it is already, since |negate_edges| isn't needed very often. The simpler-to-state condition, ``keep the |sorted| list fully sorted,'' is therefore being preserved at the cost of extra computation. @= u:=sorted_loc(p); q:=link(u); r:=q; s:=link(r); {|q=sorted(p)|} loop@+ if info(s)>info(r) then begin link(u):=q; if s=sentinel then goto done; u:=r; q:=s; r:=q; s:=link(r); end else begin t:=s; s:=link(t); link(t):=q; q:=t; end; done: link(r):=sentinel @ The |unsorted| edges of a row are merged into the |sorted| ones by a subroutine called |sort_edges|. It uses simple insertion sort, followed by a merge, because the unsorted list is supposedly quite short. However, the unsorted list is assumed to be nonempty. @p procedure sort_edges(@!h:pointer); {|h| is a row header} label done; var @!k:halfword; {key register that we compare to |info(q)|} @!p,@!q,@!r,@!s:pointer; begin r:=unsorted(h); unsorted(h):=null; p:=link(r); link(r):=sentinel; link(temp_head):=r; while p>void do {sort node |p| into the list that starts at |temp_head|} begin k:=info(p); q:=temp_head; repeat r:=q; q:=link(r); until k<=info(q); link(r):=p; r:=link(p); link(p):=q; p:=r; end; @; end; @ In this step we use the fact that |sorted(h)=link(sorted_loc(h))|. @= begin r:=sorted_loc(h); q:=link(r); p:=link(temp_head); loop@+ begin k:=info(p); while k>info(q) do begin r:=q; q:=link(r); end; link(r):=p; s:=link(p); link(p):=q; if s=sentinel then goto done; r:=p; p:=s; end; done:end @ The |cull_edges| procedure ``optimizes'' an edge structure by making all the pixel weights either |w_out| or~|w_in|. The weight will be~|w_in| after the operation if and only if it was in the closed interval |[w_lo,w_hi]| before, where |w_lo<=w_hi|. Either |w_out| or |w_in| is zero, while the other is $\pm1$, $\pm2$, or $\pm3$. The parameters will be such that zero-weight pixels will remain of weight zero. (This is fortunate, because there are infinitely many of them.) The procedure also computes the tightest possible bounds on the resulting data, by updating |m_min|, |m_max|, |n_min|, and~|n_max|. @p procedure cull_edges(@!w_lo,@!w_hi,@!w_out,@!w_in:integer); label done; var @!p,@!q,@!r,@!s:pointer; {for list manipulation} @!w:integer; {new weight after culling} @!d:integer; {data register for unpacking} @!m:integer; {the previous column number, including |m_offset|} @!mm:integer; {the next column number, including |m_offset|} @!ww:integer; {accumulated weight before culling} @!prev_w:integer; {value of |w| before column |m|} @!n,@!min_n,@!max_n:pointer; {current and extreme row numbers} @!min_d,@!max_d:pointer; {extremes of the new edge-and-weight data} begin min_d:=max_halfword; max_d:=min_halfword; min_n:=max_halfword; max_n:=min_halfword;@/ p:=link(cur_edges); n:=n_min(cur_edges); while p<>cur_edges do begin if unsorted(p)>void then sort_edges(p); if sorted(p)<>sentinel then @; p:=link(p); incr(n); end; @; last_window_time(cur_edges):=0; end; @ The entire |sorted| list is returned to available memory in this step; a new list is built starting (temporarily) at |temp_head|. Since several edges can occur at the same column, we need to be looking ahead of where the actual culling takes place. This means that it's slightly tricky to get the iteration started and stopped. @= begin r:=temp_head; q:=sorted(p); ww:=0; m:=1000000; prev_w:=0; loop@+ begin if q=sentinel then mm:=1000000 else begin d:=ho(info(q)); mm:=d div 8; ww:=ww+(d mod 8)-zero_w; end; if mm>m then begin @; if q=sentinel then goto done; end; m:=mm; if ww>=w_lo then if ww<=w_hi then w:=w_in else w:=w_out else w:=w_out; s:=link(q); free_avail(q); q:=s; end; done: link(r):=sentinel; sorted(p):=link(temp_head); if r<>temp_head then @; end @ @= if w<>prev_w then begin s:=get_avail; link(r):=s; info(s):=8*m+min_halfword+zero_w+w-prev_w; r:=s; prev_w:=w; end @ @= begin if min_n=max_halfword then min_n:=n; max_n:=n; if min_d>info(link(temp_head)) then min_d:=info(link(temp_head)); if max_d= if min_n>max_n then @ else begin n:=n_min(cur_edges); n_min(cur_edges):=min_n; while min_n>n do begin p:=link(cur_edges); link(cur_edges):=link(p); knil(link(p)):=cur_edges; free_node(p,row_node_size); incr(n); end; n:=n_max(cur_edges); n_max(cur_edges):=max_n; n_pos(cur_edges):=max_n+1; n_rover(cur_edges):=cur_edges; while max_n= begin p:=link(cur_edges); while p<>cur_edges do begin q:=link(p); free_node(p,row_node_size); p:=q; end; init_edges(cur_edges); end @ The last and most difficult routine for transforming an edge structure---and the most interesting one!---is |xy_swap_edges|, which interchanges the r\^^Doles of rows and columns. Its task can be viewed as the job of creating an edge structure that contains only horizontal edges, linked together in columns, given an edge structure that contains only vertical edges linked together in rows; we must do this without changing the implied pixel weights. Given any two adjacent rows of an edge structure, it is not difficult to determine the horizontal edges that lie ``between'' them: We simply look for vertically adjacent pixels that have different weight, and insert a horizontal edge containing the difference in weights. Every horizontal edge determined in this way should be put into an appropriate linked list. Since random access to these linked lists is desirable, we use the |move| array to hold the list heads. If we work through the given edge structure from top to bottom, the constructed lists will not need to be sorted, since they will already be in order. The following algorithm makes use of some ideas suggested by John Hobby. @^Hobby, John Douglas@> It assumes that the edge structure is non-null, i.e., that |link(cur_edges) <>cur_edges|, hence |m_max(cur_edges)>=m_min(cur_edges)|. @p procedure xy_swap_edges; {interchange |x| and |y| in |cur_edges|} label done; var @!m_magic,@!n_magic:integer; {special values that account for offsets} @!p,@!q,@!r,@!s:pointer; {pointers that traverse the given structure} @@; begin @; @; @; repeat q:=knil(p);@+if unsorted(q)>void then sort_edges(q); @; p:=q; n_magic:=n_magic-8; until knil(p)=cur_edges; free_node(p,row_node_size); {now all original rows have been recycled} @; end; @ Here we don't bother to keep the |link| entries up to date, since the procedure looks only at the |knil| fields as it destroys the former edge structure. @= p:=get_node(row_node_size); sorted(p):=sentinel; unsorted(p):=null;@/ knil(p):=cur_edges; knil(link(cur_edges)):=p; {the new bottom row} p:=get_node(row_node_size); sorted(p):=sentinel; knil(p):=knil(cur_edges); {the new top row} @ The new lists will become |sorted| lists later, so we initialize empty lists to |sentinel|. @= m_spread:=m_max(cur_edges)-m_min(cur_edges); {this is |>=0| by assumption} if m_spread>move_size then overflow("move table size",move_size); @:METAFONT capacity exceeded move table size}{\quad move table size@> for j:=0 to m_spread do move[j]:=sentinel @ @= @!m_spread:integer; {the difference between |m_max| and |m_min|} @!j,@!jj:0..move_size; {indices into |move|} @!m,@!mm:integer; {|m| values at vertical edges} @!pd,@!rd:integer; {data fields from edge-and-weight nodes} @!pm,@!rm:integer; {|m| values from edge-and-weight nodes} @!w:integer; {the difference in accumulated weight} @!ww:integer; {as much of |w| that can be stored in a single node} @!dw:integer; {an increment to be added to |w|} @ At the point where we test |w<>0|, variable |w| contains the accumulated weight from edges already passed in row~|p| minus the accumulated weight from edges already passed in row~|q|. @= r:=sorted(p); free_node(p,row_node_size); p:=r;@/ pd:=ho(info(p)); pm:=pd div 8;@/ r:=sorted(q); rd:=ho(info(r)); rm:=rd div 8; w:=0; loop@+ begin if pm0 then @; if pd; end else begin if r=sentinel then goto done; {|rd=pd=ho(max_halfword)|} dw:=-((rd mod 8)-zero_w); @; end; m:=mm; w:=w+dw; end; done: @ @= r:=link(r); rd:=ho(info(r)); rm:=rd div 8 @ @= s:=link(p); free_avail(p); p:=s; pd:=ho(info(p)); pm:=pd div 8 @ Certain ``magic'' values are needed to make the following code work, due to the various offsets in our data structure. For now, let's not worry about their precise values; we shall compute |m_magic| and |n_magic| later, after we see what the code looks like. @ @= if m<>mm then begin if mm-m_magic>=move_size then confusion("xy"); @:this can't happen xy}{\quad xy@> extras:=(abs(w)-1) div 3; if extras>0 then begin if w>0 then xw:=+3@+else xw:=-3; ww:=w-extras*xw; end else ww:=w; repeat j:=m-m_magic; for k:=1 to extras do begin s:=get_avail; info(s):=n_magic+xw; link(s):=move[j]; move[j]:=s; end; s:=get_avail; info(s):=n_magic+ww; link(s):=move[j]; move[j]:=s;@/ incr(m); until m=mm; end @ @= @!extras:integer; {the number of additional nodes to make weights |>3|} @!xw:-3..3; {the additional weight in extra nodes} @!k:integer; {loop counter for inserting extra nodes} @ At the beginning of this step, |move[m_spread]=sentinel|, because no horizontal edges will extend to the right of column |m_max(cur_edges)|. @= move[m_spread]:=0; j:=0; while move[j]=sentinel do incr(j); if j=m_spread then init_edges(cur_edges) {all edge weights are zero} else begin mm:=m_min(cur_edges); m_min(cur_edges):=n_min(cur_edges); m_max(cur_edges):=n_max(cur_edges)+1; m_offset(cur_edges):=zero_field; jj:=m_spread-1; while move[jj]=sentinel do decr(jj); n_min(cur_edges):=j+mm; n_max(cur_edges):=jj+mm; q:=cur_edges; repeat p:=get_node(row_node_size); link(q):=p; knil(p):=q; sorted(p):=move[j]; unsorted(p):=null; incr(j); q:=p; until j>jj; link(q):=cur_edges; knil(cur_edges):=q; n_pos(cur_edges):=n_max(cur_edges)+1; n_rover(cur_edges):=cur_edges; last_window_time(cur_edges):=0; end; @ The values of |m_magic| and |n_magic| can be worked out by trying the code above on a small example; if they work correctly in simple cases, they should work in general. @= m_magic:=m_min(cur_edges)+m_offset(cur_edges)-zero_field; n_magic:=8*n_max(cur_edges)+8+zero_w+min_halfword @ Now let's look at the subroutine that merges the edges from a given edge structure into |cur_edges|. The given edge structure loses all its edges. @p procedure merge_edges(@!h:pointer); label done; var @!p,@!q,@!r,@!pp,@!qq,@!rr:pointer; {list manipulation registers} @!n:integer; {row number} @!k:halfword; {key register that we compare to |info(q)|} @!delta:integer; {change to the edge/weight data} begin if link(h)<>h then begin if (m_min(h)m_max(cur_edges))or@| (n_min(h)n_max(cur_edges)) then edge_prep(m_min(h)-zero_field,m_max(h)-zero_field, n_min(h)-zero_field,n_max(h)-zero_field+1); if m_offset(h)<>m_offset(cur_edges) then @; n:=n_min(cur_edges); p:=link(cur_edges); pp:=link(h); while n; pp:=link(pp); p:=link(p); until pp=h; end; end; @ @= begin pp:=link(h); delta:=8*(m_offset(cur_edges)-m_offset(h)); repeat qq:=sorted(pp); while qq<>sentinel do begin info(qq):=info(qq)+delta; qq:=link(qq); end; qq:=unsorted(pp); while qq>void do begin info(qq):=info(qq)+delta; qq:=link(qq); end; pp:=link(pp); until pp=h; end @ The |sorted| and |unsorted| lists are merged separately. After this step, row~|pp| will have no edges remaining, since they will all have been merged into row~|p|. @= qq:=unsorted(pp); if qq>void then if unsorted(p)<=void then unsorted(p):=qq else begin while link(qq)>void do qq:=link(qq); link(qq):=unsorted(p); unsorted(p):=unsorted(pp); end; unsorted(pp):=null; qq:=sorted(pp); if qq<>sentinel then begin if unsorted(p)=void then unsorted(p):=null; sorted(pp):=sentinel; r:=sorted_loc(p); q:=link(r); {|q=sorted(p)|} if q=sentinel then sorted(p):=qq else loop@+begin k:=info(qq); while k>info(q) do begin r:=q; q:=link(r); end; link(r):=qq; rr:=link(qq); link(qq):=q; if rr=sentinel then goto done; r:=qq; qq:=rr; end; end; done: @ The |total_weight| routine computes the total of all pixel weights in a given edge structure. It's not difficult to prove that this is the sum of $(-w)$ times $x$ taken over all edges, where $w$ and~$x$ are the weight and $x$~coordinates stored in an edge. It's not necessary to worry that this quantity will overflow the size of an |integer| register, because it will be less than~$2^{31}$ unless the edge structure has more than 174,762 edges. However, we had better not try to compute it as a |scaled| integer, because a total weight of almost $12\times 2^{12}$ can be produced by only four edges. @p function total_weight(@!h:pointer):integer; {|h| is an edge header} var @!p,@!q:pointer; {variables that traverse the given structure} @!n:integer; {accumulated total so far} @!m:0..65535; {packed $x$ and $w$ values, including offsets} begin n:=0; p:=link(h); while p<>h do begin q:=sorted(p); while q<>sentinel do @; q:=unsorted(p); while q>void do @; p:=link(p); end; total_weight:=n; end; @ It's not necessary to add the offsets to the $x$ coordinates, because an entire edge structure can be shifted without affecting its total weight. Similarly, we don't need to subtract |zero_field|. @= begin m:=ho(info(q)); n:=n-((m mod 8)-zero_w)*(m div 8); q:=link(q); end @ So far we've done lots of things to edge structures assuming that edges are actually present, but we haven't seen how edges get created in the first place. Let's turn now to the problem of generating new edges. \MF\ will display new edges as they are being computed, if |tracing_edges| is positive. In order to keep such data reasonably compact, only the points at which the path makes a $90^\circ$ or $180^\circ$ turn are listed. The tracing algorithm must remember some past history in order to suppress unnecessary data. Three variables |trace_x|, |trace_y|, and |trace_yy| provide this history: The last coordinates printed were |(trace_x,trace_y)|, and the previous edge traced ended at |(trace_x,trace_yy)|. Before anything at all has been traced, |trace_x=-4096|. @= @!trace_x:integer; {$x$~coordinate most recently shown in a trace} @!trace_y:integer; {$y$~coordinate most recently shown in a trace} @!trace_yy:integer; {$y$~coordinate most recently encountered} @ Edge tracing is initiated by the |begin_edge_tracing| routine, continued by the |trace_a_corner| routine, and terminated by the |end_edge_tracing| routine. @p procedure begin_edge_tracing; begin print_diagnostic("Tracing edges","",true); print(" (weight "); print_int(cur_wt); print_char(")"); trace_x:=-4096; end; @# procedure trace_a_corner; begin if file_offset>max_print_line-13 then print_nl(""); print_char("("); print_int(trace_x); print_char(","); print_int(trace_yy); print_char(")"); trace_y:=trace_yy; end; @# procedure end_edge_tracing; begin if trace_x=-4096 then print_nl("(No new edges added.)") @.No new edges added@> else begin trace_a_corner; print_char("."); end; end_diagnostic(true); end; @ Just after a new edge weight has been put into the |info| field of node~|r|, in row~|n|, the following routine continues an ongoing trace. @p procedure trace_new_edge(@!r:pointer;@!n:integer); var @!d:integer; {temporary data register} @!w:-3..3; {weight associated with an edge transition} @!m,@!n0,@!n1:integer; {column and row numbers} begin d:=ho(info(r)); w:=(d mod 8)-zero_w; m:=(d div 8)-m_offset(cur_edges); if w=cur_wt then begin n0:=n+1; n1:=n; end else begin n0:=n; n1:=n+1; end; {the edges run from |(m,n0)| to |(m,n1)|} if m<>trace_x then begin if trace_x=-4096 then begin print_nl(""); trace_yy:=n0; end else if trace_yy<>n0 then print_char("?") {shouldn't happen} else trace_a_corner; trace_x:=m; trace_a_corner; end else begin if n0<>trace_yy then print_char("!"); {shouldn't happen} if ((n0trace_yy))or((n0>n1)and(trace_yn1 then begin m0:=round_unscaled(x0); m1:=round_unscaled(x1); delx:=x1-x0; dely:=y1-y0; yt:=n0*unity-half_unit; y0:=y0-yt; y1:=y1-yt; if n0 else @; n_rover(cur_edges):=p; n_pos(cur_edges):=n+zero_field; end; end; @ Here we are careful to cancel any effect of rounding error. @= begin base:=8*m_offset(cur_edges)+min_halfword+zero_w-cur_wt; if m0<=m1 then edge_prep(m0,m1,n0,n1)@+else edge_prep(m1,m0,n0,n1); @; y0:=unity-y0; loop@+ begin r:=get_avail; link(r):=unsorted(p); unsorted(p):=r;@/ tx:=take_fraction(delx,make_fraction(y0,dely)); if ab_vs_cd(delx,y0,dely,tx)<0 then decr(tx); {now $|tx|=\lfloor|y0|\cdot|delx|/|dely|\rfloor$} info(r):=8*round_unscaled(x0+tx)+base;@/ y1:=y1-unity; if internal[tracing_edges]>0 then trace_new_edge(r,n); if y1= begin base:=8*m_offset(cur_edges)+min_halfword+zero_w+cur_wt; if m0<=m1 then edge_prep(m0,m1,n1,n0)@+else edge_prep(m1,m0,n1,n0); decr(n0); @; loop@+ begin r:=get_avail; link(r):=unsorted(p); unsorted(p):=r;@/ tx:=take_fraction(delx,make_fraction(y0,dely)); if ab_vs_cd(delx,y0,dely,tx)<0 then incr(tx); {now $|tx|=\lceil|y0|\cdot|delx|/|dely|\rceil$, since |dely<0|} info(r):=8*round_unscaled(x0-tx)+base;@/ y1:=y1+unity; if internal[tracing_edges]>0 then trace_new_edge(r,n); if y1>=0 then goto done1; p:=knil(p); y0:=y0+unity; decr(n); end; done1: end @ @= n:=n_pos(cur_edges)-zero_field; p:=n_rover(cur_edges); if n<>n0 then if nm1-m0 then confusion("0");@+gubed@;@/ @:this can't happen 0}{\quad 0@> @; fast_case_up:@; fast_case_down:@; slow_case_up:@; slow_case_down:@; done: n_pos(cur_edges):=n+zero_field; n_rover(cur_edges):=p; end; @ The current octant code appears in a global variable. If, for example, |octant=third_octant|, it means that a curve traveling in a north to north-westerly direction has been rotated for the purposes of internal calculations so that the |move| data travels in an east to north-easterly direction. We want to update the edge structure as if the rotation had not taken place. @= @!octant:first_octant..sixth_octant; {the current octant of interest} @ @= case octant of first_octant:begin dx:=8; edge_prep(m0,m1,n0,n1); goto fast_case_up; end; second_octant:begin dx:=8; edge_prep(n0,n1,m0,m1); goto slow_case_up; end; third_octant:begin dx:=-8; edge_prep(-n1,-n0,m0,m1); negate(n0); goto slow_case_up; end; fourth_octant:begin dx:=-8; edge_prep(-m1,-m0,n0,n1); negate(m0); goto fast_case_up; end; fifth_octant:begin dx:=-8; edge_prep(-m1,-m0,-n1,-n0); negate(m0); goto fast_case_down; end; sixth_octant:begin dx:=-8; edge_prep(-n1,-n0,-m1,-m0); negate(n0); goto slow_case_down; end; seventh_octant:begin dx:=8; edge_prep(n0,n1,-m1,-m0); goto slow_case_down; end; eighth_octant:begin dx:=8; edge_prep(m0,m1,-n1,-n0); goto fast_case_down; end; end; {there are only eight octants} @ @= @; if delta>0 then begin k:=0; edge_and_weight:=8*(m0+m_offset(cur_edges))+min_halfword+zero_w-cur_wt; repeat edge_and_weight:=edge_and_weight+dx*move[k]; fast_get_avail(r); link(r):=unsorted(p); info(r):=edge_and_weight; if internal[tracing_edges]>0 then trace_new_edge(r,n); unsorted(p):=r; p:=link(p); incr(k); incr(n); until k=delta; end; goto done @ @= n0:=-n0-1; @; if delta>0 then begin k:=0; edge_and_weight:=8*(m0+m_offset(cur_edges))+min_halfword+zero_w+cur_wt; repeat edge_and_weight:=edge_and_weight+dx*move[k]; fast_get_avail(r); link(r):=unsorted(p); info(r):=edge_and_weight; if internal[tracing_edges]>0 then trace_new_edge(r,n); unsorted(p):=r; p:=knil(p); incr(k); decr(n); until k=delta; end; goto done @ @= edge_and_weight:=8*(n0+m_offset(cur_edges))+min_halfword+zero_w-cur_wt; n0:=m0; k:=0; @; repeat j:=move[k]; while j>0 do begin fast_get_avail(r); link(r):=unsorted(p); info(r):=edge_and_weight; if internal[tracing_edges]>0 then trace_new_edge(r,n); unsorted(p):=r; p:=link(p); decr(j); incr(n); end; edge_and_weight:=edge_and_weight+dx; incr(k); until k>delta; goto done @ @= edge_and_weight:=8*(n0+m_offset(cur_edges))+min_halfword+zero_w+cur_wt; n0:=-m0-1; k:=0; @; repeat j:=move[k]; while j>0 do begin fast_get_avail(r); link(r):=unsorted(p); info(r):=edge_and_weight; if internal[tracing_edges]>0 then trace_new_edge(r,n); unsorted(p):=r; p:=knil(p); decr(j); decr(n); end; edge_and_weight:=edge_and_weight+dx; incr(k); until k>delta; goto done @ All the hard work of building an edge structure is undone by the following subroutine. @= procedure toss_edges(@!h:pointer); var @!p,@!q:pointer; {for list manipulation} begin q:=link(h); while q<>h do begin flush_list(sorted(q)); if unsorted(q)>void then flush_list(unsorted(q)); p:=q; q:=link(q); free_node(p,row_node_size); end; free_node(h,edge_header_size); end; @* \[21] Subdivision into octants. When \MF\ digitizes a path, it reduces the problem to the special case of paths that travel in ``first octant'' directions; i.e., each cubic $z(t)=\bigl(x(t),y(t)\bigr)$ being digitized will have the property that $0\L y'(t)\L x'(t)$. This assumption makes digitizing simpler and faster than if the direction of motion has to be tested repeatedly. When $z(t)$ is cubic, $x'(t)$ and $y'(t)$ are quadratic, hence the four polynomials $x'(t)$, $y'(t)$, $x'(t)-y'(t)$, and $x'(t)+y'(t)$ cross through~0 at most twice each. If we subdivide the given cubic at these places, we get at most nine subintervals in each of which $x'(t)$, $y'(t)$, $x'(t)-y'(t)$, and $x'(t)+y'(t)$ all have a constant sign. The curve can be transformed in each of these subintervals so that it travels entirely in first octant directions, if we reflect $x\swap-x$, $y\swap-y$, and/or $x\swap y$ as necessary. (Incidentally, it can be shown that a cubic such that $x'(t)=16(2t-1)^2+2(2t-1)-1$ and $y'(t)=8(2t-1)^2+4(2t-1)$ does indeed split into nine subintervals.) @ The transformation that rotates coordinates, so that first octant motion can be assumed, is defined by the |skew| subroutine, which sets global variables |cur_x| and |cur_y| to the values that are appropriate in a given octant. (Octants are encoded as they were in the |n_arg| subroutine.) This transformation is ``skewed'' by replacing |(x,y)| by |(x-y,y)|, once first octant motion has been established. It turns out that skewed coordinates are somewhat better to work with when curves are actually digitized. @d set_two_end(#)==cur_y:=#;@+end @d set_two(#)==begin cur_x:=#; set_two_end @p procedure skew(@!x,@!y:scaled;@!octant:small_number); begin case octant of first_octant: set_two(x-y)(y); second_octant: set_two(y-x)(x); third_octant: set_two(y+x)(-x); fourth_octant: set_two(-x-y)(y); fifth_octant: set_two(-x+y)(-y); sixth_octant: set_two(-y+x)(-x); seventh_octant: set_two(-y-x)(x); eighth_octant: set_two(x+y)(-y); end; {there are no other cases} end; @ Conversely, the following subroutine sets |cur_x| and |cur_y| to the original coordinate values of a point, given an octant code and the point's coordinates |(x,y)| after they have been mapped into the first octant and skewed. @= procedure unskew(@!x,@!y:scaled;@!octant:small_number); begin case octant of first_octant: set_two(x+y)(y); second_octant: set_two(y)(x+y); third_octant: set_two(-y)(x+y); fourth_octant: set_two(-x-y)(y); fifth_octant: set_two(-x-y)(-y); sixth_octant: set_two(-y)(-x-y); seventh_octant: set_two(y)(-x-y); eighth_octant: set_two(x+y)(-y); end; {there are no other cases} end; @ @= @!cur_x,@!cur_y:scaled; {outputs of |rotate|, |unrotate|, and a few other routines} @ The conversion to skewed and rotated coordinates takes place in stages, and at one point in the transformation we will have negated the $x$ and/or $y$ coordinates so as to make curves travel in the first {\sl quadrant}. At this point the relevant ``octant'' code will be either |first_octant| (when no transformation has been done), or |fourth_octant=first_octant+negate_x| (when $x$ has been negated), or |fifth_octant=first_octant+negate_x+negate_y| (when both have been negated), or |eighth_octant=first_octant+negate_y| (when $y$ has been negated). The |abnegate| routine is sometimes needed to convert from one of these transformations to another. @p procedure abnegate(@!x,@!y:scaled; @!octant_before,@!octant_after:small_number); begin if odd(octant_before)=odd(octant_after) then cur_x:=x else cur_x:=-x; if (octant_before>negate_y)=(octant_after>negate_y) then cur_y:=y else cur_y:=-y; end; @ Now here's a subroutine that's handy for subdivision: Given a quadratic polynomial $B(a,b,c;t)$, the |crossing_point| function returns the unique |fraction| value |t| between 0 and~1 at which $B(a,b,c;t)$ changes from positive to negative, or returns |t=fraction_one+1| if no such value exists. If |a<0| (so that $B(a,b,c;t)$ is already negative at |t=0|), |crossing_point| returns zero. @d no_crossing==begin crossing_point:=fraction_one+1; return; end @d one_crossing==begin crossing_point:=fraction_one; return; end @d zero_crossing==begin crossing_point:=0; return; end @p function crossing_point(@!a,@!b,@!c:integer):fraction; label exit; var @!d:integer; {recursive counter} @!x,@!xx,@!x0,@!x1,@!x2:integer; {temporary registers for bisection} begin if a<0 then zero_crossing; if c>=0 then begin if b>=0 then if c>0 then no_crossing else if (a=0)and(b=0) then no_crossing else one_crossing; if a=0 then zero_crossing; end else if a=0 then if b<=0 then zero_crossing; @; exit:end; @ The general bisection method is quite simple when $n=2$, hence |crossing_point| does not take much time. At each stage in the recursion we have a subinterval defined by |l| and~|j| such that $B(a,b,c;2^{-l}(j+t))=B(x_0,x_1,x_2;t)$, and we want to ``zero in'' on the subinterval where $x_0\G0$ and $\min(x_1,x_2)<0$. It is convenient for purposes of calculation to combine the values of |l| and~|j| in a single variable $d=2^l+j$, because the operation of bisection then corresponds simply to doubling $d$ and possibly adding~1. Furthermore it proves to be convenient to modify our previous conventions for bisection slightly, maintaining the variables $X_0=2^lx_0$, $X_1=2^l(x_0-x_1)$, and $X_2=2^l(x_1-x_2)$. With these variables the conditions $x_0>=0$ and $\min(x_1,x_2)<0$ are equivalent to $\max(X_1,X_1+X_2)>X_0\G0$. The following code maintains the invariant relations $0\L|x0|<\max(|x1|,|x1|+|x2|)$, $\vert|x1|\vert<2^{30}$, $\vert|x2|\vert<2^{30}$; it has been constructed in such a way that no arithmetic overflow will occur if the inputs satisfy $a<2^{30}$, $\vert a-b\vert<2^{30}$, and $\vert b-c\vert<2^{30}$. @= d:=1; x0:=a; x1:=a-b; x2:=b-c; repeat x:=half(x1+x2); if x1-x0>x0 then begin x2:=x; double(x0); double(d); end else begin xx:=x1+x-x0; if xx>x0 then begin x2:=x; double(x0); double(d); end else begin x0:=x0-xx; if x<=x0 then if x+x2<=x0 then no_crossing; x1:=x; d:=d+d+1; end; end; until d>=fraction_one; crossing_point:=d-fraction_one @ Octant subdivision is applied only to cycles, i.e., to closed paths. A ``cycle spec'' is a data structure that contains specifications of @!@^cycle spec@> cubic curves and octant mappings for the cycle that has been subdivided into segments belonging to single octants. It is composed entirely of knot nodes, similar to those in the representation of paths; but the |explicit| type indications have been replaced by positive numbers that give further information. Additional |endpoint| data is also inserted at the octant boundaries. Recall that a cubic polynomial is represented by four control points that appear in adjacent nodes |p| and~|q| of a knot list. The |x|~coordinates are |x_coord(p)|, |right_x(p)|, |left_x(q)|, and |x_coord(q)|; the |y|~coordinates are similar. We shall call this ``the cubic following~|p|'' or ``the cubic between |p| and~|q|'' or ``the cubic preceding~|q|.'' Cycle specs are circular lists of cubic curves mixed with octant boundaries. Like cubics, the octant boundaries are represented in consecutive knot nodes |p| and~|q|. In such cases |right_type(p)= left_type(q)=endpoint|, and the fields |right_x(p)|, |right_y(p)|, |left_x(q)|, and |left_y(q)| are replaced by other fields called |right_octant(p)|, |right_transition(p)|, |left_octant(q)|, and |left_transition(q)|, respectively. For example, when the curve direction moves from the third octant to the fourth octant, the nodes say |right_octant(p)=third_octant|, |left_octant(q)=fourth_octant|, and |right_transition(p)=left_transition(q)=diagonal|. A |diagonal| transition occurs when moving between octants 1~\AM~2, 3~\AM~4, 5~\AM~6, or 7~\AM~8; an |axis| transition occurs when moving between octants 8~\AM~1, 2~\AM~3, 4~\AM~5, 6~\AM~7. (Such transition information is redundant but convenient.) Fields |x_coord(p)| and |y_coord(p)| will contain coordinates of the transition point after rotation from third octant to first octant; i.e., if the true coordinates are $(x,y)$, the coordinates $(y,\bar x)$ will appear in node~|p|. Similarly, a fourth-octant transformation will have been applied after the transition, so we will have |x_coord(q)=@t$\bar x$@>| and |y_coord(q)=y|. The cubic between |p| and |q| will contain positive numbers in the fields |right_type(p)| and |left_type(q)|; this makes cubics distinguishable from octant boundaries, because |endpoint=0|. The value of |right_type(p)| will be the current octant code, during the time that cycle specs are being constructed; it will refer later to a pen offset position, if the envelope of a cycle is being computed. A cubic that comes from some subinterval of the $k$th step in the original cyclic path will have |left_type(q)=k|. @d right_octant==right_x {the octant code before a transition} @d left_octant==left_x {the octant after a transition} @d right_transition==right_y {the type of transition} @d left_transition==left_y {ditto, either |axis| or |diagonal|} @d axis=0 {a transition across the $x'$- or $y'$-axis} @d diagonal=1 {a transition where $y'=\pm x'$} @ Here's a routine that prints a cycle spec in symbolic form, so that it is possible to see what subdivision has been made. The point coordinates are converted back from \MF's internal ``rotated'' form to the external ``true'' form. The global variable~|cur_spec| should point to a knot just after the beginning of an octant boundary, i.e., such that |left_type(cur_spec)=endpoint|. @d print_two_true(#)==unskew(#,octant); print_two(cur_x,cur_y) @p procedure print_spec(@!s:str_number); label not_found,done; var @!p,@!q:pointer; {for list traversal} @!octant:small_number; {the current octant code} begin print_diagnostic("Cycle spec",s,true); @.Cycle spec at line...@> p:=cur_spec; octant:=left_octant(p); print_ln; print_two_true(x_coord(cur_spec),y_coord(cur_spec)); print(" % beginning in octant `"); loop@+ begin print(octant_dir[octant]); print_char("'"); loop@+ begin q:=link(p); if right_type(p)=endpoint then goto not_found; @; p:=q; end; not_found: if q=cur_spec then goto done; p:=q; octant:=left_octant(p); print_nl("% entering octant `"); end; @.entering the nth octant@> done: print_nl(" & cycle"); end_diagnostic(true); end; @ Symbolic octant direction names are kept in the |octant_dir| array. @= @!octant_dir:array[first_octant..sixth_octant] of str_number; @ @= octant_dir[first_octant]:="ENE"; octant_dir[second_octant]:="NNE"; octant_dir[third_octant]:="NNW"; octant_dir[fourth_octant]:="WNW"; octant_dir[fifth_octant]:="WSW"; octant_dir[sixth_octant]:="SSW"; octant_dir[seventh_octant]:="SSE"; octant_dir[eighth_octant]:="ESE"; @ @= begin print_nl(" ..controls "); print_two_true(right_x(p),right_y(p)); print(" and "); print_two_true(left_x(q),left_y(q)); print_nl(" .."); print_two_true(x_coord(q),y_coord(q)); print(" % segment "); print_int(left_type(q)-1); end @ A much more compact version of a spec is printed to help users identify ``strange paths.'' @p procedure print_strange(@!s:str_number); var @!p:pointer; {for list traversal} @!f:pointer; {starting point in the cycle} @!q:pointer; {octant boundary to be printed} @!t:integer; {segment number, plus 1} begin if interaction=error_stop_mode then wake_up_terminal; print_nl(">"); @.>\relax@> @; @; t:=0; repeat if left_type(p)<>endpoint then begin if left_type(p)<>t then begin t:=left_type(p); print_char(" "); print_int(t-1); end; if q<>null then begin @; print_char(" "); print(octant_dir[left_octant(q)]); q:=null; end; end else if q=null then q:=p; p:=link(p); until p=f; print_char(" "); print_int(left_type(p)-1); if q<>null then @; print_err(s); end; @ If the segment numbers on the cycle are $t_1$, $t_2$, \dots, $t_m$, we have $t_{k-1}\L t_k$ except for at most one value of~$k$. If there are no exceptions, $f$ will point to $t_1$; otherwise it will point to the exceptional~$t_k$. There is at least one segment number (i.e., we always have $m>0$), because |print_strange| is never called upon to display an entirely ``dead'' cycle. @= p:=cur_spec; t:=max_quarterword+1; repeat p:=link(p); if left_type(p)<>endpoint then begin if left_type(p)= p:=cur_spec; q:=p; repeat p:=link(p); if left_type(p)=endpoint then q:=p; until p=f @ When two octant boundaries are adjacent, the path is simply changing direction without moving. Such octant directions are shown in parentheses. @= if left_type(link(q))=endpoint then begin print(" ("); print(octant_dir[left_octant(q)]); q:=link(q); while left_type(link(q))=endpoint do begin print_char(" "); print(octant_dir[left_octant(q)]); q:=link(q); end; print_char(")"); end @ The |make_spec| routine is what subdivides paths into octants: Given a pointer |cur_spec| to a cyclic path, |make_spec| mungs the path data and returns a pointer to the corresponding cyclic spec. All ``dead'' cubics (i.e., cubics that don't move at all from their starting points) will have been removed from the result. @!@^dead cubics@> The idea of |make_spec| is fairly simple: Each cubic is first subdivided, if necessary, into pieces belonging to single octants; then the octant boundaries are inserted. But some of the details of this transformation are not quite obvious. If |autorounding>0|, the path will be adjusted so that critical tangent directions occur at ``good'' points with respect to the pen called |cur_pen|. The result spec will have all scaled |x| and |y| coordinates at most $2^{28}-|half_unit|-1-|safety_margin|$ in absolute value. The pointer that is returned will start some octant, as required by |print_spec|. @p @t\4@>@@; function make_spec(@!h:pointer; @!safety_margin:scaled;@!tracing:integer):pointer; {converts a path to a cycle spec} label continue,done; var @!p,@!q,@!r,@!s:pointer; {for traversing the lists} @!k:integer; {serial number of path segment, or octant code} @!chopped:boolean; {have we truncated any of the data?} @@; begin cur_spec:=h; if tracing>0 then print_path(cur_spec,", before subdivision into octants",true); max_allowed:=fraction_one-half_unit-1-safety_margin; @; quadrant_subdivide; {subdivide each cubic into pieces belonging to quadrants} if internal[autorounding]>0 then xy_round; octant_subdivide; {complete the subdivision} if internal[autorounding]>unity then diag_round; @; @; while left_type(cur_spec)<>endpoint do cur_spec:=link(cur_spec); if tracing>0 then if internal[autorounding]<=0 then print_spec(", after subdivision") else if internal[autorounding]>unity then print_spec(", after subdivision and double autorounding") else print_spec(", after subdivision and autorounding"); make_spec:=cur_spec; end; @ The |make_spec| routine has an interesting side effect, namely to set the global variable |turning_number| to the number of times the tangent vector of the given cyclic path winds around the origin. Another global variable |cur_spec| points to the specification as it is being made, since several subroutines must go to work on it. And there are two global variables that affect the rounding decisions, as we'll see later: |cur_pen| and |cur_path_type|. The latter will be |double_path_code| if |make_spec| is being applied to a double path. @d double_path_code=0 {command modifier for `\&{doublepath}'} @d contour_code=1 {command modifier for `\&{contour}'} @d also_code=2 {command modifier for `\&{also}'} @= @!cur_spec:pointer; {the principal output of |make_spec|} @!turning_number:integer; {another output of |make_spec|} @!cur_pen:pointer; {an implicit input of |make_spec|, used in autorounding} @!cur_path_type:double_path_code..also_code; {likewise} @!max_allowed:scaled; {coordinates must be at most this big} @ First we do a simple preprocessing step. The segment numbers inserted here will propagate to all descendants of cubics that are split into subintervals. These numbers must be nonzero, but otherwise they are present merely for diagnostic purposes. The cubic from |p| to~|q| that represents ``time interval'' |(t-1)..t| usually has |right_type(q)=t|, except when |t| is too large to be stored in a quarterword. @d procrustes(#)==if abs(#)>max_allowed then begin chopped:=true; if #>0 then #:=max_allowed@+else #:=-max_allowed; end @= p:=cur_spec; k:=1; chopped:=false; repeat procrustes(left_x(p)); procrustes(left_y(p)); procrustes(x_coord(p)); procrustes(y_coord(p)); procrustes(right_x(p)); procrustes(right_y(p));@/ p:=link(p); left_type(p):=k; if k help4("At least one of the coordinates in the path I'm about to")@/ ("digitize was really huge (potentially bigger than 4095).")@/ ("So I've cut it back to the maximum size.")@/ ("The results will probably be pretty wild."); put_get_error; end @ We may need to get rid of constant ``dead'' cubics that clutter up the data structure and interfere with autorounding. @= procedure remove_cubic(@!p:pointer); {removes the cubic following~|p|} var @!q:pointer; {the node that disappears} begin q:=link(p); right_type(p):=right_type(q); link(p):=link(q);@/ x_coord(p):=x_coord(q); y_coord(p):=y_coord(q);@/ right_x(p):=right_x(q); right_y(p):=right_y(q);@/ free_node(q,knot_node_size); end; @ The subdivision process proceeds by first swapping $x\swap-x$, if necessary, to ensure that $x'\G0$; then swapping $y\swap-y$, if necessary, to ensure that $y'\G0$; and finally swapping $x\swap y$, if necessary, to ensure that $x'\G y'$. Recall that the octant codes have been defined in such a way that, for example, |third_octant=first_octant+negate_x+switch_x_and_y|. The program uses the fact that |negate_xnegate_y|. Octant codes are always positive. The first step is to subdivide on |x| and |y| only, so that horizontal and vertical autorounding can be done before we compare $x'$ to $y'$. @= @t\4@>@@; procedure quadrant_subdivide; label continue,exit; var @!p,@!q,@!r,@!s,@!pp,@!qq:pointer; {for traversing the lists} @!first_x,@!first_y:scaled; {unnegated coordinates of node |cur_spec|} @!del1,@!del2,@!del3,@!del,@!dmax:scaled; {proportional to the control points of a quadratic derived from a cubic} @!t:fraction; {where a quadratic crosses zero} @!dest_x,@!dest_y:scaled; {final values of |x| and |y| in the current cubic} @!constant_x:boolean; {is |x| constant between |p| and |q|?} begin p:=cur_spec; first_x:=x_coord(cur_spec); first_y:=y_coord(cur_spec); repeat continue: q:=link(p); @; @; p:=q; until p=cur_spec; exit:end; @ All three subdivision processes are similar, so it's possible to get the general idea by studying the first one (which is the simplest). The calculation makes use of the fact that the derivatives of Bernshte{\u\i}n polynomials satisfy $B'(z_0,z_1,\ldots,z_n;t)=nB(z_1-z_0,\ldots,z_n-z_{n-1};t)$. At the beginning of this step, |right_type(p)=explicit|; we should set |right_type(p):=first_octant|. However, no assignment is made, because |explicit=first_octant|. The author apologizes for using such trickery here; it is really hard to do redundant computations just for the sake of purity. @= if q=cur_spec then begin dest_x:=first_x; dest_y:=first_y; end else begin dest_x:=x_coord(q); dest_y:=y_coord(q); end; del1:=right_x(p)-x_coord(p); del2:=left_x(q)-right_x(p); del3:=dest_x-left_x(q); @; if del=0 then constant_x:=true else begin constant_x:=false; if del<0 then @; t:=crossing_point(del1,del2,del3); if t; end @ If |del1=del2=del3=0|, it's impossible to obey the title of this section. We just set |del=0| in that case. @^inner loop@> @= if del1<>0 then del:=del1 else if del2<>0 then del:=del2 else del:=del3; if del<>0 then begin dmax:=abs(del1); if abs(del2)>dmax then dmax:=abs(del2); if abs(del3)>dmax then dmax:=abs(del3); while dmax= begin negate(x_coord(p)); negate(right_x(p)); negate(left_x(q));@/ negate(del1); negate(del2); negate(del3);@/ negate(dest_x); right_type(p):=first_octant+negate_x; end @ When a cubic is split at a |fraction| value |t|, we obtain two cubics whose B\'ezier control points are obtained by a generalization of the bisection process: The formula `$z_k^{(j+1)}={1\over2}(z_k^{(j)}+z\k^{(j)})$' becomes `$z_k^{(j+1)}=t[z_k^{(j)},z\k^{(j)}]$'. It is convenient to define a \.{WEB} macro |t_of_the_way| such that |t_of_the_way(a)(b)| expands to |a-(a-b)*t|, i.e., to |t[a,b]|. @d t_of_the_way_end(#)==#,t@=)@> @d t_of_the_way(#)==#-take_fraction@=(@>#-t_of_the_way_end @= procedure split_cubic(@!p:pointer;@!t:fraction; @!xq,@!yq:scaled); {splits the cubic after |p|} var @!v:scaled; {an intermediate value} @!q,@!r:pointer; {for list manipulation} begin q:=link(p); r:=get_node(knot_node_size); link(p):=r; link(r):=q;@/ left_type(r):=left_type(q); right_type(r):=right_type(p);@# v:=t_of_the_way(right_x(p))(left_x(q)); right_x(p):=t_of_the_way(x_coord(p))(right_x(p)); left_x(q):=t_of_the_way(left_x(q))(xq); left_x(r):=t_of_the_way(right_x(p))(v); right_x(r):=t_of_the_way(v)(left_x(q)); x_coord(r):=t_of_the_way(left_x(r))(right_x(r));@# v:=t_of_the_way(right_y(p))(left_y(q)); right_y(p):=t_of_the_way(y_coord(p))(right_y(p)); left_y(q):=t_of_the_way(left_y(q))(yq); left_y(r):=t_of_the_way(right_y(p))(v); right_y(r):=t_of_the_way(v)(left_y(q)); y_coord(r):=t_of_the_way(left_y(r))(right_y(r)); end; @ Since $x'(t)$ is a quadratic equation, it can cross through zero at~most twice. When it does cross zero, we make doubly sure that the derivative is really zero at the splitting point, in case rounding errors have caused the split cubic to have an apparently nonzero derivative. @= begin split_cubic(p,t,dest_x,dest_y); r:=link(p); if right_type(r)>negate_x then right_type(r):=first_octant else right_type(r):=first_octant+negate_x; if x_coord(r)0 then del2:=0; t:=crossing_point(0,-del2,-del3); if t else if x_coord(r)>dest_x then x_coord(r):=dest_x; end @ @= begin split_cubic(r,t,dest_x,dest_y); s:=link(r); if x_coord(s)= pp:=p; repeat qq:=link(pp); abnegate(x_coord(qq),y_coord(qq),right_type(qq),right_type(pp)); dest_x:=cur_x; dest_y:=cur_y;@/ del1:=right_y(pp)-y_coord(pp); del2:=left_y(qq)-right_y(pp); del3:=dest_y-left_y(qq); @; if del<>0 then {they weren't all zero} begin if del<0 then @; t:=crossing_point(del1,del2,del3); if t; end else @; pp:=qq; until pp=q; if constant_x then @ @ @= begin negate(y_coord(pp)); negate(right_y(pp)); negate(left_y(qq));@/ negate(del1); negate(del2); negate(del3);@/ negate(dest_y); right_type(pp):=right_type(pp)+negate_y; end @ @= begin split_cubic(pp,t,dest_x,dest_y); r:=link(pp); if right_type(r)>negate_y then right_type(r):=right_type(r)-negate_y else right_type(r):=right_type(r)+negate_y; if x_coord(r)>dest_x then x_coord(r):=dest_x else if x_coord(r)0 then del2:=0; t:=crossing_point(0,-del2,-del3); if t else if y_coord(r)>dest_y then y_coord(r):=dest_y; end @ @= begin split_cubic(r,t,dest_x,dest_y); s:=link(r);@/ if x_coord(s)>dest_x then x_coord(s):=dest_x else if x_coord(s)= if constant_x then {|p=pp|, |q=qq|, and the cubic is dead} begin if q<>p then begin remove_cubic(p); {remove the dead cycle and recycle node |q|} if cur_spec<>q then goto continue else begin cur_spec:=p; return; end; {the final cubic was dead and is gone} end; end else if not odd(right_type(pp)) then {the $x$ coordinates were negated} @ @ A similar correction to octant codes deserves to be made when |x| is constant and |y| is decreasing. @= begin pp:=p; repeat qq:=link(pp); if right_type(pp)>negate_y then {the $y$ coordinates were negated} begin right_type(pp):=right_type(pp)+negate_x; negate(x_coord(pp)); negate(right_x(pp)); negate(left_x(qq)); end; pp:=qq; until pp=q; end @ Finally, the process of subdividing to make $x'\G y'$ is like the other two subdivisions, with a few new twists. We skew the coordinates at this time. @= procedure octant_subdivide; var @!p,@!q,@!r,@!s:pointer; {for traversing the lists} @!del1,@!del2,@!del3,@!del,@!dmax:scaled; {proportional to the control points of a quadratic derived from a cubic} @!t:fraction; {where a quadratic crosses zero} @!dest_x,@!dest_y:scaled; {final values of |x| and |y| in the current cubic} begin p:=cur_spec; repeat q:=link(p);@/ x_coord(p):=x_coord(p)-y_coord(p); right_x(p):=right_x(p)-right_y(p); left_x(q):=left_x(q)-left_y(q);@/ @; p:=q; until p=cur_spec; end; @ @= @; @; if del<>0 then {they weren't all zero} begin if del<0 then @; t:=crossing_point(del1,del2,del3); if t; end @ @= if q=cur_spec then begin unskew(x_coord(q),y_coord(q),right_type(q)); skew(cur_x,cur_y,right_type(p)); dest_x:=cur_x; dest_y:=cur_y; end else begin abnegate(x_coord(q),y_coord(q),right_type(q),right_type(p)); dest_x:=cur_x-cur_y; dest_y:=cur_y; end; del1:=right_x(p)-x_coord(p); del2:=left_x(q)-right_x(p); del3:=dest_x-left_x(q) @ The swapping here doesn't simply interchange |x| and |y| values, because the coordinates are skewed. It turns out that this is easier than ordinary swapping, because it can be done in two assignment statements rather than three. @ @= begin y_coord(p):=x_coord(p)+y_coord(p); negate(x_coord(p));@/ right_y(p):=right_x(p)+right_y(p); negate(right_x(p));@/ left_y(q):=left_x(q)+left_y(q); negate(left_x(q));@/ negate(del1); negate(del2); negate(del3);@/ dest_y:=dest_x+dest_y; negate(dest_x);@/ right_type(p):=right_type(p)+switch_x_and_y; end @ @= begin split_cubic(p,t,dest_x,dest_y); r:=link(p); if right_type(r)>switch_x_and_y then right_type(r):=right_type(r)-switch_x_and_y else right_type(r):=right_type(r)+switch_x_and_y; if y_coord(r)>dest_y then y_coord(r):=dest_y else if y_coord(r)0 then del2:=0; t:=crossing_point(0,-del2,-del3); if t else if x_coord(r)>dest_x then x_coord(r):=dest_x end @ @= begin split_cubic(r,t,dest_x,dest_y); s:=link(r);@/ if y_coord(s)>dest_y then y_coord(s):=dest_y else if y_coord(s)= procedure make_safe; var @!k:0..max_wiggle; {runs through the list of inputs} @!all_safe:boolean; {does everything look OK so far?} @!next_a:scaled; {|after[k]| before it might have changed} @!delta_a,@!delta_b:scaled; {|after[k+1]-after[k]| and |before[k+1]-before[k]|} begin before[cur_rounding_ptr]:=before[0]; {wrap around} node_to_round[cur_rounding_ptr]:=node_to_round[0]; repeat after[cur_rounding_ptr]:=after[0]; all_safe:=true; next_a:=after[0]; for k:=0 to cur_rounding_ptr-1 do begin delta_b:=before[k+1]-before[k]; if delta_b>=0 then delta_a:=after[k+1]-next_a else delta_a:=next_a-after[k+1]; next_a:=after[k+1]; if (delta_a<0)or(delta_a>abs(delta_b+delta_b)) then begin all_safe:=false; after[k]:=before[k]; if k=cur_rounding_ptr-1 then after[0]:=before[0] else after[k+1]:=before[k+1]; end; end; until all_safe; end; @ The global arrays used by |make_safe| are accompanied by an array of pointers into the current knot list. @= @!before,@!after:array[0..max_wiggle] of scaled; {data for |make_safe|} @!node_to_round:array[0..max_wiggle] of pointer; {reference back to the path} @!cur_rounding_ptr:0..max_wiggle; {how many are being used} @!max_rounding_ptr:0..max_wiggle; {how many have been used} @ @= max_rounding_ptr:=0; @ New entries go into the tables via the |before_and_after| routine: @= procedure before_and_after(@!b,@!a:scaled;@!p:pointer); begin if cur_rounding_ptr=max_rounding_ptr then if max_rounding_ptr after[cur_rounding_ptr]:=a; before[cur_rounding_ptr]:=b; node_to_round[cur_rounding_ptr]:=p; incr(cur_rounding_ptr); end; @ A global variable called |cur_gran| is used instead of |internal[ granularity]|, because we want to work with a number that's guaranteed to be positive. @= @!cur_gran:scaled; {the current granularity (which normally is |unity|)} @ The |good_val| function computes a number |a| that's as close as possible to~|b|, with the property that |a+o| is a multiple of |cur_gran|. If we assume that |cur_gran| is even (since it will in fact be a multiple of |unity| in all reasonable applications), we have the identity |good_val(-b-1,-o)=-good_val(b,o)|. @= function good_val(@!b,@!o:scaled):scaled; var @!a:scaled; {accumulator} begin a:=b+o; if a>=0 then a:=a-(a mod cur_gran)-o else a:=a+((-(a+1)) mod cur_gran)-cur_gran+1-o; if b-a modulo |cur_gran| is relevant in our applications, so the result turns out to be essentially symmetric in |u| and~|v|.) @= function compromise(@!u,@!v:scaled):scaled; begin compromise:=half(good_val(u+u,-u-v)); end; @ Here, then, is the procedure that rounds $x$ coordinates as described; it does the same for $y$ coordinates too, independently. @= procedure xy_round; var @!p,@!q:pointer; {list manipulation registers} @!b,@!a:scaled; {before and after values} @!pen_edge:scaled; {offset that governs rounding} @!alpha:fraction; {coefficient of linear transformation} begin cur_gran:=abs(internal[granularity]); if cur_gran=0 then cur_gran:=unity; p:=cur_spec; cur_rounding_ptr:=0; repeat q:=link(p); @; p:=q; until p=cur_spec; if cur_rounding_ptr>0 then @; p:=cur_spec; cur_rounding_ptr:=0; repeat q:=link(p); @; p:=q; until p=cur_spec; if cur_rounding_ptr>0 then @; end; @ When |x| has been negated, the |octant| codes are even. We allow for an error of up to .01 pixel (i.e., 655 |scaled| units) in the derivative calculations at transition nodes. @= if odd(right_type(p))<>odd(right_type(q)) then begin if odd(right_type(q)) then b:=x_coord(q)@+else b:=-x_coord(q); if (abs(x_coord(q)-right_x(q))<655)or@| (abs(x_coord(q)+left_x(q))<655) then @ else a:=b; if abs(a)>max_allowed then if a>0 then a:=max_allowed@+else a:=-max_allowed; before_and_after(b,a,q); end @ When we study the data representation for pens, we'll learn that the |x|~coordinate of the current pen's west edge is $$\hbox{|y_coord(link(cur_pen+seventh_octant))|},$$ and that there are similar ways to address other important offsets. An ``|east_west_edge|'' is computed as a compromise between east and west, for use in doublepaths, in case the two edges have conflicting tendencies. @d north_edge(#)==y_coord(link(#+fourth_octant)) @d south_edge(#)==y_coord(link(#+first_octant)) @d east_edge(#)==y_coord(link(#+second_octant)) @d west_edge(#)==y_coord(link(#+seventh_octant)) @d north_south_edge(#)==mem[#+10].int {compromise between north and south} @d east_west_edge(#)==mem[#+11].int {compromise between east and west} @d NE_SW_edge(#)==mem[#+12].int {compromise between northeast and southwest} @d NW_SE_edge(#)==mem[#+13].int {compromise between northwest and southeast} @= begin if cur_pen=null_pen then pen_edge:=0 else if cur_path_type=double_path_code then pen_edge:=compromise(east_edge(cur_pen),west_edge(cur_pen)) else if odd(right_type(q)) then pen_edge:=west_edge(cur_pen) else pen_edge:=east_edge(cur_pen); a:=good_val(b,pen_edge); end @ @= begin make_safe; repeat decr(cur_rounding_ptr); if (after[cur_rounding_ptr]<>before[cur_rounding_ptr])or@| (after[cur_rounding_ptr+1]<>before[cur_rounding_ptr+1]) then begin p:=node_to_round[cur_rounding_ptr]; if odd(right_type(p)) then begin b:=before[cur_rounding_ptr]; a:=after[cur_rounding_ptr]; end else begin b:=-before[cur_rounding_ptr]; a:=-after[cur_rounding_ptr]; end; if before[cur_rounding_ptr]=before[cur_rounding_ptr+1] then alpha:=fraction_one else alpha:=make_fraction(after[cur_rounding_ptr+1]-after[cur_rounding_ptr],@| before[cur_rounding_ptr+1]-before[cur_rounding_ptr]); repeat x_coord(p):=take_fraction(alpha,x_coord(p)-b)+a; right_x(p):=take_fraction(alpha,right_x(p)-b)+a; p:=link(p); left_x(p):=take_fraction(alpha,left_x(p)-b)+a; until p=node_to_round[cur_rounding_ptr+1]; end; until cur_rounding_ptr=0; end @ When |y| has been negated, the |octant| codes are |>negate_y|. Otherwise these routines are essentially identical to the routines for |x| coordinates that we have just seen. @= if (right_type(p)>negate_y)<>(right_type(q)>negate_y) then begin if right_type(q)<=negate_y then b:=y_coord(q)@+else b:=-y_coord(q); if (abs(y_coord(q)-right_y(q))<655)or@| (abs(y_coord(q)+left_y(q))<655) then @ else a:=b; if abs(a)>max_allowed then if a>0 then a:=max_allowed@+else a:=-max_allowed; before_and_after(b,a,q); end @ @= begin if cur_pen=null_pen then pen_edge:=0 else if cur_path_type=double_path_code then pen_edge:=compromise(north_edge(cur_pen),south_edge(cur_pen)) else if right_type(q)<=negate_y then pen_edge:=south_edge(cur_pen) else pen_edge:=north_edge(cur_pen); a:=good_val(b,pen_edge); end @ @= begin make_safe; repeat decr(cur_rounding_ptr); if (after[cur_rounding_ptr]<>before[cur_rounding_ptr])or@| (after[cur_rounding_ptr+1]<>before[cur_rounding_ptr+1]) then begin p:=node_to_round[cur_rounding_ptr]; if right_type(p)<=negate_y then begin b:=before[cur_rounding_ptr]; a:=after[cur_rounding_ptr]; end else begin b:=-before[cur_rounding_ptr]; a:=-after[cur_rounding_ptr]; end; if before[cur_rounding_ptr]=before[cur_rounding_ptr+1] then alpha:=fraction_one else alpha:=make_fraction(after[cur_rounding_ptr+1]-after[cur_rounding_ptr],@| before[cur_rounding_ptr+1]-before[cur_rounding_ptr]); repeat y_coord(p):=take_fraction(alpha,y_coord(p)-b)+a; right_y(p):=take_fraction(alpha,right_y(p)-b)+a; p:=link(p); left_y(p):=take_fraction(alpha,left_y(p)-b)+a; until p=node_to_round[cur_rounding_ptr+1]; end; until cur_rounding_ptr=0; end @ Rounding at diagonal tangents takes place after the subdivision into octants is complete, hence after the coordinates have been skewed. The details are somewhat tricky, because we want to round to points whose skewed coordinates are halfway between integer multiples of the granularity. Furthermore, both coordinates change when they are rounded; this means we need a generalization of the |make_safe| routine, ensuring safety in both |x| and |y|. In spite of these extra complications, we can take comfort in the fact that the basic structure of the routine is the same as before. @= procedure diag_round; var @!p,@!q,@!pp:pointer; {list manipulation registers} @!b,@!a,@!bb,@!aa,@!d,@!c,@!dd,@!cc:scaled; {before and after values} @!pen_edge:scaled; {offset that governs rounding} @!alpha,@!beta:fraction; {coefficients of linear transformation} @!next_a:scaled; {|after[k]| before it might have changed} @!all_safe:boolean; {does everything look OK so far?} @!k:0..max_wiggle; {runs through before-and-after values} @!first_x,@!first_y:scaled; {coordinates before rounding} begin p:=cur_spec; cur_rounding_ptr:=0; repeat q:=link(p); @; p:=q; until p=cur_spec; if cur_rounding_ptr>0 then @; end; @ We negate the skewed |x| coordinates in the before-and-after table when the octant code is greater than |switch_x_and_y|. @= if right_type(p)<>right_type(q) then begin if right_type(q)>switch_x_and_y then b:=-x_coord(q) else b:=x_coord(q); if abs(right_type(q)-right_type(p))=switch_x_and_y then if (abs(x_coord(q)-right_x(q))<655)or(abs(x_coord(q)+left_x(q))<655) then @ else a:=b else a:=b; before_and_after(b,a,q); end @ In octants whose code number is even, $x$~has been negated; we want to round ambiguous cases downward instead of upward, so that the rounding will be consistent with octants whose code number is odd. This downward bias can be achieved by subtracting~1 from the first argument of |good_val|. @d diag_offset(#)==x_coord(knil(link(cur_pen+#))) @= begin if cur_pen=null_pen then pen_edge:=0 else if cur_path_type=double_path_code then @ else if right_type(q)<=switch_x_and_y then pen_edge:=diag_offset(right_type(q)) else pen_edge:=-diag_offset(right_type(q)); if odd(right_type(q)) then a:=good_val(b,pen_edge+half(cur_gran)) else a:=good_val(b-1,pen_edge+half(cur_gran)); end @ (It seems a shame to compute these compromise offsets repeatedly. The author would have stored them directly in the pen data structure, if the granularity had been constant.) @= case right_type(q) of first_octant,second_octant:pen_edge:=compromise(diag_offset(first_octant),@| -diag_offset(fifth_octant)); fifth_octant,sixth_octant:pen_edge:=-compromise(diag_offset(first_octant),@| -diag_offset(fifth_octant)); third_octant,fourth_octant:pen_edge:=compromise(diag_offset(fourth_octant),@| -diag_offset(eighth_octant)); seventh_octant,eighth_octant:pen_edge:=-compromise(diag_offset(fourth_octant),@| -diag_offset(eighth_octant)); end {there are no other cases} @ @= begin p:=node_to_round[0]; first_x:=x_coord(p); first_y:=y_coord(p); @; for k:=0 to cur_rounding_ptr-1 do begin a:=after[k]; b:=before[k]; aa:=after[k+1]; bb:=before[k+1]; if (a<>b)or(aa<>bb) then begin p:=node_to_round[k]; pp:=node_to_round[k+1]; @; if b=bb then alpha:=fraction_one else alpha:=make_fraction(aa-a,bb-b); if d=dd then beta:=fraction_one else beta:=make_fraction(cc-c,dd-d); repeat x_coord(p):=take_fraction(alpha,x_coord(p)-b)+a; y_coord(p):=take_fraction(beta,y_coord(p)-d)+c; right_x(p):=take_fraction(alpha,right_x(p)-b)+a; right_y(p):=take_fraction(beta,right_y(p)-d)+c; p:=link(p); left_x(p):=take_fraction(alpha,left_x(p)-b)+a; left_y(p):=take_fraction(beta,left_y(p)-d)+c; until p=pp; end; end; end @ In node |p|, the coordinates |(b,d)| will be rounded to |(a,c)|; in node |pp|, the coordinates |(bb,dd)| will be rounded to |(aa,cc)|. (We transform the values from node |pp| so that they agree with the conventions of node |p|.) If |aa<>bb|, we know that |abs(right_type(p)-right_type(pp))=switch_x_and_y|. @= if aa=bb then begin if pp=node_to_round[0] then unskew(first_x,first_y,right_type(pp)) else unskew(x_coord(pp),y_coord(pp),right_type(pp)); skew(cur_x,cur_y,right_type(p)); bb:=cur_x; aa:=bb; dd:=cur_y; cc:=dd; if right_type(p)>switch_x_and_y then begin b:=-b; a:=-a; end; end else begin if right_type(p)>switch_x_and_y then begin bb:=-bb; aa:=-aa; b:=-b; a:=-a; end; if pp=node_to_round[0] then dd:=first_y-bb@+else dd:=y_coord(pp)-bb; if odd(aa-bb) then if right_type(p)>switch_x_and_y then cc:=dd-half(aa-bb+1) else cc:=dd-half(aa-bb-1) else cc:=dd-half(aa-bb); end; d:=y_coord(p); if odd(a-b) then if right_type(p)>switch_x_and_y then c:=d-half(a-b-1) else c:=d-half(a-b+1) else c:=d-half(a-b) @ @= before[cur_rounding_ptr]:=before[0]; {cf.~|make_safe|} node_to_round[cur_rounding_ptr]:=node_to_round[0]; repeat after[cur_rounding_ptr]:=after[0]; all_safe:=true; next_a:=after[0]; for k:=0 to cur_rounding_ptr-1 do begin a:=next_a; b:=before[k]; next_a:=after[k+1]; aa:=next_a; bb:=before[k+1]; if (a<>b)or(aa<>bb) then begin p:=node_to_round[k]; pp:=node_to_round[k+1]; @; if (aa2*(bb-b))or(cc-c>2*(dd-d)) then begin all_safe:=false; after[k]:=before[k]; if k=cur_rounding_ptr-1 then after[0]:=before[0] else after[k+1]:=before[k+1]; end; end; end; until all_safe @ Here we get rid of ``dead'' cubics, i.e., polynomials that don't move at all when |t|~changes, since the subdivision process might have introduced such things. If the cycle reduces to a single point, however, we are left with a single dead cubic that will not be removed until later. @= p:=cur_spec; repeat continue: q:=link(p); if p<>q then begin if x_coord(p)=right_x(p) then if y_coord(p)=right_y(p) then if x_coord(p)=left_x(q) then if y_coord(p)=left_y(q) then begin unskew(x_coord(q),y_coord(q),right_type(q)); skew(cur_x,cur_y,right_type(p)); if x_coord(p)=cur_x then if y_coord(p)=cur_y then begin remove_cubic(p); {remove the cubic following |p|} if q<>cur_spec then goto continue; cur_spec:=p; q:=p; end; end; end; p:=q; until p=cur_spec; @ Finally we come to the last steps of |make_spec|, when boundary nodes are inserted between cubics that move in different octants. The main complication remaining arises from consecutive cubics whose octants are not adjacent; we should insert more than one octant boundary at such sharp turns, so that the envelope-forming routine will work. For this purpose, conversion tables between numeric and Gray codes for octants are desirable. @= @!octant_number:array[first_octant..sixth_octant] of 1..8; @!octant_code:array[1..8] of first_octant..sixth_octant; @ @= octant_code[1]:=first_octant; octant_code[2]:=second_octant; octant_code[3]:=third_octant; octant_code[4]:=fourth_octant; octant_code[5]:=fifth_octant; octant_code[6]:=sixth_octant; octant_code[7]:=seventh_octant; octant_code[8]:=eighth_octant; for k:=1 to 8 do octant_number[octant_code[k]]:=k; @ The main loop for boundary insertion deals with three consecutive nodes |p,q,r|. @= turning_number:=0; p:=cur_spec; q:=link(p); repeat r:=link(q); if (right_type(p)<>right_type(q))or(q=r) then @; p:=q; q:=r; until p=cur_spec; @ The |new_boundary| subroutine comes in handy at this point. It inserts a new boundary node just after a given node |p|, using a given octant code to transform the new node's coordinates. The ``transition'' fields are not computed here. @= procedure new_boundary(@!p:pointer;@!octant:small_number); var @!q,@!r:pointer; {for list manipulation} begin q:=link(p); {we assume that |right_type(q)<>endpoint|} r:=get_node(knot_node_size); link(r):=q; link(p):=r; left_type(r):=left_type(q); {but possibly |left_type(q)=endpoint|} left_x(r):=left_x(q); left_y(r):=left_y(q); right_type(r):=endpoint; left_type(q):=endpoint; right_octant(r):=octant; left_octant(q):=right_type(q); unskew(x_coord(q),y_coord(q),right_type(q)); skew(cur_x,cur_y,octant); x_coord(r):=cur_x; y_coord(r):=cur_y; end; @ The case |q=r| occurs if and only if |p=q=r=cur_spec|, when we want to turn $360^\circ$ in eight steps and then remove a solitary dead cubic. The program below happens to work in that case, but the reader isn't expected to understand why. @= begin new_boundary(p,right_type(p)); s:=link(p); o1:=octant_number[right_type(p)]; o2:=octant_number[right_type(q)]; case o2-o1 of 1,-7,7,-1: goto done; 2,-6: clockwise:=false; 3,-5,4,-4,5,-3: @; 6,-2: clockwise:=true; 0:clockwise:=rev_turns; end; {there are no other cases} @; done: if q=r then begin q:=link(q); r:=q; p:=s; link(s):=q; left_octant(q):=right_octant(q); left_type(q):=endpoint; free_node(cur_spec,knot_node_size); cur_spec:=q; end; @; end @ @= @!o1,@!o2:small_number; {octant numbers} @!clockwise:boolean; {should we turn clockwise?} @!dx1,@!dy1,@!dx2,@!dy2:integer; {directions of travel at a cusp} @!dmax,@!del:integer; {temporary registers} @ A tricky question arises when a path jumps four octants. We want the direction of turning to be counterclockwise if the curve has changed direction by $180^\circ$, or by something so close to $180^\circ$ that the difference is probably due to rounding errors; otherwise we want to turn through an angle of less than $180^\circ$. This decision needs to be made even when a curve seems to have jumped only three octants, since a curve may approach direction $(-1,0)$ from the fourth octant, then it might leave from direction $(+1,0)$ into the first. The following code solves the problem by analyzing the incoming direction |(dx1,dy1)| and the outgoing direction |(dx2,dy2)|. @= begin @; unskew(dx1,dy1,right_type(p)); del:=pyth_add(cur_x,cur_y);@/ dx1:=make_fraction(cur_x,del); dy1:=make_fraction(cur_y,del); {$\cos\theta_1$ and $\sin\theta_1$} unskew(dx2,dy2,right_type(q)); del:=pyth_add(cur_x,cur_y);@/ dx2:=make_fraction(cur_x,del); dy2:=make_fraction(cur_y,del); {$\cos\theta_2$ and $\sin\theta_2$} del:=take_fraction(dx1,dy2)-take_fraction(dx2,dy1); {$\sin(\theta_2-\theta_1)$} if del>4684844 then clockwise:=false else if del<-4684844 then clockwise:=true {$2^{28}\cdot\sin 1^\circ\approx4684844.68$} else clockwise:=rev_turns; end @ Actually the turnarounds are clockwise, not counterclockwise, if the global variable |rev_turns| is |true|; it is usually |false|. @= @!rev_turns:boolean; {should we make U-turns in the English manner?} @ @= rev_turns:=false; @ @= dx1:=x_coord(s)-left_x(s); dy1:=y_coord(s)-left_y(s); if dx1=0 then if dy1=0 then begin dx1:=x_coord(s)-right_x(p); dy1:=y_coord(s)-right_y(p); if dx1=0 then if dy1=0 then begin dx1:=x_coord(s)-x_coord(p); dy1:=y_coord(s)-y_coord(p); end; {and they {\sl can't} both be zero} end; dmax:=abs(dx1);@+if abs(dy1)>dmax then dmax:=abs(dy1); while dmaxdmax then dmax:=abs(dy2); while dmax= loop@+ begin if clockwise then if o1=1 then o1:=8@+else decr(o1) else if o1=8 then o1:=1@+else incr(o1); if o1=o2 then goto done; new_boundary(s,octant_code[o1]); s:=link(s); left_octant(s):=right_octant(s); end @ Now it remains to insert the redundant information into the |left_transition| and |right_transition| fields between adjacent octants, in the octant boundary nodes that have just been inserted between |link(p)| and~|q|. The turning number is easily computed from these transitions. @= p:=link(p); repeat s:=link(p); o1:=octant_number[right_octant(p)]; o2:=octant_number[left_octant(s)]; if abs(o1-o2)=1 then begin if o2= @!y_corr,@!xy_corr,@!z_corr:array[first_octant..sixth_octant] of 0..1; @!x_corr:array[first_octant..sixth_octant] of -1..1; @ Here |xy_corr| is 1 if and only if the $x$ component of a skewed coordinate is to be decreased by an infinitesimal amount; |y_corr| is similar, but for the $y$ components. The other tables are set up so that the condition $$(x+y+|half_unit|)\bmod|unity|\G(y+|half_unit|)\bmod|unity|$$ is properly perturbed to the condition $$(x+y+|half_unit|-|x_corr|-|y_corr|)\bmod|unity|\G (y+|half_unit|-|y_corr|)\bmod|unity|+|z_corr|.$$ @= x_corr[first_octant]:=0; y_corr[first_octant]:=0; xy_corr[first_octant]:=0;@/ x_corr[second_octant]:=0; y_corr[second_octant]:=0; xy_corr[second_octant]:=1;@/ x_corr[third_octant]:=-1; y_corr[third_octant]:=1; xy_corr[third_octant]:=0;@/ x_corr[fourth_octant]:=1; y_corr[fourth_octant]:=0; xy_corr[fourth_octant]:=1;@/ x_corr[fifth_octant]:=0; y_corr[fifth_octant]:=1; xy_corr[fifth_octant]:=1;@/ x_corr[sixth_octant]:=0; y_corr[sixth_octant]:=1; xy_corr[sixth_octant]:=0;@/ x_corr[seventh_octant]:=1; y_corr[seventh_octant]:=0; xy_corr[seventh_octant]:=1;@/ x_corr[eighth_octant]:=-1; y_corr[eighth_octant]:=1; xy_corr[eighth_octant]:=0;@/ for k:=1 to 8 do z_corr[k]:=xy_corr[k]-x_corr[k]; @ Here's a procedure that handles the details of rounding at the endpoints: Given skewed coordinates |(x,y)|, it sets |(m1,n1)| to the corresponding rounded lattice points, taking the current |octant| into account. Global variable |d1| is also set to 1 if $(x+y+{1\over2})\bmod1\G(y+{1\over2})\bmod1$. @p procedure end_round(@!x,@!y:scaled); begin y:=y+half_unit-y_corr[octant]; x:=x+y-x_corr[octant]; m1:=floor_unscaled(x); n1:=floor_unscaled(y); if x-unity*m1>=y-unity*n1+z_corr[octant] then d1:=1@+else d1:=0; end; @ The outputs |(m1,n1,d1)| of |end_round| will sometimes be moved to |(m0,n0,d0)|. @= @!m0,@!n0,@!m1,@!n1:integer; {lattice point coordinates} @!d0,@!d1:0..1; {displacement corrections} @ We're ready now to fill the pixels enclosed by a given cycle spec~|h|; the knot list that represents the cycle is destroyed in the process. The edge structure that gets all the resulting data is |cur_edges|, and the edges are weighted by |cur_wt|. @p procedure fill_spec(@!h:pointer); var @!p,@!q,@!r,@!s:pointer; {for list traversal} begin if internal[tracing_edges]>0 then begin_edge_tracing; p:=h; {we assume that |left_type(h)=endpoint|} repeat octant:=left_octant(p); @; if q<>p then begin @; @; move_to_edges(m0,n0,m1,n1); end; p:=link(q); until p=h; toss_knot_list(h); if internal[tracing_edges]>0 then end_edge_tracing; end; @ @= q:=p; while right_type(q)<>endpoint do q:=link(q) @ @= end_round(x_coord(p),y_coord(p)); m0:=m1; n0:=n1; d0:=d1;@/ end_round(x_coord(q),y_coord(q)) @ Here we perform the five-step process that was explained at the very beginning of this part of the program. @= if n1-n0>=move_size then overflow("move table size",move_size); @:METAFONT capacity exceeded move table size}{\quad move table size@> move[0]:=d0; move_ptr:=0; r:=p; repeat s:=link(r);@/ make_moves(x_coord(r),right_x(r),left_x(s),x_coord(s),@| y_coord(r)+half_unit,right_y(r)+half_unit,left_y(s)+half_unit, y_coord(s)+half_unit,@| xy_corr[octant],y_corr[octant]); r:=s; until r=q; move[move_ptr]:=move[move_ptr]-d1; if internal[smoothing]>0 then smooth_moves(0,move_ptr) @* \[23] Polygonal pens. The next few parts of the program deal with the additional complications associated with ``envelopes,'' leading up to an algorithm that fills a contour with respect to a pen whose boundary is a convex polygon. The mathematics underlying this algorithm is based on simple aspects of the theory of tracings developed by Leo Guibas, Lyle Ramshaw, and Jorge Stolfi [``A kinetic framework for computational geometry,'' {\sl Proc.\ IEEE Symp.\ Foundations of Computer Science\/ \bf24} (1983), 100--111]. @^Guibas, Leonidas Ioannis@> @^Ramshaw, Lyle Harold@> @^Stolfi, Jorge@> If the vertices of the polygon are $w_0$, $w_1$, \dots, $w_{n-1}$, $w_n=w_0$, in counterclockwise order, the convexity condition requires that ``left turns'' are made at each vertex when a person proceeds from $w_0$ to $w_1$ to $\cdots$ to~$w_n$. The envelope is obtained if we offset a given curve $z(t)$ by $w_k$ when that curve is traveling in a direction $z'(t)$ lying between the directions $w_k-w_{k-1}$ and $w\k-w_k$. At times~$t$ when the curve direction $z'(t)$ increases past $w\k-w_k$, we temporarily stop plotting the offset curve and we insert a straight line from $z(t)+w_k$ to $z(t)+w\k$; notice that this straight line is tangent to the offset curve. Similarly, when the curve direction decreases past $w_k-w_{k-1}$, we stop plotting and insert a straight line from $z(t)+w_k$ to $z(t)+w_{k-1}$; the latter line is actually a ``retrograde'' step, which won't be part of the final envelope under \MF's assumptions. The result of this construction is a continuous path that consists of alternating curves and straight line segments. The segments are usually so short, in practice, that they blend with the curves; after all, it's possible to represent any digitized path as a sequence of digitized straight lines. The nicest feature of this approach to envelopes is that it blends perfectly with the octant subdivision process we have already developed. The envelope travels in the same direction as the curve itself, as we plot it, and we need merely be careful what offset is being added. Retrograde motion presents a problem, but we will see that there is a decent way to handle it. @ We shall represent pens by maintaining eight lists of offsets, one for each octant direction. The offsets at the boundary points where a curve turns into a new octant will appear in the lists for both octants. This means that we can restrict consideration to segments of the original polygon whose directions aim in the first octant, as we have done in the simpler case when envelopes were not required. An example should help to clarify this situation: Consider the quadrilateral whose vertices are $w_0=(0,-1)$, $w_1=(3,-1)$, $w_2=(6,1)$, and $w_3=(1,2)$. A curve that travels in the first octant will be offset by $w_1$ or $w_2$, unless its slope drops to zero enroute to the eighth octant; in the latter case we should switch to $w_0$ as we cross the octant boundary. Our list for the first octant will contain the three offsets $w_0$, $w_1$,~$w_2$. By convention we will duplicate a boundary offset if the angle between octants doesn't explicitly appear; in this case there is no explicit line of slope~1 at the end of the list, so the full list is $$w_0\;w_1\;w_2\;w_2\;=\;(0,-1)\;(3,-1)\;(6,1)\;(6,1).$$ With skewed coordinates $(u-v,v)$ instead of $(u,v)$ we obtain the list $$w_0\;w_1\;w_2\;w_2\;\mapsto\;(1,-1)\;(4,-1)\;(5,1)\;(5,1),$$ which is what actually appears in the data structure. In the second octant there's only one offset; we list it three times (with coordinates interchanged, so as to make the second octant look like the first), and skew those coordinates, obtaining $$\tabskip\centering \halign to\hsize{$\hfil#\;\mapsto\;{}$\tabskip=0pt& $#\hfil$&\quad in the #\hfil\tabskip\centering\cr w_2\;w_2\;w_2&(-5,6)\;(-5,6)\;(-5,6)\cr \noalign{\vskip\belowdisplayskip \vbox{\noindent\strut as the list of transformed and skewed offsets to use when curves that travel in the second octant. Similarly, we will have\strut} \vskip\abovedisplayskip} w_2\;w_2\;w_2&(7,-6)\;(7,-6)\;(7,-6)&third;\cr w_2\;w_2\;w_3\;w_3&(-7,1)\;(-7,1)\;(-3,2)\;(-3,2)&fourth;\cr w_3\;w_3\;w_3&(3,-2)\;(3,-2)\;(3,-2)&fifth;\cr w_3\;w_3\;w_0\;w_0&(-3,1)\;(-3,1)\;(1,0)\;(1,0)&sixth;\cr w_0\;w_0\;w_0&(1,0)\;(1,0)\;(1,0)&seventh;\cr w_0\;w_0\;w_0&(-1,1)\;(-1,1)\;(-1,1)&eighth.\cr}$$ Notice that $w_1$ is considered here to be internal to the first octant; it's not part of the eighth. We could equally well have taken $w_0$ out of the first octant list and put it into the eighth; then the first octant list would have been $$w_1\;w_1\;w_2\;w_2\;\mapsto\;(4,-1)\;(4,-1)\;(5,1)\;(5,1)$$ and the eighth octant list would have been $$w_0\;w_0\;w_1\;\mapsto\;(-1,1)\;(-1,1)\;(2,1).$$ Actually, there's one more complication: The order of offsets is reversed in even-numbered octants, because the transformation of coordinates has reversed counterclockwise and clockwise orientations in those octants. The offsets in the fourth octant, for example, are really $w_3$, $w_3$, $w_2$,~$w_2$, not $w_2$, $w_2$, $w_3$,~$w_3$. @ In general, the list of offsets for an octant will have the form $$w_0\;\;w_1\;\;\ldots\;\;w_n\;\;w_{n+1}$$ (if we renumber the subscripts in each list), where $w_0$ and $w_{n+1}$ are offsets common to the neighboring lists. We'll often have $w_0=w_1$ and/or $w_n=w_{n+1}$, but the other $w$'s will be distinct. Curves that travel between slope~0 and direction $w_2-w_1$ will use offset~$w_1$; curves that travel between directions $w_k-w_{k-1}$ and $w\k-w_k$ will use offset~$w_k$, for $1= procedure print_pen(@!p:pointer;@!s:str_number;@!nuline:boolean); var @!nothing_printed:boolean; {has there been any action yet?} @!k:1..8; {octant number} @!h:pointer; {offset list head} @!m,@!n:integer; {offset indices} @!w,@!ww:pointer; {pointers that traverse the offset list} begin print_diagnostic("Pen polygon",s,nuline); nothing_printed:=true; print_ln; for k:=1 to 8 do begin octant:=octant_code[k]; h:=p+octant; n:=info(h); w:=link(h); if not odd(k) then w:=knil(w); {in even octants, start at $w_{n+1}$} for m:=1 to n+1 do begin if odd(k) then ww:=link(w)@+else ww:=knil(w); if (x_coord(ww)<>x_coord(w))or(y_coord(ww)<>y_coord(w)) then @; w:=ww; end; end; if nothing_printed then begin w:=link(p+first_octant); print_two(x_coord(w)+y_coord(w),y_coord(w)); end; print_nl(" .. cycle"); end_diagnostic(true); end; @ @= begin if nothing_printed then nothing_printed:=false else print_nl(" .. "); print_two_true(x_coord(ww),y_coord(ww)); end @ A null pen polygon, which has just one vertex $(0,0)$, is predeclared for error recovery. It doesn't need a proper reference count, because the |toss_pen| procedure below will never delete it from memory. @= ref_count(null_pen):=null; link(null_pen):=null;@/ info(null_pen+1):=1; link(null_pen+1):=null_coords; for k:=null_pen+2 to null_pen+8 do mem[k]:=mem[null_pen+1]; max_offset(null_pen):=0;@/ link(null_coords):=null_coords; knil(null_coords):=null_coords;@/ x_coord(null_coords):=0; y_coord(null_coords):=0; @ Here's a trivial subroutine that inserts a copy of an offset on the |link| side of its clone in the doubly linked list. @p procedure dup_offset(@!w:pointer); var @!r:pointer; {the new node} begin r:=get_node(coord_node_size); x_coord(r):=x_coord(w); y_coord(r):=y_coord(w); link(r):=link(w); knil(link(w)):=r; knil(r):=w; link(w):=r; end; @ The following algorithm is somewhat more interesting: It converts a knot list for a cyclic path into a pen polygon, ignoring everything but the |x_coord|, |y_coord|, and |link| fields. If the given path vertices do not define a convex polygon, an error message is issued and the null pen is returned. @p function make_pen(@!h:pointer):pointer; label done,done1,not_found,found; var @!o,@!oo,@!k:small_number; {octant numbers---old, new, and current} @!p:pointer; {top-level node for the new pen} @!q,@!r,@!s,@!w,@!hh:pointer; {for list manipulation} @!n:integer; {offset counter} @!dx,@!dy:scaled; {polygon direction} @!mc:scaled; {the largest coordinate} begin @; if mc>=fraction_one-half_unit then goto not_found; p:=get_node(pen_node_size); q:=hh; max_offset(p):=mc; ref_count(p):=null; if link(q)<>q then link(p):=null+1; for k:=1 to 8 do @; goto found; not_found:p:=null_pen; @; found: if internal[tracing_pens]>0 then print_pen(p," (newly created)",true); make_pen:=p; end; @ @= if mc>=fraction_one-half_unit then begin print_err("Pen too large"); @.Pen too large@> help2("The cycle you specified has a coordinate of 4095.5 or more.")@/ ("So I've replaced it by the trivial path `(0,0)..cycle'.");@/ end else begin print_err("Pen cycle must be convex"); @.Pen cycle must be convex@> help3("The cycle you specified either has consecutive equal points")@/ ("or turns right or turns through more than 360 degrees.")@/ ("So I've replaced it by the trivial path `(0,0)..cycle'.");@/ end; put_get_error @ There should be exactly one node whose octant number is less than its predecessor in the cycle; that is node~|hh|. The loop here will terminate in all cases, but the proof is somewhat tricky: If there are at least two distinct $y$~coordinates in the cycle, we will have |o>4| and |o<=4| at different points of the cycle. Otherwise there are at least two distinct $x$~coordinates, and we will have |o>2| somewhere, |o<=2| somewhere. @= q:=h; r:=link(q); mc:=abs(x_coord(h)); if q=r then begin hh:=h; right_type(h):=0; {this trick is explained below} if mc; right_type(q):=octant; oo:=octant_number[octant]; if o>oo then begin if hh<>null then goto not_found; {$>360^\circ$} hh:=q; end; o:=oo; if (q=h)and(hh<>null) then goto done; q:=r; r:=s; end; done:end @ We want the octant for |(-dx,-dy)| to be exactly opposite the octant for |(dx,dy)|. @= if dx>0 then octant:=first_octant else if dx=0 then if dy>0 then octant:=first_octant@+else octant:=first_octant+negate_x else begin negate(dx); octant:=first_octant+negate_x; end; if dy<0 then begin negate(dy); octant:=octant+negate_y; end else if dy=0 then if octant>first_octant then octant:=first_octant+negate_x+negate_y; if dx= begin octant:=octant_code[k]; n:=0; h:=p+octant; loop@+ begin r:=get_node(coord_node_size); skew(x_coord(q),y_coord(q),octant); x_coord(r):=cur_x; y_coord(r):=cur_y; if n=0 then link(h):=r else @; w:=r; if right_type(q)<>octant then goto done1; q:=link(q); incr(n); end; done1: @; if n>=max_quarterword then overflow("pen polygon size",max_quarterword); @:METAFONT capacity exceeded pen polygon size}{\quad pen polygon size@> info(h):=n; end @ Now |w| points to the node that was inserted most recently, and |k| is the current octant number. @= if odd(k) then begin link(w):=r; knil(r):=w; end else begin knil(w):=r; link(r):=w; end @ We have inserted |n+1| nodes; it remains to duplicate the nodes at the ends, if slopes 0 and~$\infty$ aren't already represented. At the end of this section the total number of offset nodes should be |n+2| (since we call them $w_0$, $w_1$, \dots,~$w_{n+1}$). @= r:=link(h); if odd(k) then begin link(w):=r; knil(r):=w; end else begin knil(w):=r; link(r):=w; link(h):=w; r:=w; end; if (y_coord(r)<>y_coord(link(r)))or(n=0) then begin dup_offset(r); incr(n); end; r:=knil(r); if x_coord(r)<>x_coord(knil(r)) then dup_offset(r) else decr(n) @ Conversely, |make_path| goes back from a pen to a cyclic path that might have generated it. The structure of this subroutine is essentially the same as |print_pen|. @p @t\4@>@@; function make_path(@!pen_head:pointer):pointer; var @!p:pointer; {the most recently copied knot} @!k:1..8; {octant number} @!h:pointer; {offset list head} @!m,@!n:integer; {offset indices} @!w,@!ww:pointer; {pointers that traverse the offset list} begin p:=temp_head; for k:=1 to 8 do begin octant:=octant_code[k]; h:=pen_head+octant; n:=info(h); w:=link(h); if not odd(k) then w:=knil(w); {in even octants, start at $w_{n+1}$} for m:=1 to n+1 do begin if odd(k) then ww:=link(w)@+else ww:=knil(w); if (x_coord(ww)<>x_coord(w))or(y_coord(ww)<>y_coord(w)) then @; w:=ww; end; end; if p=temp_head then begin w:=link(pen_head+first_octant); p:=trivial_knot(x_coord(w)+y_coord(w),y_coord(w)); link(temp_head):=p; end; link(p):=link(temp_head); make_path:=link(temp_head); end; @ @= begin unskew(x_coord(ww),y_coord(ww),octant); link(p):=trivial_knot(cur_x,cur_y); p:=link(p); end @ @= function trivial_knot(@!x,@!y:scaled):pointer; var @!p:pointer; {a new knot for explicit coordinates |x| and |y|} begin p:=get_node(knot_node_size); left_type(p):=explicit; right_type(p):=explicit;@/ x_coord(p):=x; left_x(p):=x; right_x(p):=x;@/ y_coord(p):=y; left_y(p):=y; right_y(p):=y;@/ trivial_knot:=p; end; @ That which can be created can be destroyed. @d add_pen_ref(#)==incr(ref_count(#)) @d delete_pen_ref(#)==if ref_count(#)=null then toss_pen(#) else decr(ref_count(#)) @= procedure toss_pen(@!p:pointer); var @!k:1..8; {relative header locations} @!w,@!ww:pointer; {pointers to offset nodes} begin if p<>null_pen then begin for k:=1 to 8 do begin w:=link(p+k); repeat ww:=link(w); free_node(w,coord_node_size); w:=ww; until w=link(p+k); end; free_node(p,pen_node_size); end; end; @ The |find_offset| procedure sets |(cur_x,cur_y)| to the offset associated with a given direction~|(x,y)| and a given pen~|p|. If |x=y=0|, the result is |(0,0)|. If two different offsets apply, one of them is chosen arbitrarily. @p procedure find_offset(@!x,@!y:scaled; @!p:pointer); label done,exit; var @!octant:first_octant..sixth_octant; {octant code for |(x,y)|} @!s:-1..+1; {sign of the octant} @!n:integer; {number of offsets remaining} @!h,@!w,@!ww:pointer; {list traversal registers} begin @; if odd(octant_number[octant]) then s:=-1@+else s:=+1; h:=p+octant; w:=link(link(h)); ww:=link(w); n:=info(h); while n>1 do begin if ab_vs_cd(x,y_coord(ww)-y_coord(w),@| y,x_coord(ww)-x_coord(w))<>s then goto done; w:=ww; ww:=link(w); decr(n); end; done:unskew(x_coord(w),y_coord(w),octant); exit:end; @ @= if x>0 then octant:=first_octant else if x=0 then if y<=0 then if y=0 then begin cur_x:=0; cur_y:=0; return; end else octant:=first_octant+negate_x else octant:=first_octant else begin x:=-x; if y=0 then octant:=first_octant+negate_x+negate_y else octant:=first_octant+negate_x; end; if y<0 then begin octant:=octant+negate_y; y:=-y; end; if x>=y then x:=x-y else begin octant:=octant+switch_x_and_y; x:=y-x; y:=y-x; end @* \[24] Filling an envelope. We are about to reach the culmination of \MF's digital plotting routines: Almost all of the previous algorithms will be brought to bear on \MF's most difficult task, which is to fill the envelope of a given cyclic path with respect to a given pen polygon. But we still must complete some of the preparatory work before taking such a big plunge. @ Given a pointer |c| to a nonempty list of cubics, and a pointer~|h| to the header information of a pen polygon segment, the |offset_prep| routine changes the list into cubics that are associated with particular pen offsets. Namely, the cubic between |p| and~|q| should be associated with the |k|th offset when |right_type(p)=k|. List |c| is actually part of a cycle spec, so it terminates at the first node whose |right_type| is |endpoint|. The cubics all have monotone-nondecreasing $x'(t)$ and $y'(t)$. @p @t\4@>@@; procedure offset_prep(@!c,@!h:pointer); label done,not_found; var @!n:halfword; {the number of pen offsets} @!p,@!q,@!r,@!lh,@!ww:pointer; {for list manipulation} @!k:halfword; {the current offset index} @!w:pointer; {a pointer to offset $w_k$} @@; begin p:=c; n:=info(h); lh:=link(h); {now |lh| points to $w_0$} while right_type(p)<>endpoint do begin q:=link(p); @; @; end; end; @ @= repeat r:=link(p); if x_coord(p)=right_x(p) then if y_coord(p)=right_y(p) then if x_coord(p)=left_x(r) then if y_coord(p)=left_y(r) then if x_coord(p)=x_coord(r) then if y_coord(p)=y_coord(r) then begin remove_cubic(p); if r=q then q:=p; r:=p; end; p:=r; until p=q @ The splitting process uses a subroutine like |split_cubic|, but (for ``bulletproof'' operation) we check to make sure that the resulting (skewed) coordinates satisfy $\Delta x\G0$ and $\Delta y\G0$ after splitting; |make_spec| has made sure that these relations hold before splitting. @= procedure split_for_offset(@!p:pointer;@!t:fraction); var @!q:pointer; {the successor of |p|} @!r:pointer; {the new node} begin q:=link(p); split_cubic(p,t,x_coord(q),y_coord(q)); r:=link(p); if y_coord(r)y_coord(q) then y_coord(r):=y_coord(q); if x_coord(r)x_coord(q) then x_coord(r):=x_coord(q); end; @ If the pen polygon has |n| offsets, and if $w_k=(u_k,v_k)$ is the $k$th of these, the $k$th pen slope is defined by the formula $$s_k={v\k-v_k\over u\k-u_k},\qquad\hbox{for $0= if n<=1 then right_type(p):=1 {this case is easy} else begin @; @; if dx=0 then @ else begin @; @; end; not_found: end @ The slope of a cubic $B(z_0,z_1,z_2,z_3;t)=\bigl(x(t),y(t)\bigr)$ can be calculated from the quadratic polynomials ${1\over3}x'(t)=B(x_1-x_0,x_2-x_1,x_3-x_2;t)$ and ${1\over3}y'(t)=B(y_1-y_0,y_2-y_1,y_3-y_2;t)$. Since we may be calculating slopes from several cubics split from the current one, it is desirable to do these calculations without losing too much precision. ``Scaled up'' values of the derivatives, which will be less tainted by accumulated errors than derivatives found from the cubics themselves, are maintained in local variables |x0|, |x1|, and |x2|, representing $X_0=2^l(x_1-x_0)$, $X_1=2^l(x_2-x_1)$, and $X_2=2^l(x_3-x_2)$; similarly |y0|, |y1|, and~|y2| represent $Y_0=2^l(y_1-y_0)$, $Y_1=2^l(y_2-y_1)$, and $Y_2=2^l(y_3-y_2)$. To test whether the slope of the cubic is $>=s$ or $<=s$, we will test the sign of the quadratic ${1\over3}2^l\bigl(y'(t)-sx'(t)\bigr)$ if $s<=1$, or ${1\over3}2^l\bigl(y'(t)/s-x'(t)\bigr)$ if $s>1$. @= @!x0,@!x1,@!x2,@!y0,@!y1,@!y2:integer; {representatives of derivatives} @!t0,@!t1,@!t2:integer; {coefficients of polynomial for slope testing} @!du,@!dv,@!dx,@!dy:integer; {for slopes of the pen and the curve} @!max_coef:integer; {used while scaling} @!x0a,@!x1a,@!x2a,@!y0a,@!y1a,@!y2a:integer; {intermediate values} @!t:fraction; {where the derivative passes through zero} @!s:fraction; {slope or reciprocal slope} @ @= x0:=right_x(p)-x_coord(p); {should be |>=0|} x2:=x_coord(q)-left_x(q); {likewise} x1:=left_x(q)-right_x(p); {but this might be negative} y0:=right_y(p)-y_coord(p); y2:=y_coord(q)-left_y(q); y1:=left_y(q)-right_y(p); max_coef:=abs(x0); {we take |abs| just to make sure} if abs(x1)>max_coef then max_coef:=abs(x1); if abs(x2)>max_coef then max_coef:=abs(x2); if abs(y0)>max_coef then max_coef:=abs(y0); if abs(y1)>max_coef then max_coef:=abs(y1); if abs(y2)>max_coef then max_coef:=abs(y2); if max_coef=0 then goto not_found; while max_coefs_{k-1}$. Then, in a sense, we're halfway done, since one of the two inequalities in $(*)$ is satisfied, and the other couldn't be satisfied for any other value of~|k|. The |fin_offset_prep| subroutine solves the stated subproblem. It has a boolean parameter called |rising| that is |true| in case~(i), |false| in case~(ii). When |rising=false|, parameters |x0| through |y2| represent the negative of the derivative of the cubic following |p|; otherwise they represent the actual derivative. The |w| parameter should point to offset~$w_k$. @= procedure fin_offset_prep(@!p:pointer;@!k:halfword;@!w:pointer; @!x0,@!x1,@!x2,@!y0,@!y1,@!y2:integer;@!rising:boolean;@!n:integer); label exit; var @!ww:pointer; {for list manipulation} @!du,@!dv:scaled; {for slope calculation} @!t0,@!t1,@!t2:integer; {test coefficients} @!t:fraction; {place where the derivative passes a critical slope} @!s:fraction; {slope or reciprocal slope} @!v:integer; {intermediate value for updating |x0..y2|} begin loop begin right_type(p):=k; if rising then if k=n then return else ww:=link(w) {a pointer to $w\k$} else if k=1 then return else ww:=knil(w); {a pointer to $w_{k-1}$} @; t:=crossing_point(t0,t1,t2); if t>=fraction_one then return; @; if rising then incr(k)@+else decr(k); w:=ww; end; exit:end; @ @= du:=x_coord(ww)-x_coord(w); dv:=y_coord(ww)-y_coord(w); if abs(du)>=abs(dv) then {$s_{k\pm1}<=1$} begin s:=make_fraction(dv,du); t0:=take_fraction(x0,s)-y0; t1:=take_fraction(x1,s)-y1; t2:=take_fraction(x2,s)-y2; end else begin s:=make_fraction(du,dv); t0:=x0-take_fraction(y0,s); t1:=x1-take_fraction(y1,s); t2:=x2-take_fraction(y2,s); end @ The curve has crossed $s_k$ or $s_{k-1}$; its initial segment satisfies $(*)$, and it might cross again and return towards $s_k$, yielding another solution of $(*)$. @= begin split_for_offset(p,t); right_type(p):=k; p:=link(p);@/ v:=t_of_the_way(x0)(x1); x1:=t_of_the_way(x1)(x2); x0:=t_of_the_way(v)(x1);@/ v:=t_of_the_way(y0)(y1); y1:=t_of_the_way(y1)(y2); y0:=t_of_the_way(v)(y1);@/ t1:=t_of_the_way(t1)(t2); if t1>0 then t1:=0; {without rounding error, |t1| would be |<=0|} t:=crossing_point(0,-t1,-t2); if t= dx:=x0; dy:=y0; if dx=0 then if dy=0 then begin dx:=x1; dy:=y1; if dx=0 then if dy=0 then begin dx:=x2; dy:=y2; end; end @ The next step is to bracket the initial slope between consecutive slopes of the pen polygon. The most important invariant relation in the following loop is that |dy/dx>=@t$s_{k-1}$@>|. @= k:=1; w:=link(lh); loop@+ begin if k=n then goto done; ww:=link(w); if ab_vs_cd(dy,abs(x_coord(ww)-x_coord(w)),@| dx,abs(y_coord(ww)-y_coord(w)))>=0 then begin incr(k); w:=ww; end else goto done; end; done: @ Finally we want to reduce the general problem to situations that |fin_offset_prep| can handle. If |k=1|, we already are in the desired situation. Otherwise we can split the cubic into at most three parts with respect to $s_{k-1}$, and apply |fin_offset_prep| to each part. @= if k=1 then t:=fraction_one+1 else begin ww:=knil(w); @; t:=crossing_point(-t0,-t1,-t2); end; if t>=fraction_one then fin_offset_prep(p,k,w,x0,x1,x2,y0,y1,y2,true,n) else begin split_for_offset(p,t); r:=link(p);@/ x1a:=t_of_the_way(x0)(x1); x1:=t_of_the_way(x1)(x2); x2a:=t_of_the_way(x1a)(x1);@/ y1a:=t_of_the_way(y0)(y1); y1:=t_of_the_way(y1)(y2); y2a:=t_of_the_way(y1a)(y1);@/ fin_offset_prep(p,k,w,x0,x1a,x2a,y0,y1a,y2a,true,n); x0:=x2a; y0:=y2a; t1:=t_of_the_way(t1)(t2); if t1<0 then t1:=0; t:=crossing_point(0,t1,t2); if t; fin_offset_prep(r,k-1,ww,-x0,-x1,-x2,-y0,-y1,-y2,false,n); end @ @= begin split_for_offset(r,t);@/ x1a:=t_of_the_way(x1)(x2); x1:=t_of_the_way(x0)(x1); x0a:=t_of_the_way(x1)(x1a);@/ y1a:=t_of_the_way(y1)(y2); y1:=t_of_the_way(y0)(y1); y0a:=t_of_the_way(y1)(y1a);@/ fin_offset_prep(link(r),k,w,x0a,x1a,x2,y0a,y1a,y2,true,n); x2:=x0a; y2:=y0a; end @ @= fin_offset_prep(p,n,knil(knil(lh)),-x0,-x1,-x2,-y0,-y1,-y2,false,n) @ OK, it's time now for the biggie. The |fill_envelope| routine generalizes |fill_spec| to polygonal envelopes. Its outer structure is essentially the same as before, except that octants with no cubics do contribute to the envelope. @p @t\4@>@@; @t\4@>@@; procedure fill_envelope(@!spec_head:pointer); label done, done1; var @!p,@!q,@!r,@!s:pointer; {for list traversal} @!h:pointer; {head of pen offset list for current octant} @!www:pointer; {a pen offset of temporary interest} @@; begin if internal[tracing_edges]>0 then begin_edge_tracing; p:=spec_head; {we assume that |left_type(spec_head)=endpoint|} repeat octant:=left_octant(p); h:=cur_pen+octant; @; @; offset_prep(p,h); {this may clobber node~|q|, if it becomes ``dead''} @; @; p:=link(q); until p=spec_head; if internal[tracing_edges]>0 then end_edge_tracing; toss_knot_list(spec_head); end; @ In even-numbered octants we have reflected the coordinates an odd number of times, hence clockwise and counterclockwise are reversed; this means that the envelope is being formed in a ``dual'' manner. For the time being, let's concentrate on odd-numbered octants, since they're easier to understand. After we have coded the program for odd-numbered octants, the changes needed to dualize it will not be so mysterious. It is convenient to assume that we enter an odd-numbered octant with an |axis| transition (where the skewed slope is zero) and leave at a |diagonal| one (where the skewed slope is infinite). Then all of the offset points $z(t)+w(t)$ will lie in a rectangle whose lower left and upper right corners are the initial and final offset points. If this assumption doesn't hold we can implicitly change the curve so that it does. For example, if the entering transition is diagonal, we can draw a straight line from $z_0+w_{n+1}$ to $z_0+w_0$ and continue as if the curve were moving rightward. The effect of this on the envelope is simply to ``doubly color'' the region enveloped by a section of the pen that goes from $w_0$ to $w_1$ to $\cdots$ to $w_{n+1}$ to~$w_0$. The additional straight line at the beginning (and a similar one at the end, where it may be necessary to go from $z_1+w_{n+1}$ to $z_1+w_0$) can be drawn by the |line_edges| routine; we are thereby saved from the embarrassment that these lines travel backwards from the current octant direction. Once we have established the assumption that the curve goes from $z_0+w_0$ to $z_1+w_{n+1}$, any further retrograde moves that might occur within the octant can be essentially ignored; we merely need to keep track of the rightmost edge in each row, in order to compute the envelope. Envelope moves consist of offset cubics intermixed with straight line segments. We record them in a separate |env_move| array, which is something like |move| but it keeps track of the rightmost position of the envelope in each row. @= @!env_move:array[0..move_size] of integer; @ @= w:=link(h);@+if left_transition(p)=diagonal then w:=knil(w); @!stat if internal[tracing_edges]>unity then @; tats@;@/ ww:=link(h); www:=ww; {starting and ending offsets} if odd(octant_number[octant]) then www:=knil(www)@+else ww:=knil(ww); if w<>ww then skew_line_edges(p,w,ww); end_round(x_coord(p)+x_coord(ww),y_coord(p)+y_coord(ww)); m0:=m1; n0:=n1; d0:=d1;@/ end_round(x_coord(q)+x_coord(www),y_coord(q)+y_coord(www)); if n1-n0>=move_size then overflow("move table size",move_size) @:METAFONT capacity exceeded move table size}{\quad move table size@> @ @= begin print_nl("@@ Octant "); print(octant_dir[octant]); @:]]]\AT!_Octant}{\.{\AT! Octant...}@> print(" ("); print_int(info(h)); print(" offset"); if info(h)<>1 then print_char("s"); print("), from "); print_two_true(x_coord(p)+x_coord(w),y_coord(p)+y_coord(w)); ww:=link(h);@+if right_transition(q)=diagonal then ww:=knil(ww); print(" to "); print_two_true(x_coord(q)+x_coord(ww),y_coord(q)+y_coord(ww)); end @ A slight variation of the |line_edges| procedure comes in handy when we must draw the retrograde lines for nonstandard entry and exit conditions. @= procedure skew_line_edges(@!p,@!w,@!ww:pointer); var @!x0,@!y0,@!x1,@!y1:scaled; {from and to} begin if (x_coord(w)<>x_coord(ww))or(y_coord(w)<>y_coord(ww)) then begin x0:=x_coord(p)+x_coord(w); y0:=y_coord(p)+y_coord(w);@/ x1:=x_coord(p)+x_coord(ww); y1:=y_coord(p)+y_coord(ww);@/ unskew(x0,y0,octant); {unskew and unrotate the coordinates} x0:=cur_x; y0:=cur_y;@/ unskew(x1,y1,octant);@/ @!stat if internal[tracing_edges]>unity then begin print_nl("@@ retrograde line from "); @:]]]\AT!_retro_}{\.{\AT! retrograde line...}@> @.retrograde line...@> print_two(x0,y0); print(" to "); print_two(cur_x,cur_y); print_nl(""); end;@+tats@;@/ line_edges(x0,y0,cur_x,cur_y); {then draw a straight line} end; end; @ The envelope calculations require more local variables than we needed in the simpler case of |fill_spec|. At critical points in the computation, |w| will point to offset $w_k$; |m| and |n| will record the current lattice positions. The values of |move_ptr| after the initial and before the final offset adjustments are stored in |smooth_bot| and |smooth_top|, respectively. @= @!m,@!n:integer; {current lattice position} @!mm0,@!mm1:integer; {skewed equivalents of |m0| and |m1|} @!k:integer; {current offset number} @!w,@!ww:pointer; {pointers to the current offset and its neighbor} @!smooth_bot,@!smooth_top:0..move_size; {boundaries of smoothing} @!xx,@!yy,@!xp,@!yp,@!delx,@!dely,@!tx,@!ty:scaled; {registers for coordinate calculations} @ @= if odd(octant_number[octant]) then begin @; r:=p; right_type(q):=info(h)+1; loop@+ begin if r=q then smooth_top:=move_ptr; while right_type(r)<>k do @; if r=p then smooth_bot:=move_ptr; if r=q then goto done; move[move_ptr]:=1; n:=move_ptr; s:=link(r);@/ make_moves(x_coord(r)+x_coord(w),right_x(r)+x_coord(w), left_x(s)+x_coord(w),x_coord(s)+x_coord(w),@| y_coord(r)+y_coord(w)+half_unit,right_y(r)+y_coord(w)+half_unit, left_y(s)+y_coord(w)+half_unit,y_coord(s)+y_coord(w)+half_unit,@| xy_corr[octant],y_corr[octant]); @; r:=s; end; done: @; end else dual_moves(h,p,q); right_type(q):=endpoint @ @= k:=0; w:=link(h); ww:=knil(w); mm0:=floor_unscaled(x_coord(p)+x_coord(w)-xy_corr[octant]); mm1:=floor_unscaled(x_coord(q)+x_coord(ww)-xy_corr[octant]); for n:=0 to n1-n0 do env_move[n]:=mm0; env_move[n1-n0]:=mm1; move_ptr:=0; m:=mm0 @ At this point |n| holds the value of |move_ptr| that was current when |make_moves| began to record its moves. @= repeat m:=m+move[n]-1; if m>env_move[n] then env_move[n]:=m; incr(n); until n>move_ptr @ Retrograde lines (when |k| decreases) do not need to be recorded in |env_move| because their edges are not the furthest right in any row. @= begin xx:=x_coord(r)+x_coord(w); yy:=y_coord(r)+y_coord(w)+half_unit; @!stat if internal[tracing_edges]>unity then begin print_nl("@@ transition line "); print_int(k); print(", from "); @:]]]\AT!_trans_}{\.{\AT! transition line...}@> @.transition line...@> print_two_true(xx,yy-half_unit); end;@+tats@;@/ if right_type(r)>k then begin incr(k); w:=link(w); xp:=x_coord(r)+x_coord(w); yp:=y_coord(r)+y_coord(w)+half_unit; if yp<>yy then @; end else begin decr(k); w:=knil(w); xp:=x_coord(r)+x_coord(w); yp:=y_coord(r)+y_coord(w)+half_unit; end; stat if internal[tracing_edges]>unity then begin print(" to "); print_two_true(xp,yp-half_unit); print_nl(""); end;@+tats@;@/ m:=floor_unscaled(xp-xy_corr[octant]); move_ptr:=floor_unscaled(yp-y_corr[octant])-n0; if m>env_move[move_ptr] then env_move[move_ptr]:=m; end @ In this step we have |xp>=xx| and |yp>=yy|. @= begin ty:=floor_scaled(yy-y_corr[octant]); dely:=yp-yy; yy:=yy-ty; ty:=yp-y_corr[octant]-ty; if ty>=unity then begin delx:=xp-xx; yy:=unity-yy; loop@+ begin tx:=take_fraction(delx,make_fraction(yy,dely)); if ab_vs_cd(tx,dely,delx,yy)+xy_corr[octant]>0 then decr(tx); m:=floor_unscaled(xx+tx); if m>env_move[move_ptr] then env_move[move_ptr]:=m; ty:=ty-unity; if ty= debug if (m<>mm1)or(move_ptr<>n1-n0) then confusion("1");@+gubed@;@/ move[0]:=d0+env_move[0]-mm0; for n:=1 to move_ptr do move[n]:=env_move[n]-env_move[n-1]+1; move[move_ptr]:=move[move_ptr]-d1; if internal[smoothing]>0 then smooth_moves(smooth_bot,smooth_top); move_to_edges(m0,n0,m1,n1); if right_transition(q)=axis then begin w:=link(h); skew_line_edges(q,knil(w),w); end @ OK, now we've done it all in the odd-octant case; the only thing remaining is to repeat the same ideas, upside down and/or backwards. The following code has been split off as a subprocedure of |fill_envelope|, because some \PASCAL\ compilers cannot handle procedures as large as |fill_envelope| would otherwise be. @= procedure dual_moves(@!h,@!p,@!q:pointer); label done,done1; var @!r,@!s:pointer; {for list traversal} @@; begin @; r:=p; {recall that |right_type(q)=endpoint=0| now} loop@+ begin if r=q then smooth_top:=move_ptr; while right_type(r)<>k do @; if r=p then smooth_bot:=move_ptr; if r=q then goto done; move[move_ptr]:=1; n:=move_ptr; s:=link(r);@/ make_moves(x_coord(r)+x_coord(w),right_x(r)+x_coord(w), left_x(s)+x_coord(w),x_coord(s)+x_coord(w),@| y_coord(r)+y_coord(w)+half_unit,right_y(r)+y_coord(w)+half_unit, left_y(s)+y_coord(w)+half_unit,y_coord(s)+y_coord(w)+half_unit,@| xy_corr[octant],y_corr[octant]); @; r:=s; end; done:@; end; @ In the dual case the normal situation is to arrive with a |diagonal| transition and to leave at the |axis|. The leftmost edge in each row is relevant instead of the rightmost one. @= k:=info(h)+1; ww:=link(h); w:=knil(ww);@/ mm0:=floor_unscaled(x_coord(p)+x_coord(w)-xy_corr[octant]); mm1:=floor_unscaled(x_coord(q)+x_coord(ww)-xy_corr[octant]); for n:=1 to n1-n0+1 do env_move[n]:=mm1; env_move[0]:=mm0; move_ptr:=0; m:=mm0 @ @= repeat if mmove_ptr @ Dual retrograde lines occur when |k| increases; the edges of such lines are not the furthest left in any row. @= begin xx:=x_coord(r)+x_coord(w); yy:=y_coord(r)+y_coord(w)+half_unit; @!stat if internal[tracing_edges]>unity then begin print_nl("@@ transition line "); print_int(k); print(", from "); @:]]]\AT!_trans_}{\.{\AT! transition line...}@> @.transition line...@> print_two_true(xx,yy-half_unit); end;@+tats@;@/ if right_type(r)yy then @; end else begin incr(k); w:=link(w); xp:=x_coord(r)+x_coord(w); yp:=y_coord(r)+y_coord(w)+half_unit; end; stat if internal[tracing_edges]>unity then begin print(" to "); print_two_true(xp,yp-half_unit); print_nl(""); end;@+tats@;@/ m:=floor_unscaled(xp-xy_corr[octant]); move_ptr:=floor_unscaled(yp-y_corr[octant])-n0; if m=xx| and |yp>=yy|; but this time we are interested in the {\sl smallest\/} |m| that belongs to a given |move_ptr| position, instead of the largest~|m|. @= begin ty:=floor_scaled(yy-y_corr[octant]); dely:=yp-yy; yy:=yy-ty; ty:=yp-y_corr[octant]-ty; if ty>=unity then begin delx:=xp-xx; yy:=unity-yy; loop@+ begin if m0 then decr(tx); m:=floor_unscaled(xx+tx); ty:=ty-unity; incr(move_ptr); if ty= debug if (m<>mm1)or(move_ptr<>n1-n0) then confusion("2");@+gubed@;@/ move[0]:=d0+env_move[1]-mm0; for n:=1 to move_ptr do move[n]:=env_move[n+1]-env_move[n]+1; move[move_ptr]:=move[move_ptr]-d1; if internal[smoothing]>0 then smooth_moves(smooth_bot,smooth_top); move_to_edges(m0,n0,m1,n1); if right_transition(q)=diagonal then begin w:=link(h); skew_line_edges(q,w,knil(w)); end @* \[25] Elliptical pens. To get the envelope of a cyclic path with respect to an ellipse, \MF\ calculates the envelope with respect to a polygonal approximation to the ellipse, using an approach due to John Hobby (Ph.D. thesis, Stanford University, 1985). @^Hobby, John Douglas@> This has two important advantages over trying to obtain the ``exact'' envelope: \yskip\textindent{1)}It gives better results, because the polygon has been designed to counteract problems that arise from digitization; the polygon includes sub-pixel corrections to an exact ellipse that make the results essentially independent of where the path falls on the raster. For example, the exact envelope with respect to a pen of diameter~1 blackens a pixel if and only if the path intersects a circle of diameter~1 inscribed in that pixel; the resulting pattern has ``blots'' when the path is travelling diagonally in unfortunate raster positions. A much better result is obtained when pixels are blackened only when the path intersects an inscribed {\sl diamond\/} of diameter~1. Such a diamond is precisely the polygon that \MF\ uses in the special case of a circle whose diameter is~1. \yskip\textindent{2)}Polygonal envelopes of cubic splines are cubic splines, hence it isn't necessary to introduce completely different routines. By contrast, exact envelopes of cubic splines with respect to circles are complicated curves, more difficult to plot than cubics. @ Hobby's construction involves some interesting number theory. If $u$ and~$v$ are relatively prime integers, we divide the set of integer points $(m,n)$ into equivalence classes by saying that $(m,n)$ belongs to class $um+vn$. Then any two integer points that lie on a line of slope $-u/v$ belong to the same class, because such points have the form $(m+tv,n-tu)$. Neighboring lines of slope $-u/v$ that go through integer points are separated by distance $1/\psqrt{u^2+v^2}$ from each other, and these lines are perpendicular to lines of slope~$v/u$. If we start at the origin and travel a distance $k/\psqrt{u^2+v^2}$ in direction $(u,v)$, we reach the line of slope~$-u/v$ whose points belong to class~$k$. For example, let $u=2$ and $v=3$. Then the points $(0,0)$, $(3,-2)$, $\ldots$ belong to class~0; the points $(-1,1)$, $(2,-1)$, $\ldots$ belong to class~1; and the distance between these two lines is $1/\sqrt{13}$. The point $(2,3)$ itself belongs to class~13, hence its distance from the origin is $13/\sqrt{13}=\sqrt{13}$ (which we already knew). Suppose we wish to plot envelopes with respect to polygons with integer vertices. Then the best polygon for curves that travel in direction $(v,-u)$ will contain the points of class~$k$ such that $k/\psqrt{u^2+v^2}$ is as close as possible to~$d$, where $d$ is the maximum distance of the given ellipse from the line $ux+vy=0$. The |fillin| correction assumes that a diagonal line has an apparent thickness $2f\cdot\min(\vert u\vert,\vert v\vert)/\psqrt{u^2+v^2}$ greater than what would be obtained with truly square pixels. (If a white pixel at an exterior corner is assumed to have apparent darkness $f_1$ and a black pixel at an interior corner is assumed to have apparent darkness $1-f_2$, then $f=f_1-f_2$ is the |fillin| parameter.) Under this assumption we want to choose $k$ so that $\bigl(k+2f\cdot\min(\vert u\vert,\vert v\vert)\bigr)\big/\psqrt{u^2+v^2}$ is as close as possible to $d$. Integer coordinates for the vertices work nicely because the thickness of the envelope at any given slope is independent of the position of the path with respect to the raster. It turns out, in fact, that the same property holds for polygons whose vertices have coordinates that are integer multiples of~$1\over2$, because ellipses are symmetric about the origin. It's convenient to double all dimensions and require the resulting polygon to have vertices with integer coordinates. For example, to get a circle of {\sl diameter}~$r$, we shall compute integer coordinates for a circle of {\sl radius}~$r$. The circle of radius~$r$ will want to be represented by a polygon that contains the boundary points $(0,\pm r)$ and~$(\pm r,0)$; later we will divide everything by~2 and get a polygon with $(0,\pm{1\over2}r)$ and $(\pm{1\over2}r,0)$ on its boundary. @ In practice the important slopes are those having small values of $u$ and~$v$; these make regular patterns in which our eyes quickly spot irregularities. For example, horizontal and vertical lines (when $u=0$ and $\vert v\vert=1$, or $\vert u\vert=1$ and $v=0$) are the most important; diagonal lines (when $\vert u\vert=\vert v\vert=1$) are next; and then come lines with slope $\pm2$ or $\pm1/2$. The nicest way to generate all rational directions having small numerators and denominators is to generalize the Stern-Peirce tree [cf.~{\sl The Art of Computer Programming\/ \bf2}, exercise 4.5.3--40] @^Peirce, Charles Santiago Sanders@> @^Stern, Moriz Abraham@> to a ``Stern-Peirce wreath'' as follows: Begin with four nodes arranged in a circle, containing the respective directions $(u,v)=(1,0)$, $(0,1)$, $(-1,0)$, and~$(0,-1)$. Then between pairs of consecutive terms $(u,v)$ and $(u',v')$ of the wreath, insert the direction $(u+u',v+v')$; continue doing this until some stopping criterion is fulfilled. It is not difficult to verify that, regardless of the stopping criterion, consecutive directions $(u,v)$ and $(u',v')$ of this wreath will always satisfy the relation $uv'-u'v=1$. Such pairs of directions have a nice property with respect to the equivalence classes described above. Let $l$ be a line of equivalent integer points $(m+tv,n-tu)$ with respect to~$(u,v)$, and let $l'$ be a line of equivalent integer points $(m'+tv',n'-tu')$ with respect to~$(u',v')$. Then $l$ and~$l'$ intersect in an integer point $(m'',n'')$, because the determinant of the linear equations for intersection is $uv'-u'v=1$. Notice that the class number of $(m'',n'')$ with respect to $(u+u',v+v')$ is the sum of its class numbers with respect to $(u,v)$ and~$(u',v')$. Moreover, consecutive points on~$l$ and~$l'$ belong to classes that differ by exactly~1 with respect to $(u+u',v+v')$. This leads to a nice algorithm in which we construct a polygon having ``correct'' class numbers for as many small-integer directions $(u,v)$ as possible: Assuming that lines $l$ and~$l'$ contain points of the correct class for $(u,v)$ and~$(u',v')$, respectively, we determine the intersection $(m'',n'')$ and compute its class with respect to $(u+u',v+v')$. If the class is too large to be the best approximation, we move back the proper number of steps from $(m'',n'')$ toward smaller class numbers on both $l$ and~$l'$, unless this requires moving to points that are no longer in the polygon; in this we arrive at two points that determine a line~$l''$ having the appropriate class. The process continues recursively, until it cannot proceed without removing the last remaining point from the class for $(u,v)$ or the class for $(u',v')$. @ The |make_ellipse| subroutine produces a pointer to a cyclic path whose vertices define a polygon suitable for envelopes. The control points on this path will be ignored; in fact, the fields in knot nodes that are usually reserved for control points are occupied by other data that helps |make_ellipse| compute the desired polygon. Parameters |major_axis| and |minor_axis| define the axes of the ellipse; and parameter |theta| is an angle by which the ellipse is rotated counterclockwise. If |theta=0|, the ellipse has the equation $(x/a)^2+(y/b)^2=1$, where |a=major_axis/2| and |b=minor_axis/2|. In general, the points of the ellipse are generated in the complex plane by the formula $e^{i\theta}(a\cos t+ib\sin t)$, as $t$~ranges over all angles. Notice that if |major_axis=minor_axis=d|, we obtain a circle of diameter~|d|, regardless of the value of |theta|. The method sketched above is used to produce the elliptical polygon, except that the main work is done only in the halfplane obtained from the three starting directions $(0,-1)$, $(1,0)$,~$(0,1)$. Since the ellipse has circular symmetry, we use the fact that the last half of the polygon is simply the negative of the first half. Furthermore, we need to compute only one quarter of the polygon if the ellipse has axis symmetry. @p function make_ellipse(@!major_axis,@!minor_axis:scaled; @!theta:angle):pointer; label done,done1,found; var @!p,@!q,@!r,@!s:pointer; {for list manipulation} @!h:pointer; {head of the constructed knot list} @!alpha,@!beta,@!gamma,@!delta:integer; {special points} @!c,@!d:integer; {class numbers} @!u,@!v:integer; {directions} @!symmetric:boolean; {should the result be symmetric about the axes?} begin @; @; if symmetric then @; @; make_ellipse:=h; end; @ A special data structure is used only with |make_ellipse|: The |right_x|, |left_x|, |right_y|, and |left_y| fields of knot nodes are renamed |right_u|, |left_v|, |right_class|, and |left_length|, in order to store information that simplifies the necessary computations. If |p| and |q| are consecutive knots in this data structure, the |x_coord| and |y_coord| fields of |p| and~|q| contain current vertices of the polygon; their values are integer multiples of |half_unit|. Both of these vertices belong to equivalence class |right_class(p)| with respect to the direction $\bigl($|right_u(p),left_v(q)|$\bigr)$. The number of points of this class on the line from vertex~|p| to vertex~|q| is |1+left_length(q)|. In particular, |left_length(q)=0| means that |x_coord(p)=x_coord(q)| and |y_coord(p)=y_coord(q)|; such duplicate vertices will be discarded during the course of the algorithm. The contents of |right_u(p)| and |left_v(q)| are integer multiples of |half_unit|, just like the coordinate fields. Hence, for example, the point $\bigl($|x_coord(p)-left_v(q),y_coord(p)+right_u(q)|$\bigr)$ also belongs to class number |right_class(p)|. This point is one step closer to the vertex in node~|q|; it equals that vertex if and only if |left_length(q)=1|. The |left_type| and |right_type| fields are not used, but |link| has its normal meaning. To start the process, we create four nodes for the three directions $(0,-1)$, $(1,0)$, and $(0,1)$. The corresponding vertices are $(-\alpha,-\beta)$, $(\gamma,-\beta)$, $(\gamma,\beta)$, and $(\alpha,\beta)$, where $(\alpha,\beta)$ is a half-integer approximation to where the ellipse rises highest above the $x$-axis, and where $\gamma$ is a half-integer approximation to the maximum $x$~coordinate of the ellipse. The fourth of these nodes is not actually calculated if the ellipse has axis symmetry. @d right_u==right_x {|u| value for a pen edge} @d left_v==left_x {|v| value for a pen edge} @d right_class==right_y {equivalence class number of a pen edge} @d left_length==left_y {length of a pen edge} @= @; p:=get_node(knot_node_size); q:=get_node(knot_node_size); r:=get_node(knot_node_size); if symmetric then s:=null@+else s:=get_node(knot_node_size); h:=p; link(p):=q; link(q):=r; link(r):=s; {|s=null| or |link(s)=null|} @; x_coord(p):=-alpha*half_unit; y_coord(p):=-beta*half_unit; x_coord(q):=gamma*half_unit;@/ y_coord(q):=y_coord(p); x_coord(r):=x_coord(q);@/ right_u(p):=0; left_v(q):=-half_unit;@/ right_u(q):=half_unit; left_v(r):=0;@/ right_u(r):=0; right_class(p):=beta; right_class(q):=gamma; right_class(r):=beta;@/ left_length(q):=gamma+alpha; if symmetric then begin y_coord(r):=0; left_length(r):=beta; end else begin y_coord(r):=-y_coord(p); left_length(r):=beta+beta;@/ x_coord(s):=-x_coord(p); y_coord(s):=y_coord(r);@/ left_v(s):=half_unit; left_length(s):=gamma-alpha; end @ One of the important invariants of the pen data structure is that the points are distinct. We may need to correct the pen specification in order to avoid this. (The result of \&{pencircle} will always be at least one pixel wide and one pixel tall, although \&{makepen} is capable of producing smaller pens.) @= if beta=0 then beta:=1; if gamma=0 then gamma:=1; if gamma<=abs(alpha) then if alpha>0 then alpha:=gamma-1 else alpha:=1-gamma @ If $a$ and $b$ are the semi-major and semi-minor axes, the given ellipse rises highest above the $y$-axis at the point $\bigl((a^2-b^2)\sin\theta\cos\theta/\rho\bigr)+i\rho$, where $\rho=\sqrt{(a\sin\theta)^2+(b\cos\theta)^2}$. It reaches furthest to the right of~the $x$-axis at the point $\sigma+i(a^2-b^2)\sin\theta\cos\theta/\sigma$, where $\sigma=\sqrt{(a\cos\theta)^2+(b\sin\theta)^2}$. @= if (major_axis=minor_axis)or(theta mod ninety_deg=0) then begin symmetric:=true; alpha:=0; if odd(theta div ninety_deg) then begin beta:=major_axis; gamma:=minor_axis; n_sin:=fraction_one; n_cos:=0; {|n_sin| and |n_cos| are used later} end else begin beta:=minor_axis; gamma:=major_axis; end; {|n_sin| and |n_cos| aren't needed in this case} end else begin symmetric:=false; n_sin_cos(theta); {set up $|n_sin|=\sin\theta$ and $|n_cos|=\cos\theta$} gamma:=take_fraction(major_axis,n_sin); delta:=take_fraction(minor_axis,n_cos); beta:=pyth_add(gamma,delta); alpha:=take_fraction(take_fraction(major_axis, make_fraction(gamma,beta)),n_cos)@| -take_fraction(take_fraction(minor_axis, make_fraction(delta,beta)),n_sin); alpha:=(alpha+half_unit) div unity; gamma:=pyth_add(take_fraction(major_axis,n_cos), take_fraction(minor_axis,n_sin)); end; beta:=(beta+half_unit) div unity; gamma:=(gamma+half_unit) div unity @ Now |p|, |q|, and |r| march through the list, always representing three consecutive vertices and two consecutive slope directions. When a new slope is interpolated, we back up slightly, until further refinement is impossible; then we march forward again. The somewhat magical operations performed in this part of the algorithm are justified by the theory sketched earlier. Complications arise only from the need to keep zero-length lines out of the final data structure. @= loop@+ begin u:=right_u(p)+right_u(q); v:=left_v(q)+left_v(r); c:=right_class(p)+right_class(q);@/ @; delta:=c-d; {we want to move |delta| steps back from the intersection vertex~|q|} if delta>0 then begin if delta>left_length(r) then delta:=left_length(r); if delta>=left_length(q) then @ else @; end else p:=q; @; end; done: @ The appearance of a zero-length line means that we should advance |p| past it. We must not try to straddle a missing direction, because the algorithm works only on consecutive pairs of directions. @= loop@+ begin q:=link(p); if q=null then goto done; if left_length(q)=0 then begin link(p):=link(q); right_class(p):=right_class(q); right_u(p):=right_u(q); free_node(q,knot_node_size); end else begin r:=link(q); if r=null then goto done; if left_length(r)=0 then begin link(p):=r; free_node(q,knot_node_size); p:=r; end else goto found; end; end; found: @ The `\&{div} 8' near the end of this step comes from the fact that |delta| is scaled by~$2^{15}$ and $d$~by~$2^{16}$, while |take_fraction| removes a scale factor of~$2^{28}$. We also make sure that $d\G\max(\vert u\vert,\vert v\vert)$, so that the pen will always include a circular pen of diameter~1 as a subset; then it won't be possible to get disconnected path envelopes. @= delta:=pyth_add(u,v); if major_axis=minor_axis then d:=major_axis {circles are easy} else begin if theta=0 then begin alpha:=u; beta:=v; end else begin alpha:=take_fraction(u,n_cos)+take_fraction(v,n_sin); beta:=take_fraction(v,n_cos)-take_fraction(u,n_sin); end; alpha:=make_fraction(alpha,delta); beta:=make_fraction(beta,delta); d:=pyth_add(take_fraction(major_axis,alpha), take_fraction(minor_axis,beta)); end; alpha:=abs(u); beta:=abs(v); if alpha0 then d:=d-take_fraction(internal[fillin],make_fraction(beta+beta,delta)); d:=take_fraction((d+4) div 8,delta); alpha:=alpha div half_unit; if d=delta| from vertex~|q| to to vertex~|r|, orthogonal to direction $\bigl($|right_u(q),left_v(r)|$\bigr)$. The best line to direction $(u,v)$ should replace the line from |p| to~|q|; this new line will have the same length as the old. @= begin delta:=left_length(q);@/ right_class(p):=c-delta; right_u(p):=u; left_v(q):=v;@/ x_coord(q):=x_coord(q)-delta*left_v(r); y_coord(q):=y_coord(q)+delta*right_u(q);@/ left_length(r):=left_length(r)-delta; end @ Here is the main case, now that we have dealt with the exception: We insert a new line of length |delta| for direction |(u,v)|, decreasing each of the adjacent lines by |delta| steps. @= begin s:=get_node(knot_node_size); link(p):=s; link(s):=q;@/ x_coord(s):=x_coord(q)+delta*left_v(q); y_coord(s):=y_coord(q)-delta*right_u(p);@/ x_coord(q):=x_coord(q)-delta*left_v(r); y_coord(q):=y_coord(q)+delta*right_u(q);@/ left_v(s):=left_v(q); right_u(s):=u; left_v(q):=v;@/ right_class(s):=c-delta;@/ left_length(s):=left_length(q)-delta; left_length(q):=delta; left_length(r):=left_length(r)-delta; end @ Only the coordinates need to be copied, not the class numbers and other stuff. @= begin s:=null; q:=h; loop@+ begin r:=get_node(knot_node_size); link(r):=s; s:=r;@/ x_coord(s):=x_coord(q); y_coord(s):=-y_coord(q); if q=p then goto done1; q:=link(q); if y_coord(q)=0 then goto done1; end; done1: link(p):=s; beta:=-y_coord(h); while y_coord(p)<>beta do p:=link(p); q:=link(p); end @ Now we use a somewhat tricky fact: The pointer |q| will be null if and only if the line for the final direction $(0,1)$ has been removed. If that line still survives, it should be combined with a possibly surviving line in the initial direction $(0,-1)$. @= if q<>null then begin if right_u(h)=0 then begin p:=h; h:=link(h); free_node(p,knot_node_size);@/ x_coord(q):=-x_coord(h); end; p:=q; end else q:=p; r:=link(h); {now |p=q|, |x_coord(p)=-x_coord(h)|, |y_coord(p)=-y_coord(h)|} repeat s:=get_node(knot_node_size); link(p):=s; p:=s;@/ x_coord(p):=-x_coord(r); y_coord(p):=-y_coord(r); r:=link(r); until r=q; link(p):=h @* \[26] Direction and intersection times. A path of length $n$ is defined parametrically by functions $x(t)$ and $y(t)$, for |0<=t<=n|; we can regard $t$ as the ``time'' at which the path reaches the point $\bigl(x(t),y(t)\bigr)$. In this section of the program we shall consider operations that determine special times associated with given paths: the first time that a path travels in a given direction, and a pair of times at which two paths cross each other. @ Let's start with the easier task. The function |find_direction_time| is given a direction |(x,y)| and a path starting at~|h|. If the path never travels in direction |(x,y)|, the direction time will be~|-1|; otherwise it will be nonnegative. Certain anomalous cases can arise: If |(x,y)=(0,0)|, so that the given direction is undefined, the direction time will be~0. If $\bigl(x'(t), y'(t)\bigr)=(0,0)$, so that the path direction is undefined, it will be assumed to match any given direction at time~|t|. The routine solves this problem in nondegenerate cases by rotating the path and the given direction so that |(x,y)=(1,0)|; i.e., the main task will be to find when a given path first travels ``due east.'' @p function find_direction_time(@!x,@!y:scaled;@!h:pointer):scaled; label exit,found,not_found,done; var @!max:scaled; {$\max\bigl(\vert x\vert,\vert y\vert\bigr)$} @!p,@!q:pointer; {for list traversal} @!n:scaled; {the direction time at knot |p|} @!tt:scaled; {the direction time within a cubic} @@; begin @; n:=0; p:=h; loop@+ begin if right_type(p)=endpoint then goto not_found; q:=link(p); @; p:=q; n:=n+unity; end; not_found: find_direction_time:=-unity; return; found: find_direction_time:=n+tt; exit:end; @ @= if abs(x)0 then y:=fraction_one@+else y:=-fraction_one; end else if x=0 then begin find_direction_time:=0; return; end else begin y:=make_fraction(y,abs(x)); if x>0 then x:=fraction_one@+else x:=-fraction_one; end @ Since we're interested in the tangent directions, we work with the derivative $3B'(x_0,x_1,x_2,x_3;t)=B(x_1-x_0,x_2-x_1,x_3-x_2;t)$ instead of $B(x_0,x_1,x_2,x_3;t)$ itself. The derived coefficients are also scaled up in order to achieve better accuracy. The given path may turn abruptly at a knot, and it might pass the critical tangent direction at such a time. Therefore we remember the direction |phi| in which the previous rotated cubic was traveling. (The value of |phi| will be undefined on the first cubic, i.e., when |n=0|.) @= tt:=0; @; if y1=0 then if x1>=0 then goto found; if n>0 then begin @; if p=h then goto not_found; end; if (x3<>0)or(y3<>0) then phi:=n_arg(x3,y3); @ @ @= @!x1,@!x2,@!x3,@!y1,@!y2,@!y3:scaled; {multiples of rotated derivatives} @!theta,@!phi:angle; {angles of exit and entry at a knot} @!t:fraction; {temp storage} @ @= x1:=right_x(p)-x_coord(p); x2:=left_x(q)-right_x(p); x3:=x_coord(q)-left_x(q);@/ y1:=right_y(p)-y_coord(p); y2:=left_y(q)-right_y(p); y3:=y_coord(q)-left_y(q);@/ max:=abs(x1); if abs(x2)>max then max:=abs(x2); if abs(x3)>max then max:=abs(x3); if abs(y1)>max then max:=abs(y1); if abs(y2)>max then max:=abs(y2); if abs(y3)>max then max:=abs(y3); if max=0 then goto found; while max= theta:=n_arg(x1,y1); if theta>=0 then if phi<=0 then if phi>=theta-one_eighty_deg then goto found; if theta<=0 then if phi>=0 then if phi<=theta+one_eighty_deg then goto found @ In this step we want to use the |crossing_point| routine to find the roots of the quadratic equation $B(y_1,y_2,y_3;t)=0$. Several complications arise: If the quadratic equation has a double root, the curve never crosses zero, and |crossing_point| will find nothing; this case occurs iff $y_1y_3=y_2^2$ and $y_1y_2<0$. If the quadratic equation has simple roots, or only one root, we may have to negate it so that $B(y_1,y_2,y_3;t)$ crosses from positive to negative at its first root. And finally, we need to do special things if $B(y_1,y_2,y_3;t)$ is identically zero. @ @= if x1<0 then if x2<0 then if x3<0 then goto done; if ab_vs_cd(y1,y3,y2,y2)=0 then @; if y1<=0 then if y1<0 then begin y1:=-y1; y2:=-y2; y3:=-y3; end else if y2>0 then begin y2:=-y2; y3:=-y3; end; @=0$@>; done: @ The quadratic polynomial $B(y_1,y_2,y_3;t)$ begins |>=0| and has at most two roots, because we know that it isn't identically zero. It must be admitted that the |crossing_point| routine is not perfectly accurate; rounding errors might cause it to find a root when $y_1y_3>y_2^2$, or to miss the roots when $y_1y_3= t:=crossing_point(y1,y2,y3); if t>fraction_one then goto done; y2:=t_of_the_way(y2)(y3); x1:=t_of_the_way(x1)(x2); x2:=t_of_the_way(x2)(x3); x1:=t_of_the_way(x1)(x2); if x1>=0 then we_found_it; if y2>0 then y2:=0; tt:=t; t:=crossing_point(0,-y2,-y3); if t>fraction_one then goto done; x1:=t_of_the_way(x1)(x2); x2:=t_of_the_way(x2)(x3); if t_of_the_way(x1)(x2)>=0 then begin t:=t_of_the_way(tt)(fraction_one); we_found_it; end @ @= begin if ab_vs_cd(y1,y2,0,0)<0 then begin t:=make_fraction(y1,y1-y2); x1:=t_of_the_way(x1)(x2); x2:=t_of_the_way(x2)(x3); if t_of_the_way(x1)(x2)>=0 then we_found_it; end else if y3=0 then if y1=0 then @=0|@> else if x3>=0 then begin tt:=unity; goto found; end; goto done; end @ At this point we know that the derivative of |y(t)| is identically zero, and that |x1<0|; but either |x2>=0| or |x3>=0|, so there's some hope of traveling east. @=0|...@>= begin t:=crossing_point(-x1,-x2,-x3); if t<=fraction_one then we_found_it; if ab_vs_cd(x1,x3,x2,x2)<=0 then begin t:=make_fraction(x1,x1-x2); we_found_it; end; end @ The intersection of two cubics can be found by an interesting variant of the general bisection scheme described in the introduction to |make_moves|.\ Given $w(t)=B(w_0,w_1,w_2,w_3;t)$ and $z(t)=B(z_0,z_1,z_2,z_3;t)$, we wish to find a pair of times $(t_1,t_2)$ such that $w(t_1)=z(t_2)$, if an intersection exists. First we find the smallest rectangle that encloses the points $\{w_0,w_1,w_2,w_3\}$ and check that it overlaps the smallest rectangle that encloses $\{z_0,z_1,z_2,z_3\}$; if not, the cubics certainly don't intersect. But if the rectangles do overlap, we bisect the intervals, getting new cubics $w'$ and~$w''$, $z'$~and~$z''$; the intersection routine first tries for an intersection between $w'$ and~$z'$, then (if unsuccessful) between $w'$ and~$z''$, then (if still unsuccessful) between $w''$ and~$z'$, finally (if thrice unsuccessful) between $w''$ and~$z''$. After $l$~successful levels of bisection we will have determined the intersection times $t_1$ and~$t_2$ to $l$~bits of accuracy. \def\submin{_{\rm min}} \def\submax{_{\rm max}} As before, it is better to work with the numbers $W_k=2^l(w_k-w_{k-1})$ and $Z_k=2^l(z_k-z_{k-1})$ rather than the coefficients $w_k$ and $z_k$ themselves. We also need one other quantity, $\Delta=2^l(w_0-z_0)$, to determine when the enclosing rectangles overlap. Here's why: The $x$~coordinates of~$w(t)$ are between $u\submin$ and $u\submax$, and the $x$~coordinates of~$z(t)$ are between $x\submin$ and $x\submax$, if we write $w_k=(u_k,v_k)$ and $z_k=(x_k,y_k)$ and $u\submin= \min(u_0,u_1,u_2,u_3)$, etc. These intervals of $x$~coordinates overlap if and only if $u\submin\L x\submax$ and $x\submin\L u\submax$. Letting $$U\submin=\min(0,U_1,U_1+U_2,U_1+U_2+U_3),\; U\submax=\max(0,U_1,U_1+U_2,U_1+U_2+U_3),$$ we have $u\submin=2^lu_0+U\submin$, etc.; the condition for overlap reduces to $$X\submin-U\submax\L 2^l(u_0-x_0)\L X\submax-U\submin.$$ Thus we want to maintain the quantity $2^l(u_0-x_0)$; similarly, the quantity $2^l(v_0-y_0)$ accounts for the $y$~coordinates. The coordinates of $\Delta=2^l(w_0-z_0)$ must stay bounded as $l$ increases, because of the overlap condition; i.e., we know that $X\submin$, $X\submax$, and their relatives are bounded, hence $X\submax- U\submin$ and $X\submin-U\submax$ are bounded. @ Incidentally, if the given cubics intersect more than once, the process just sketched will not necessarily find the lexicographically smallest pair $(t_1,t_2)$. The solution actually obtained will be smallest in ``shuffled order''; i.e., if $t_1=(.a_1a_2\ldots a_{16})_2$ and $t_2=(.b_1b_2\ldots b_{16})_2$, then we will minimize $a_1b_1a_2b_2\ldots a_{16}b_{16}$, not $a_1a_2\ldots a_{16}b_1b_2\ldots b_{16}$. Shuffled order agrees with lexicographic order if all pairs of solutions $(t_1,t_2)$ and $(t_1',t_2')$ have the property that $t_1= @!tol_step:0..6; {either 0 or 3, usually} @ We shall use an explicit stack to implement the recursive bisection method described above. In fact, the |bisect_stack| array is available for this purpose. It will contain numerous 5-word packets like $(U_1,U_2,U_3,U\submin,U\submax)$, as well as 20-word packets comprising the 5-word packets for $U$, $V$, $X$, and~$Y$. The following macros define the allocation of stack positions to the quantities needed for bisection-intersection. @d stack_1(#)==bisect_stack[#] {$U_1$, $V_1$, $X_1$, or $Y_1$} @d stack_2(#)==bisect_stack[#+1] {$U_2$, $V_2$, $X_2$, or $Y_2$} @d stack_3(#)==bisect_stack[#+2] {$U_3$, $V_3$, $X_3$, or $Y_3$} @d stack_min(#)==bisect_stack[#+3] {$U\submin$, $V\submin$, $X\submin$, or $Y\submin$} @d stack_max(#)==bisect_stack[#+4] {$U\submax$, $V\submax$, $X\submax$, or $Y\submax$} @d int_packets=20 {number of words to represent $U_k$, $V_k$, $X_k$, and $Y_k$} @# @d u_packet(#)==#-5 @d v_packet(#)==#-10 @d x_packet(#)==#-15 @d y_packet(#)==#-20 @d l_packets==bisect_ptr-int_packets @d r_packets==bisect_ptr @d ul_packet==u_packet(l_packets) {base of $U'_k$ variables} @d vl_packet==v_packet(l_packets) {base of $V'_k$ variables} @d xl_packet==x_packet(l_packets) {base of $X'_k$ variables} @d yl_packet==y_packet(l_packets) {base of $Y'_k$ variables} @d ur_packet==u_packet(r_packets) {base of $U''_k$ variables} @d vr_packet==v_packet(r_packets) {base of $V''_k$ variables} @d xr_packet==x_packet(r_packets) {base of $X''_k$ variables} @d yr_packet==y_packet(r_packets) {base of $Y''_k$ variables} @# @d u1l==stack_1(ul_packet) {$U'_1$} @d u2l==stack_2(ul_packet) {$U'_2$} @d u3l==stack_3(ul_packet) {$U'_3$} @d v1l==stack_1(vl_packet) {$V'_1$} @d v2l==stack_2(vl_packet) {$V'_2$} @d v3l==stack_3(vl_packet) {$V'_3$} @d x1l==stack_1(xl_packet) {$X'_1$} @d x2l==stack_2(xl_packet) {$X'_2$} @d x3l==stack_3(xl_packet) {$X'_3$} @d y1l==stack_1(yl_packet) {$Y'_1$} @d y2l==stack_2(yl_packet) {$Y'_2$} @d y3l==stack_3(yl_packet) {$Y'_3$} @d u1r==stack_1(ur_packet) {$U''_1$} @d u2r==stack_2(ur_packet) {$U''_2$} @d u3r==stack_3(ur_packet) {$U''_3$} @d v1r==stack_1(vr_packet) {$V''_1$} @d v2r==stack_2(vr_packet) {$V''_2$} @d v3r==stack_3(vr_packet) {$V''_3$} @d x1r==stack_1(xr_packet) {$X''_1$} @d x2r==stack_2(xr_packet) {$X''_2$} @d x3r==stack_3(xr_packet) {$X''_3$} @d y1r==stack_1(yr_packet) {$Y''_1$} @d y2r==stack_2(yr_packet) {$Y''_2$} @d y3r==stack_3(yr_packet) {$Y''_3$} @# @d stack_dx==bisect_stack[bisect_ptr] {stacked value of |delx|} @d stack_dy==bisect_stack[bisect_ptr+1] {stacked value of |dely|} @d stack_tol==bisect_stack[bisect_ptr+2] {stacked value of |tol|} @d stack_uv==bisect_stack[bisect_ptr+3] {stacked value of |uv|} @d stack_xy==bisect_stack[bisect_ptr+4] {stacked value of |xy|} @d int_increment=int_packets+int_packets+5 {number of stack words per level} @= if int_packets+17*int_increment>bistack_size then bad:=32; @ Computation of the min and max is a tedious but fairly fast sequence of instructions; exactly four comparisons are made in each branch. @d set_min_max(#)== if stack_1(#)<0 then if stack_3(#)>=0 then begin if stack_2(#)<0 then stack_min(#):=stack_1(#)+stack_2(#) else stack_min(#):=stack_1(#); stack_max(#):=stack_1(#)+stack_2(#)+stack_3(#); if stack_max(#)<0 then stack_max(#):=0; end else begin stack_min(#):=stack_1(#)+stack_2(#)+stack_3(#); if stack_min(#)>stack_1(#) then stack_min(#):=stack_1(#); stack_max(#):=stack_1(#)+stack_2(#); if stack_max(#)<0 then stack_max(#):=0; end else if stack_3(#)<=0 then begin if stack_2(#)>0 then stack_max(#):=stack_1(#)+stack_2(#) else stack_max(#):=stack_1(#); stack_min(#):=stack_1(#)+stack_2(#)+stack_3(#); if stack_min(#)>0 then stack_min(#):=0; end else begin stack_max(#):=stack_1(#)+stack_2(#)+stack_3(#); if stack_max(#)0 then stack_min(#):=0; end @ It's convenient to keep the current values of $l$, $t_1$, and $t_2$ in the integer form $2^l+2^lt_1$ and $2^l+2^lt_2$. The |cubic_intersection| routine uses global variables |cur_t| and |cur_tt| for this purpose; after successful completion, |cur_t| and |cur_tt| will contain |unity| plus the |scaled| values of $t_1$ and $t_2$. The values of |cur_t| and |cur_tt| will be set to zero if |cubic_intersection| finds no intersection. The routine gives up and gives an approximate answer if it has backtracked more than 5000 times (otherwise there are cases where several minutes of fruitless computation would be possible). @d max_patience=5000 @= @!cur_t,@!cur_tt:integer; {controls and results of |cubic_intersection|} @!time_to_go:integer; {this many backtracks before giving up} @!max_t:integer; {maximum of $2^{l+1}$ so far achieved} @ The given cubics $B(w_0,w_1,w_2,w_3;t)$ and $B(z_0,z_1,z_2,z_3;t)$ are specified in adjacent knot nodes |(p,link(p))| and |(pp,link(pp))|, respectively. @p procedure cubic_intersection(@!p,@!pp:pointer); label continue, not_found, exit; var @!q,@!qq:pointer; {|link(p)|, |link(pp)|} begin time_to_go:=max_patience; max_t:=2; @; loop@+ begin continue: if delx-tol<=stack_max(x_packet(xy))-stack_min(u_packet(uv)) then if delx+tol>=stack_min(x_packet(xy))-stack_max(u_packet(uv)) then if dely-tol<=stack_max(y_packet(xy))-stack_min(v_packet(uv)) then if dely+tol>=stack_min(y_packet(xy))-stack_max(v_packet(uv)) then begin if cur_t>=max_t then begin if max_t=two then {we've done 17 bisections} begin cur_t:=half(cur_t+1); cur_tt:=half(cur_tt+1); return; end; double(max_t); appr_t:=cur_t; appr_tt:=cur_tt; end; @; goto continue; end; if time_to_go>0 then decr(time_to_go) else begin while appr_t; end; exit:end; @ The following variables are global, although they are used only by |cubic_intersection|, because it is necessary on some machines to split |cubic_intersection| up into two procedures. @= @!delx,@!dely:integer; {the components of $\Delta=2^l(w_0-z_0)$} @!tol:integer; {bound on the uncertainly in the overlap test} @!uv,@!xy:0..bistack_size; {pointers to the current packets of interest} @!three_l:integer; {|tol_step| times the bisection level} @!appr_t,@!appr_tt:integer; {best approximations known to the answers} @ We shall assume that the coordinates are sufficiently non-extreme that integer overflow will not occur. @= q:=link(p); qq:=link(pp); bisect_ptr:=int_packets;@/ u1r:=right_x(p)-x_coord(p); u2r:=left_x(q)-right_x(p); u3r:=x_coord(q)-left_x(q); set_min_max(ur_packet);@/ v1r:=right_y(p)-y_coord(p); v2r:=left_y(q)-right_y(p); v3r:=y_coord(q)-left_y(q); set_min_max(vr_packet);@/ x1r:=right_x(pp)-x_coord(pp); x2r:=left_x(qq)-right_x(pp); x3r:=x_coord(qq)-left_x(qq); set_min_max(xr_packet);@/ y1r:=right_y(pp)-y_coord(pp); y2r:=left_y(qq)-right_y(pp); y3r:=y_coord(qq)-left_y(qq); set_min_max(yr_packet);@/ delx:=x_coord(p)-x_coord(pp); dely:=y_coord(p)-y_coord(pp);@/ tol:=0; uv:=r_packets; xy:=r_packets; three_l:=0; cur_t:=1; cur_tt:=1 @ @= stack_dx:=delx; stack_dy:=dely; stack_tol:=tol; stack_uv:=uv; stack_xy:=xy; bisect_ptr:=bisect_ptr+int_increment;@/ double(cur_t); double(cur_tt);@/ u1l:=stack_1(u_packet(uv)); u3r:=stack_3(u_packet(uv)); u2l:=half(u1l+stack_2(u_packet(uv))); u2r:=half(u3r+stack_2(u_packet(uv))); u3l:=half(u2l+u2r); u1r:=u3l; set_min_max(ul_packet); set_min_max(ur_packet);@/ v1l:=stack_1(v_packet(uv)); v3r:=stack_3(v_packet(uv)); v2l:=half(v1l+stack_2(v_packet(uv))); v2r:=half(v3r+stack_2(v_packet(uv))); v3l:=half(v2l+v2r); v1r:=v3l; set_min_max(vl_packet); set_min_max(vr_packet);@/ x1l:=stack_1(x_packet(xy)); x3r:=stack_3(x_packet(xy)); x2l:=half(x1l+stack_2(x_packet(xy))); x2r:=half(x3r+stack_2(x_packet(xy))); x3l:=half(x2l+x2r); x1r:=x3l; set_min_max(xl_packet); set_min_max(xr_packet);@/ y1l:=stack_1(y_packet(xy)); y3r:=stack_3(y_packet(xy)); y2l:=half(y1l+stack_2(y_packet(xy))); y2r:=half(y3r+stack_2(y_packet(xy))); y3l:=half(y2l+y2r); y1r:=y3l; set_min_max(yl_packet); set_min_max(yr_packet);@/ uv:=l_packets; xy:=l_packets; double(delx); double(dely);@/ tol:=tol-three_l+tol_step; double(tol); three_l:=three_l+tol_step @ @= not_found: if odd(cur_tt) then if odd(cur_t) then @ else begin incr(cur_t); delx:=delx+stack_1(u_packet(uv))+stack_2(u_packet(uv)) +stack_3(u_packet(uv)); dely:=dely+stack_1(v_packet(uv))+stack_2(v_packet(uv)) +stack_3(v_packet(uv)); uv:=uv+int_packets; {switch from |l_packet| to |r_packet|} decr(cur_tt); xy:=xy-int_packets; {switch from |r_packet| to |l_packet|} delx:=delx+stack_1(x_packet(xy))+stack_2(x_packet(xy)) +stack_3(x_packet(xy)); dely:=dely+stack_1(y_packet(xy))+stack_2(y_packet(xy)) +stack_3(y_packet(xy)); end else begin incr(cur_tt); tol:=tol+three_l; delx:=delx-stack_1(x_packet(xy))-stack_2(x_packet(xy)) -stack_3(x_packet(xy)); dely:=dely-stack_1(y_packet(xy))-stack_2(y_packet(xy)) -stack_3(y_packet(xy)); xy:=xy+int_packets; {switch from |l_packet| to |r_packet|} end @ @= begin cur_t:=half(cur_t); cur_tt:=half(cur_tt); if cur_t=0 then return; bisect_ptr:=bisect_ptr-int_increment; three_l:=three_l-tol_step; delx:=stack_dx; dely:=stack_dy; tol:=stack_tol; uv:=stack_uv; xy:=stack_xy;@/ goto not_found; end @ The |path_intersection| procedure is much simpler (and less interesting). It invokes |cubic_intersection| in lexicographic order until finding a pair of cubics that intersect. The final intersection times are placed in |cur_t| and~|cur_tt|. @p procedure path_intersection(@!h,@!hh:pointer); label exit; var @!p,@!pp:pointer; {link registers that traverse the given paths} @!n,@!nn:integer; {integer parts of intersection times, minus |unity|} begin @; tol_step:=0; repeat n:=-unity; p:=h; repeat if right_type(p)<>endpoint then begin nn:=-unity; pp:=hh; repeat if right_type(pp)<>endpoint then begin cubic_intersection(p,pp); if cur_t>0 then begin cur_t:=cur_t+n; cur_tt:=cur_tt+nn; return; end; end; nn:=nn+unity; pp:=link(pp); until pp=hh; end; n:=n+unity; p:=link(p); until p=h; tol_step:=tol_step+3; until tol_step>3; cur_t:=-unity; cur_tt:=-unity; exit:end; @ @= if right_type(h)=endpoint then begin right_x(h):=x_coord(h); left_x(h):=x_coord(h); right_y(h):=y_coord(h); left_y(h):=y_coord(h); right_type(h):=explicit; end; if right_type(hh)=endpoint then begin right_x(hh):=x_coord(hh); left_x(hh):=x_coord(hh); right_y(hh):=y_coord(hh); left_y(hh):=y_coord(hh); right_type(hh):=explicit; end; @* \[27] Online graphic output. \MF\ displays images on the user's screen by means of a few primitive operations that are defined below. These operations have deliberately been kept simple so that they can be implemented without great difficulty on a wide variety of machines. Since \PASCAL\ has no traditional standards for graphic output, some system-dependent code needs to be written in order to support this aspect of \MF; but the necessary routines are usually quite easy to write. @^system dependencies@> In fact, there are exactly four such routines: \yskip\hang |init_screen| does whatever initialization is necessary to support the other operations; it is a boolean function that returns |false| if graphic output cannot be supported (e.g., if the other three routines have not been written, or if the user doesn't have the right kind of terminal). \yskip\hang |blank_rectangle| updates a buffer area in memory so that all pixels in a specified rectangle will be set to the background color. \yskip\hang |paint_row| assigns values to specified pixels in a row of the buffer just mentioned, based on ``transition'' indices explained below. \yskip\hang |update_screen| displays the current screen buffer; the effects of |blank_rectangle| and |paint_row| commands may or may not become visible until the next |update_screen| operation is performed. (Thus, |update_screen| is analogous to |update_terminal|.) \yskip\noindent The \PASCAL\ code here is a minimum version of |init_screen| and |update_screen|, usable on \MF\ installations that don't support screen output. If |init_screen| is changed to return |true| instead of |false|, the other routines will simply log the fact that they have been called; they won't really display anything. The standard test routines for \MF\ use this log information to check that \MF\ is working properly, but the |wlog| instructions should be removed from production versions of \MF. @p function init_screen:boolean; begin init_screen:=false; end; @# procedure update_screen; {will be called only if |init_screen| returns |true|} begin @!init wlog_ln('Calling UPDATESCREEN');@+tini {for testing only} end; @ The user's screen is assumed to be a rectangular area, |screen_width| pixels wide and |screen_depth| pixels deep. The pixel in the upper left corner is said to be in column~0 of row~0; the pixel in the lower right corner is said to be in column |screen_width-1| of row |screen_depth-1|. Notice that row numbers increase from top to bottom, contrary to \MF's other coordinates. Each pixel is assumed to have two states, referred to in this documentation as |black| and |white|. The background color is called |white| and the other color is called |black|; but any two distinct pixel values can actually be used. For example, the author developed \MF\ on a system for which |white| was black and |black| was bright green. @d white=0 {background pixels} @d black=1 {visible pixels} @= @!screen_row=0..screen_depth; {a row number on the screen} @!screen_col=0..screen_width; {a column number on the screen} @!trans_spec=array[screen_col] of screen_col; {a transition spec, see below} @!pixel_color=white..black; {specifies one of the two pixel values} @ We'll illustrate the |blank_rectangle| and |paint_row| operations by pretending to declare a screen buffer called |screen_pixel|. This code is actually commented out, but it does specify the intended effects. @= @{@!screen_pixel:array[screen_row,screen_col] of pixel_color;@+@} @ The |blank_rectangle| routine simply whitens all pixels that lie in columns |left_col| through |right_col-1|, inclusive, of rows |top_row| through |bot_row-1|, inclusive, given four parameters that satisfy the relations $$\hbox{|0<=left_col<=right_col<=screen_width|,\quad |0<=top_row<=bot_row<=screen_depth|.}$$ If |left_col=right_col| or |top_row=bot_row|, nothing happens. The commented-out code in the following procedure is for illustrative purposes only. @^system dependencies@> @p procedure blank_rectangle(@!left_col,@!right_col:screen_col; @!top_row,@!bot_row:screen_row); var @!r:screen_row; @!c:screen_col; begin @{@+for r:=top_row to bot_row-1 do for c:=left_col to right_col-1 do screen_pixel[r,c]:=white;@+@}@/ @!init wlog_cr; {this will be done only after |init_screen=true|} wlog_ln('Calling BLANKRECTANGLE(',left_col:1,',', right_col:1,',',top_row:1,',',bot_row:1,')');@+tini end; @ The real work of screen display is done by |paint_row|. But it's not hard work, because the operation affects only one of the screen rows, and it affects only a contiguous set of columns in that row. There are four parameters: |r|~(the row), |b|~(the initial color), |a|~(the array of transition specifications), and |n|~(the number of transitions). The elements of~|a| will satisfy $$0\L a[0] @p procedure paint_row(@!r:screen_row;@!b:pixel_color;var @!a:trans_spec; @!n:screen_col); var @!k:screen_col; {an index into |a|} @!c:screen_col; {an index into |screen_pixel|} begin @{ k:=0; c:=a[0]; repeat incr(k); repeat screen_pixel[r,c]:=b; incr(c); until c=a[k]; b:=black-b; {$|black|\swap|white|$} until k=n;@+@}@/ @!init wlog('Calling PAINTROW(',r:1,',',b:1,';'); {this is done only after |init_screen=true|} for k:=0 to n do begin wlog(a[k]:1); if k<>n then wlog(','); end; wlog_ln(')');@+tini end; @ The remainder of \MF's screen routines are system-independent calls on the four primitives just defined. First we have a global boolean variable that tells if |init_screen| has been called, and another one that tells if |init_screen| has given a |true| response. @= @!screen_started:boolean; {have the screen primitives been initialized?} @!screen_OK:boolean; {is it legitimate to call |blank_rectangle|, |paint_row|, and |update_screen|?} @ @d start_screen==begin if not screen_started then begin screen_OK:=init_screen; screen_started:=true; end; end @= screen_started:=false; screen_OK:=false; @ \MF\ provides the user with 16 ``window'' areas on the screen, in each of which it is possible to produce independent displays. It should be noted that \MF's windows aren't really independent ``clickable'' entities in the sense of multi-window graphic workstations; \MF\ simply maps them into subsets of a single screen image that is controlled by |init_screen|, |blank_rectangle|, |paint_row|, and |update_screen| as described above. Implementations of \MF\ on a multi-window workstation probably therefore make use of only two windows in the other sense: one for the terminal output and another for the screen with \MF's 16 areas. Henceforth we shall use the term window only in \MF's sense. @= @!window_number=0..15; @ A user doesn't have to use any of the 16 windows. But when a window is ``opened,'' it is allocated to a specific rectangular portion of the screen and to a specific rectangle with respect to \MF's coordinates. The relevant data is stored in global arrays |window_open|, |left_col|, |right_col|, |top_row|, |bot_row|, |m_window|, and |n_window|. The |window_open| array is boolean, and its significance is obvious. The |left_col|, \dots, |bot_row| arrays contain screen coordinates that can be used to blank the entire window with |blank_rectangle|. And the other two arrays just mentioned handle the conversion between actual coordinates and screen coordinates: \MF's pixel in column~$m$ of row~$n$ will appear in screen column |m_window+m| and in screen row |n_window-n|, provided that these lie inside the boundaries of the window. Another array |window_time| holds the number of times this window has been updated. @= @!window_open:array[window_number] of boolean; {has this window been opened?} @!left_col:array[window_number] of screen_col; {leftmost column position on screen} @!right_col:array[window_number] of screen_col; {rightmost column position, plus~1} @!top_row:array[window_number] of screen_row; {topmost row position on screen} @!bot_row:array[window_number] of screen_row; {bottommost row position, plus~1} @!m_window:array[window_number] of integer; {offset between user and screen columns} @!n_window:array[window_number] of integer; {offset between user and screen rows} @!window_time:array[window_number] of integer; {it has been updated this often} @ @= for k:=0 to 15 do begin window_open[k]:=false; window_time[k]:=0; end; @ Opening a window isn't like opening a file, because you can open it as often as you like, and you never have to close it again. The idea is simply to define special points on the current screen display. Overlapping window specifications may cause complex effects that can be understood only by scrutinizing \MF's display algorithms; thus it has been left undefined in the \MF\ user manual, although the behavior @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> is in fact predictable. Here is a subroutine that implements the command `\&{openwindow}~|k| \&{from}~$(\\{r0},\\{c0})$ \&{to}~$(\\{r1},\\{c1})$ \&{at}~$(x,y)$'. @p procedure open_a_window(@!k:window_number;@!r0,@!c0,@!r1,@!c1:scaled; @!x,@!y:scaled); var @!m,@!n:integer; {pixel coordinates} begin @; window_open[k]:=true; incr(window_time[k]);@/ left_col[k]:=c0; right_col[k]:=c1; top_row[k]:=r0; bot_row[k]:=r1;@/ @; start_screen; if screen_OK then begin blank_rectangle(c0,c1,r0,r1); update_screen; end; end; @ A window whose coordinates don't fit the existing screen size will be truncated until they do. @= if r0<0 then r0:=0@+else r0:=round_unscaled(r0); r1:=round_unscaled(r1); if r1>screen_depth then r1:=screen_depth; if r1screen_depth then r0:=r1@+else r1:=r0; if c0<0 then c0:=0@+else c0:=round_unscaled(c0); c1:=round_unscaled(c1); if c1>screen_width then c1:=screen_width; if c1screen_width then c0:=c1@+else c1:=c0 @ Three sets of coordinates are rampant, and they must be kept straight! (i)~\MF's main coordinates refer to the edges between pixels. (ii)~\MF's pixel coordinates (within edge structures) say that the pixel bounded by $(m,n)$, $(m,n+1)$, $(m+1,n)$, and~$(m+1,n+1)$ is in pixel row number~$n$ and pixel column number~$m$. (iii)~Screen coordinates, on the other hand, have rows numbered in increasing order from top to bottom, as mentioned above. @^coordinates, explained@> The program here first computes integers $m$ and $n$ such that pixel column~$m$ of pixel row~$n$ will be at the upper left corner of the window. Hence pixel column |m-c0| of pixel row |n+r0| will be at the upper left corner of the screen. @= m:=round_unscaled(x); n:=round_unscaled(y)-1;@/ m_window[k]:=c0-m; n_window[k]:=r0+n @ Now here comes \MF's most complicated operation related to window display: Given the number~|k| of an open window, the pixels of positive weight in |cur_edges| will be shown as |black| in the window; all other pixels will be shown as |white|. @p procedure disp_edges(@!k:window_number); label done,found; var @!p,@!q:pointer; {for list manipulation} @!already_there:boolean; {is a previous incarnation in the window?} @!r:integer; {row number} @@; begin if screen_OK then if left_col[k]; p:=link(cur_edges); r:=n_window[k]-(n_min(cur_edges)-zero_field); while (p<>cur_edges)and(r>=top_row[k]) do begin if r; p:=link(p); decr(r); end; update_screen; incr(window_time[k]); last_window(cur_edges):=k; last_window_time(cur_edges):=window_time[k]; end; end; @ Since it takes some work to display a row, we try to avoid recomputation whenever we can. @= begin if unsorted(p)>void then sort_edges(p) else if unsorted(p)=void then if already_there then goto done; unsorted(p):=void; {this time we'll paint, but maybe not next time} @; paint_row(r,b,row_transition,n); done: end @ The transition-specification parameter to |paint_row| is always the same array. @= @!row_transition:trans_spec; {an array of |black|/|white| transitions} @ The job remaining is to go through the list |sorted(p)|, unpacking the |info| fields into |m| and weight, then making |black| the pixels whose accumulated weight~|w| is positive. @= @!n:screen_col; {the highest active index in |row_transition|} @!w,@!ww:integer; {old and new accumulated weights} @!b:pixel_color; {status of first pixel in the row transitions} @!m,@!mm:integer; {old and new screen column positions} @!d:integer; {edge-and-weight without |min_halfword| compensation} @!m_adjustment:integer; {conversion between edge and screen coordinates} @!right_edge:integer; {largest edge-and-weight that could affect the window} @!min_col:screen_col; {the smallest screen column number in the window} @ Some precomputed constants make the display calculations faster. @= m_adjustment:=m_window[k]-m_offset(cur_edges);@/ right_edge:=8*(right_col[k]-m_adjustment);@/ min_col:=left_col[k] @ @= n:=0; ww:=0; m:=-1; w:=0; q:=sorted(p); row_transition[0]:=min_col; loop@+ begin if q=sentinel then d:=right_edge else d:=ho(info(q)); mm:=(d div 8)+m_adjustment; if mm<>m then begin @; m:=mm; w:=ww; end; if d>=right_edge then goto found; ww:=ww+(d mod 8)-zero_w; q:=link(q); end; found:@; @ Now |m| is a screen column |= if w<=0 then begin if ww>0 then if m>min_col then begin if n=0 then if already_there then begin b:=white; incr(n); end else b:=black else incr(n); row_transition[n]:=m; end; end else if ww<=0 then if m>min_col then begin if n=0 then b:=black; incr(n); row_transition[n]:=m; end @ If the entire row is |white| in the window area, we can omit painting it when |already_there| is false, since it has already been blanked out in that case. When the following code is invoked, |row_transition[n]| will be strictly less than |right_col[k]|. @= if already_there or(ww>0) then begin if n=0 then if ww>0 then b:=black else b:=white; incr(n); row_transition[n]:=right_col[k]; end else if n=0 then goto done @* \[28] Dynamic linear equations. \MF\ users define variables implicitly by stating equations that should be satisfied; the computer is supposed to be smart enough to solve those equations. And indeed, the computer tries valiantly to do so, by distinguishing five different types of numeric values: \smallskip\hang |type(p)=known| is the nice case, when |value(p)| is the |scaled| value of the variable whose address is~|p|. \smallskip\hang |type(p)=dependent| means that |value(p)| is not present, but |dep_list(p)| points to a {\sl dependency list\/} that expresses the value of variable~|p| as a |scaled| number plus a sum of independent variables with |fraction| coefficients. \smallskip\hang |type(p)=independent| means that |value(p)=64s+m|, where |s>0| is a ``serial number'' reflecting the time this variable was first used in an equation; also |0<=m<64|, and each dependent variable that refers to this one is actually referring to the future value of this variable times~$2^m$. (Usually |m=0|, but higher degrees of scaling are sometimes needed to keep the coefficients in dependency lists from getting too large. The value of~|m| will always be even.) \smallskip\hang |type(p)=numeric_type| means that variable |p| hasn't appeared in an equation before, but it has been explicitly declared to be numeric. \smallskip\hang |type(p)=undefined| means that variable |p| hasn't appeared before. \smallskip\noindent We have actually discussed these five types in the reverse order of their history during a computation: Once |known|, a variable never again becomes |dependent|; once |dependent|, it almost never again becomes |independent|; once |independent|, it never again becomes |numeric_type|; and once |numeric_type|, it never again becomes |undefined| (except of course when the user specifically decides to scrap the old value and start again). A backward step may, however, take place: Sometimes a |dependent| variable becomes |independent| again, when one of the independent variables it depends on is reverting to |undefined|. @d s_scale=64 {the serial numbers are multiplied by this factor} @d new_indep(#)== {create a new independent variable} begin type(#):=independent; serial_no:=serial_no+s_scale; value(#):=serial_no; end @= @!serial_no:integer; {the most recent serial number, times |s_scale|} @ @=new_indep(q+s) @ But how are dependency lists represented? Simple; the linear combination $\alpha_1v_1+\cdots+\alpha_kv_k+\beta$ appears in |k+1| value nodes. If |q=dep_list(p)| points to this list, and if |k>0|, then |value(q)= @t$\alpha_1$@>| (which is a |fraction|); |info(q)| points to the location of $\alpha_1$; and |link(p)| points to the dependency list $\alpha_2v_2+\cdots+\alpha_kv_k+\beta$. On the other hand if |k=0|, then |value(q)=@t$\beta$@>| (which is |scaled|) and |info(q)=null|. The independent variables $v_1$, \dots,~$v_k$ have been sorted so that they appear in decreasing order of their |value| fields (i.e., of their serial numbers). \ (It is convenient to use decreasing order, since |value(null)=0|. If the independent variables were not sorted by serial number but by some other criterion, such as their location in |mem|, the equation-solving mechanism would be too system-dependent, because the ordering can affect the computed results.) The |link| field in the node that contains the constant term $\beta$ is called the {\sl final link\/} of the dependency list. \MF\ maintains a doubly-linked master list of all dependency lists, in terms of a permanently allocated node in |mem| called |dep_head|. If there are no dependencies, we have |link(dep_head)=dep_head| and |prev_dep(dep_head)=dep_head|; otherwise |link(dep_head)| points to the first dependent variable, say~|p|, and |prev_dep(p)=dep_head|. We have |type(p)=dependent|, and |dep_list(p)| points to its dependency list. If the final link of that dependency list occurs in location~|q|, then |link(q)| points to the next dependent variable (say~|r|); and |prev_dep(r)=q|, etc. @d dep_list(#)==link(value_loc(#)) {half of the |value| field in a |dependent| variable} @d prev_dep(#)==info(value_loc(#)) {the other half; makes a doubly linked list} @d dep_node_size=2 {the number of words per dependency node} @= serial_no:=0; link(dep_head):=dep_head; prev_dep(dep_head):=dep_head; info(dep_head):=null; dep_list(dep_head):=null; @ Actually the description above contains a little white lie. There's another kind of variable called |proto_dependent|, which is just like a |dependent| one except that the $\alpha$ coefficients in its dependency list are |scaled| instead of being fractions. Proto-dependency lists are mixed with dependency lists in the nodes reachable from |dep_head|. @ Here is a procedure that prints a dependency list in symbolic form. The second parameter should be either |dependent| or |proto_dependent|, to indicate the scaling of the coefficients. @= procedure print_dependency(@!p:pointer;@!t:small_number); label exit; var @!v:integer; {a coefficient} @!pp,@!q:pointer; {for list manipulation} begin pp:=p; loop@+ begin v:=abs(value(p)); q:=info(p); if q=null then {the constant term} begin if (v<>0)or(p=pp) then begin if value(p)>0 then if p<>pp then print_char("+"); print_scaled(value(p)); end; return; end; @; if type(q)<>independent then confusion("dep"); @:this can't happen dep}{\quad dep@> print_variable_name(q); v:=value(q) mod s_scale; while v>0 do begin print("*4"); v:=v-2; end; p:=link(p); end; exit:end; @ @= if value(p)<0 then print_char("-") else if p<>pp then print_char("+"); if t=dependent then v:=round_fraction(v); if v<>unity then print_scaled(v) @ The maximum absolute value of a coefficient in a given dependency list is returned by the following simple function. @p function max_coef(@!p:pointer):fraction; var @!x:fraction; {the maximum so far} begin x:=0; while info(p)<>null do begin if abs(value(p))>x then x:=abs(value(p)); p:=link(p); end; max_coef:=x; end; @ One of the main operations needed on dependency lists is to add a multiple of one list to the other; we call this |p_plus_fq|, where |p| and~|q| point to dependency lists and |f| is a fraction. If the coefficient of any independent variable becomes |coef_bound| or more, in absolute value, this procedure changes the type of that variable to `|independent_needing_fix|', and sets the global variable |fix_needed| to~|true|. The value of $|coef_bound|=\mu$ is chosen so that $\mu^2+\mu<8$; this means that the numbers we deal with won't get too large. (Instead of the ``optimum'' $\mu=(\sqrt{33}-1)/2\approx 2.3723$, the safer value 7/3 is taken as the threshold.) The changes mentioned in the preceding paragraph are actually done only if the global variable |watch_coefs| is |true|. But it usually is; in fact, it is |false| only when \MF\ is making a dependency list that will soon be equated to zero. Several procedures that act on dependency lists, including |p_plus_fq|, set the global variable |dep_final| to the final (constant term) node of the dependency list that they produce. @d coef_bound==@'4525252525 {|fraction| approximation to 7/3} @d independent_needing_fix=0 @= @!fix_needed:boolean; {does at least one |independent| variable need scaling?} @!watch_coefs:boolean; {should we scale coefficients that exceed |coef_bound|?} @!dep_final:pointer; {location of the constant term and final link} @ @= fix_needed:=false; watch_coefs:=true; @ The |p_plus_fq| procedure has a fourth parameter, |t|, that should be set to |proto_dependent| if |p| is a proto-dependency list. In this case |f| will be |scaled|, not a |fraction|. Similarly, the fifth parameter~|tt| should be |proto_dependent| if |q| is a proto-dependency list. List |q| is unchanged by the operation; but list |p| is totally destroyed. The final link of the dependency list or proto-dependency list returned by |p_plus_fq| is the same as the original final link of~|p|. Indeed, the constant term of the result will be located in the same |mem| location as the original constant term of~|p|. Coefficients of the result are assumed to be zero if they are less than a certain threshold. This compensates for inevitable rounding errors, and tends to make more variables `|known|'. The threshold is approximately $10^{-5}$ in the case of normal dependency lists, $10^{-4}$ for proto-dependencies. @d fraction_threshold=2685 {a |fraction| coefficient less than this is zeroed} @d half_fraction_threshold=1342 {half of |fraction_threshold|} @d scaled_threshold=8 {a |scaled| coefficient less than this is zeroed} @d half_scaled_threshold=4 {half of |scaled_threshold|} @= function p_plus_fq(@!p:pointer;@!f:integer;@!q:pointer; @!t,@!tt:small_number):pointer; label done; var @!pp,@!qq:pointer; {|info(p)| and |info(q)|, respectively} @!r,@!s:pointer; {for list manipulation} @!threshold:integer; {defines a neighborhood of zero} @!v:integer; {temporary register} begin if t=dependent then threshold:=fraction_threshold else threshold:=scaled_threshold; r:=temp_head; pp:=info(p); qq:=info(q); loop@+ if pp=qq then if pp=null then goto done else @ else if value(pp) else begin link(r):=p; r:=p; p:=link(p); pp:=info(p); end; done: if t=dependent then value(p):=slow_add(value(p),take_fraction(value(q),f)) else value(p):=slow_add(value(p),take_scaled(value(q),f)); link(r):=p; dep_final:=p; p_plus_fq:=link(temp_head); end; @ @= begin if tt=dependent then v:=value(p)+take_fraction(f,value(q)) else v:=value(p)+take_scaled(f,value(q)); value(p):=v; s:=p; p:=link(p); if abs(v)=coef_bound then if watch_coefs then begin type(qq):=independent_needing_fix; fix_needed:=true; end; link(r):=s; r:=s; end; pp:=info(p); q:=link(q); qq:=info(q); end @ @= begin if tt=dependent then v:=take_fraction(f,value(q)) else v:=take_scaled(f,value(q)); if abs(v)>half(threshold) then begin s:=get_node(dep_node_size); info(s):=qq; value(s):=v; if abs(v)>=coef_bound then if watch_coefs then begin type(qq):=independent_needing_fix; fix_needed:=true; end; link(r):=s; r:=s; end; q:=link(q); qq:=info(q); end @ It is convenient to have another subroutine for the special case of |p_plus_fq| when |f=1.0|. In this routine lists |p| and |q| are both of the same type~|t| (either |dependent| or |proto_dependent|). @p function p_plus_q(@!p:pointer;@!q:pointer;@!t:small_number):pointer; label done; var @!pp,@!qq:pointer; {|info(p)| and |info(q)|, respectively} @!r,@!s:pointer; {for list manipulation} @!threshold:integer; {defines a neighborhood of zero} @!v:integer; {temporary register} begin if t=dependent then threshold:=fraction_threshold else threshold:=scaled_threshold; r:=temp_head; pp:=info(p); qq:=info(q); loop@+ if pp=qq then if pp=null then goto done else @ else if value(pp)= begin v:=value(p)+value(q); value(p):=v; s:=p; p:=link(p); pp:=info(p); if abs(v)=coef_bound then if watch_coefs then begin type(qq):=independent_needing_fix; fix_needed:=true; end; link(r):=s; r:=s; end; q:=link(q); qq:=info(q); end @ A somewhat simpler routine will multiply a dependency list by a given constant~|v|. The constant is either a |fraction| less than |fraction_one|, or it is |scaled|. In the latter case we might be forced to convert a dependency list to a proto-dependency list. Parameters |t0| and |t1| are the list types before and after; they should agree unless |t0=dependent| and |t1=proto_dependent| and |v_is_scaled=true|. @p function p_times_v(@!p:pointer;@!v:integer; @!t0,@!t1:small_number;@!v_is_scaled:boolean):pointer; var @!r,@!s:pointer; {for list manipulation} @!w:integer; {tentative coefficient} @!threshold:integer; @!scaling_down:boolean; begin if t0<>t1 then scaling_down:=true@+else scaling_down:=not v_is_scaled; if t1=dependent then threshold:=half_fraction_threshold else threshold:=half_scaled_threshold; r:=temp_head; while info(p)<>null do begin if scaling_down then w:=take_fraction(v,value(p)) else w:=take_scaled(v,value(p)); if abs(w)<=threshold then begin s:=link(p); free_node(p,dep_node_size); p:=s; end else begin if abs(w)>=coef_bound then begin fix_needed:=true; type(info(p)):=independent_needing_fix; end; link(r):=p; r:=p; value(p):=w; p:=link(p); end; end; link(r):=p; if v_is_scaled then value(p):=take_scaled(value(p),v) else value(p):=take_fraction(value(p),v); p_times_v:=link(temp_head); end; @ Similarly, we sometimes need to divide a dependency list by a given |scaled| constant. @= function p_over_v(@!p:pointer;@!v:scaled; @!t0,@!t1:small_number):pointer; var @!r,@!s:pointer; {for list manipulation} @!w:integer; {tentative coefficient} @!threshold:integer; @!scaling_down:boolean; begin if t0<>t1 then scaling_down:=true@+else scaling_down:=false; if t1=dependent then threshold:=half_fraction_threshold else threshold:=half_scaled_threshold; r:=temp_head; while info(p)<>null do begin if scaling_down then if abs(v)<@'2000000 then w:=make_scaled(value(p),v*@'10000) else w:=make_scaled(round_fraction(value(p)),v) else w:=make_scaled(value(p),v); if abs(w)<=threshold then begin s:=link(p); free_node(p,dep_node_size); p:=s; end else begin if abs(w)>=coef_bound then begin fix_needed:=true; type(info(p)):=independent_needing_fix; end; link(r):=p; r:=p; value(p):=w; p:=link(p); end; end; link(r):=p; value(p):=make_scaled(value(p),v); p_over_v:=link(temp_head); end; @ Here's another utility routine for dependency lists. When an independent variable becomes dependent, we want to remove it from all existing dependencies. The |p_with_x_becoming_q| function computes the dependency list of~|p| after variable~|x| has been replaced by~|q|. This procedure has basically the same calling conventions as |p_plus_fq|: List~|q| is unchanged; list~|p| is destroyed; the constant node and the final link are inherited from~|p|; and the fourth parameter tells whether or not |p| is |proto_dependent|. However, the global variable |dep_final| is not altered if |x| does not occur in list~|p|. @p function p_with_x_becoming_q(@!p,@!x,@!q:pointer;@!t:small_number):pointer; var @!r,@!s:pointer; {for list manipulation} @!v:integer; {coefficient of |x|} @!sx:integer; {serial number of |x|} begin s:=p; r:=temp_head; sx:=value(x); while value(info(s))>sx do begin r:=s; s:=link(s); end; if info(s)<>x then p_with_x_becoming_q:=p else begin link(temp_head):=p; link(r):=link(s); v:=value(s); free_node(s,dep_node_size); p_with_x_becoming_q:=p_plus_fq(link(temp_head),v,q,t,dependent); end; end; @ Here's a simple procedure that reports an error when a variable has just received a known value that's out of the required range. @= procedure val_too_big(@!x:scaled); begin if internal[warning_check]>0 then begin print_err("Value is too large ("); print_scaled(x); print_char(")"); @.Value is too large@> help4("The equation I just processed has given some variable")@/ ("a value of 4096 or more. Continue and I'll try to cope")@/ ("with that big value; but it might be dangerous.")@/ ("(Set warningcheck:=0 to suppress this message.)"); error; end; end; @ When a dependent variable becomes known, the following routine removes its dependency list. Here |p| points to the variable, and |q| points to the dependency list (which is one node long). @= procedure make_known(@!p,@!q:pointer); var @!t:dependent..proto_dependent; {the previous type} begin prev_dep(link(q)):=prev_dep(p); link(prev_dep(p)):=link(q); t:=type(p); type(p):=known; value(p):=value(q); free_node(q,dep_node_size); if abs(value(p))>=fraction_one then val_too_big(value(p)); if internal[tracing_equations]>0 then if interesting(p) then begin begin_diagnostic; print_nl("#### "); @:]]]\#\#\#\#_}{\.{\#\#\#\#}@> print_variable_name(p); print_char("="); print_scaled(value(p)); end_diagnostic(false); end; if cur_exp=p then if cur_type=t then begin cur_type:=known; cur_exp:=value(p); free_node(p,value_node_size); end; end; @ The |fix_dependencies| routine is called into action when |fix_needed| has been triggered. The program keeps a list~|s| of independent variables whose coefficients must be divided by~4. In unusual cases, this fixup process might reduce one or more coefficients to zero, so that a variable will become known more or less by default. @= procedure fix_dependencies; label done; var @!p,@!q,@!r,@!s,@!t:pointer; {list manipulation registers} @!x:pointer; {an independent variable} begin r:=link(dep_head); s:=null; while r<>dep_head do begin t:=r; @; r:=link(q); if q=dep_list(t) then make_known(t,q); end; while s<>null do begin p:=link(s); x:=info(s); free_avail(s); s:=p; type(x):=independent; value(x):=value(x)+2; end; fix_needed:=false; end; @ @d independent_being_fixed=1 {this variable already appears in |s|} @= r:=value_loc(t); {|link(r)=dep_list(t)|} loop@+ begin q:=link(r); x:=info(q); if x=null then goto done; if type(x)<=independent_being_fixed then begin if type(x)28 then single_dependency:=const_dependency(0) else begin q:=get_node(dep_node_size); value(q):=two_to_the[28-m]; info(q):=p;@/ link(q):=const_dependency(0); single_dependency:=q; end; end; @ We sometimes need to make an exact copy of a dependency list. @p function copy_dep_list(@!p:pointer):pointer; label done; var @!q:pointer; {the new dependency list} begin q:=get_node(dep_node_size); dep_final:=q; loop@+ begin info(dep_final):=info(p); value(dep_final):=value(p); if info(dep_final)=null then goto done; link(dep_final):=get_node(dep_node_size); dep_final:=link(dep_final); p:=link(p); end; done:copy_dep_list:=q; end; @ But how do variables normally become known? Ah, now we get to the heart of the equation-solving mechanism. The |linear_eq| procedure is given a |dependent| or |proto_dependent| list,~|p|, in which at least one independent variable appears. It equates this list to zero, by choosing an independent variable with the largest coefficient and making it dependent on the others. The newly dependent variable is eliminated from all current dependencies, thereby possibly making other dependent variables known. The given list |p| is, of course, totally destroyed by all this processing. @p procedure linear_eq(@!p:pointer;@!t:small_number); var @!q,@!r,@!s:pointer; {for link manipulation} @!x:pointer; {the variable that loses its independence} @!n:integer; {the number of times |x| had been halved} @!v:integer; {the coefficient of |x| in list |p|} @!prev_r:pointer; {lags one step behind |r|} @!final_node:pointer; {the constant term of the new dependency list} @!w:integer; {a tentative coefficient} begin @; x:=info(q); n:=value(x) mod s_scale;@/ @; if internal[tracing_equations]>0 then @; @; @; if fix_needed then fix_dependencies; end; @ @= q:=p; r:=link(p); v:=value(q); while info(r)<>null do begin if abs(value(r))>abs(v) then begin q:=r; v:=value(r); end; r:=link(r); end @ Here we want to change the coefficients from |scaled| to |fraction|, except in the constant term. In the common case of a trivial equation like `\.{x=3.14}', we will have |v=-fraction_one|, |q=p|, and |t=dependent|. @= s:=temp_head; link(s):=p; r:=p; repeat if r=q then begin link(s):=link(r); free_node(r,dep_node_size); end else begin w:=make_fraction(value(r),v); if abs(w)<=half_fraction_threshold then begin link(s):=link(r); free_node(r,dep_node_size); end else begin value(r):=-w; s:=r; end; end; r:=link(s); until info(r)=null; if t=proto_dependent then value(r):=-make_scaled(value(r),v) else if v<>-fraction_one then value(r):=-make_fraction(value(r),v); final_node:=r; p:=link(temp_head) @ @= if interesting(x) then begin begin_diagnostic; print_nl("## "); print_variable_name(x); @:]]]\#\#_}{\.{\#\#}@> w:=n; while w>0 do begin print("*4"); w:=w-2; end; print_char("="); print_dependency(p,dependent); end_diagnostic(false); end @ @= prev_r:=dep_head; r:=link(dep_head); while r<>dep_head do begin s:=dep_list(r); q:=p_with_x_becoming_q(s,x,p,type(r)); if info(q)=null then make_known(r,q) else begin dep_list(r):=q; repeat q:=link(q); until info(q)=null; prev_r:=q; end; r:=link(prev_r); end @ @= if n>0 then @; if info(p)=null then begin type(x):=known; value(x):=value(p); if abs(value(x))>=fraction_one then val_too_big(value(x)); free_node(p,dep_node_size); if cur_exp=x then if cur_type=independent then begin cur_exp:=value(x); cur_type:=known; free_node(x,value_node_size); end; end else begin type(x):=dependent; dep_final:=final_node; new_dep(x,p); if cur_exp=x then if cur_type=independent then cur_type:=dependent; end @ @= begin s:=temp_head; link(temp_head):=p; r:=p; repeat if n>30 then w:=0 else w:=value(r) div two_to_the[n]; if (abs(w)<=half_fraction_threshold)and(info(r)<>null) then begin link(s):=link(r); free_node(r,dep_node_size); end else begin value(r):=w; s:=r; end; r:=link(s); until info(s)=null; p:=link(temp_head); end @ The |check_mem| procedure, which is used only when \MF\ is being debugged, makes sure that the current dependency lists are well formed. @= q:=dep_head; p:=link(q); while p<>dep_head do begin if prev_dep(p)<>q then begin print_nl("Bad PREVDEP at "); print_int(p); @.Bad PREVDEP...@> end; p:=dep_list(p); r:=hi_mem_min; repeat if info(p)>=r then begin print_nl("Out of order at "); print_int(p); @.Out of order...@> end; r:=info(p); q:=p; p:=link(q); until r=null; end @* \[29] Dynamic nonlinear equations. Variables of numeric type are maintained by the general scheme of independent, dependent, and known values that we have just studied; and the components of pair and transform variables are handled in the same way. But \MF\ also has five other types of values: \&{boolean}, \&{string}, \&{pen}, \&{path}, and \&{picture}; what about them? Equations are allowed between nonlinear quantities, but only in a simple form. Two variables that haven't yet been assigned values are either equal to each other, or they're not. Before a boolean variable has received a value, its type is |unknown_boolean|; similarly, there are variables whose type is |unknown_string|, |unknown_pen|, |unknown_path|, and |unknown_picture|. In such cases the value is either |null| (which means that no other variables are equivalent to this one), or it points to another variable of the same undefined type. The pointers in the latter case form a cycle of nodes, which we shall call a ``ring.'' Rings of undefined variables may include capsules, which arise as intermediate results within expressions or as \&{expr} parameters to macros. When one member of a ring receives a value, the same value is given to all the other members. In the case of paths and pictures, this implies making separate copies of a potentially large data structure; users should restrain their enthusiasm for such generality, unless they have lots and lots of memory space. @ The following procedure is called when a capsule node is being added to a ring (e.g., when an unknown variable is mentioned in an expression). @p function new_ring_entry(@!p:pointer):pointer; var q:pointer; {the new capsule node} begin q:=get_node(value_node_size); name_type(q):=capsule; type(q):=type(p); if value(p)=null then value(q):=p@+else value(q):=value(p); value(p):=q; new_ring_entry:=q; end; @ Conversely, we might delete a capsule or a variable before it becomes known. The following procedure simply detaches a quantity from its ring, without recycling the storage. @= procedure ring_delete(@!p:pointer); var @!q:pointer; begin q:=value(p); if q<>null then if q<>p then begin while value(q)<>p do q:=value(q); value(q):=value(p); end; end; @ Eventually there might be an equation that assigns values to all of the variables in a ring. The |nonlinear_eq| subroutine does the necessary propagation of values. If the parameter |flush_p| is |true|, node |p| itself needn't receive a value, because it will soon be recycled. @p procedure nonlinear_eq(@!v:integer;@!p:pointer;@!flush_p:boolean); var @!t:small_number; {the type of ring |p|} @!q,@!r:pointer; {link manipulation registers} begin t:=type(p)-unknown_tag; q:=value(p); if flush_p then type(p):=vacuous@+else p:=q; repeat r:=value(q); type(q):=t; case t of boolean_type: value(q):=v; string_type: begin value(q):=v; add_str_ref(v); end; pen_type: begin value(q):=v; add_pen_ref(v); end; path_type: value(q):=copy_path(v); picture_type: value(q):=copy_edges(v); end; {there ain't no more cases} q:=r; until q=p; end; @ If two members of rings are equated, and if they have the same type, the |ring_merge| procedure is called on to make them equivalent. @p procedure ring_merge(@!p,@!q:pointer); label exit; var @!r:pointer; {traverses one list} begin r:=value(p); while r<>p do begin if r=q then begin @; return; end; r:=value(r); end; r:=value(p); value(p):=value(q); value(q):=r; exit:end; @ @= begin print_err("Redundant equation");@/ @.Redundant equation@> help2("I already knew that this equation was true.")@/ ("But perhaps no harm has been done; let's continue.");@/ put_get_error; end @* \[30] Introduction to the syntactic routines. Let's pause a moment now and try to look at the Big Picture. The \MF\ program consists of three main parts: syntactic routines, semantic routines, and output routines. The chief purpose of the syntactic routines is to deliver the user's input to the semantic routines, while parsing expressions and locating operators and operands. The semantic routines act as an interpreter responding to these operators, which may be regarded as commands. And the output routines are periodically called on to produce compact font descriptions that can be used for typesetting or for making interim proof drawings. We have discussed the basic data structures and many of the details of semantic operations, so we are good and ready to plunge into the part of \MF\ that actually controls the activities. Our current goal is to come to grips with the |get_next| procedure, which is the keystone of \MF's input mechanism. Each call of |get_next| sets the value of three variables |cur_cmd|, |cur_mod|, and |cur_sym|, representing the next input token. $$\vbox{\halign{#\hfil\cr \hbox{|cur_cmd| denotes a command code from the long list of codes given earlier;}\cr \hbox{|cur_mod| denotes a modifier of the command code;}\cr \hbox{|cur_sym| is the hash address of the symbolic token that was just scanned,}\cr \hbox{\qquad or zero in the case of a numeric or string or capsule token.}\cr}}$$ Underlying this external behavior of |get_next| is all the machinery necessary to convert from character files to tokens. At a given time we may be only partially finished with the reading of several files (for which \&{input} was specified), and partially finished with the expansion of some user-defined macros and/or some macro parameters, and partially finished reading some text that the user has inserted online, and so on. When reading a character file, the characters must be converted to tokens; comments and blank spaces must be removed, numeric and string tokens must be evaluated. To handle these situations, which might all be present simultaneously, \MF\ uses various stacks that hold information about the incomplete activities, and there is a finite state control for each level of the input mechanism. These stacks record the current state of an implicitly recursive process, but the |get_next| procedure is not recursive. @= @!cur_cmd: eight_bits; {current command set by |get_next|} @!cur_mod: integer; {operand of current command} @!cur_sym: halfword; {hash address of current symbol} @ The |print_cmd_mod| routine prints a symbolic interpretation of a command code and its modifier. It consists of a rather tedious sequence of print commands, and most of it is essentially an inverse to the |primitive| routine that enters a \MF\ primitive into |hash| and |eqtb|. Therefore almost all of this procedure appears elsewhere in the program, together with the corresponding |primitive| calls. @= procedure print_cmd_mod(@!c,@!m:integer); begin case c of @t\4@>@@/ othercases print("[unknown command code!]") endcases; end; @ Here is a procedure that displays a given command in braces, in the user's transcript file. @d show_cur_cmd_mod==show_cmd_mod(cur_cmd,cur_mod) @p procedure show_cmd_mod(@!c,@!m:integer); begin begin_diagnostic; print_nl("{"); print_cmd_mod(c,m); print_char("}"); end_diagnostic(false); end; @* \[31] Input stacks and states. The state of \MF's input mechanism appears in the input stack, whose entries are records with five fields, called |index|, |start|, |loc|, |limit|, and |name|. The top element of this stack is maintained in a global variable for which no subscripting needs to be done; the other elements of the stack appear in an array. Hence the stack is declared thus: @= @!in_state_record = record @!index_field: quarterword; @!start_field,@!loc_field, @!limit_field, @!name_field: halfword; end; @ @= @!input_stack : array[0..stack_size] of in_state_record; @!input_ptr : 0..stack_size; {first unused location of |input_stack|} @!max_in_stack: 0..stack_size; {largest value of |input_ptr| when pushing} @!cur_input : in_state_record; {the ``top'' input state} @ We've already defined the special variable |@!loc==cur_input.loc_field| in our discussion of basic input-output routines. The other components of |cur_input| are defined in the same way: @d index==cur_input.index_field {reference for buffer information} @d start==cur_input.start_field {starting position in |buffer|} @d limit==cur_input.limit_field {end of current line in |buffer|} @d name==cur_input.name_field {name of the current file} @ Let's look more closely now at the five control variables (|index|,~|start|,~|loc|,~|limit|,~|name|), assuming that \MF\ is reading a line of characters that have been input from some file or from the user's terminal. There is an array called |buffer| that acts as a stack of all lines of characters that are currently being read from files, including all lines on subsidiary levels of the input stack that are not yet completed. We will return to the other lines when we are finished with the present input file. (Incidentally, on a machine with byte-oriented addressing, it would be appropriate to combine |buffer| with the |str_pool| array, letting the buffer entries grow downward from the top of the string pool and checking that these two tables don't bump into each other.) The line we are currently working on begins in position |start| of the buffer; the next character we are about to read is |buffer[loc]|; and |limit| is the location of the last character present. We always have |loc<=limit|. For convenience, |buffer[limit]| has been set to |"%"|, so that the end of a line is easily sensed. The |name| variable is a string number that designates the name of the current file, if we are reading a text file. It is 0 if we are reading from the terminal for normal input, or 1 if we are executing a \&{readstring} command, or 2 if we are reading a string that was moved into the buffer by \&{scantokens}. @ Additional information about the current line is available via the |index| variable, which counts how many lines of characters are present in the buffer below the current level. We have |index=0| when reading from the terminal and prompting the user for each line; then if the user types, e.g., `\.{input font}', we will have |index=1| while reading the file \.{font.mf}. However, it does not follow that |index| is the same as the input stack pointer, since many of the levels on the input stack may come from token lists. The global variable |in_open| is equal to the |index| value of the highest non-token-list level. Thus, the number of partially read lines in the buffer is |in_open+1|, and we have |in_open=index| when we are not reading a token list. If we are not currently reading from the terminal, we are reading from the file variable |input_file[index]|. We use the notation |terminal_input| as a convenient abbreviation for |name=0|, and |cur_file| as an abbreviation for |input_file[index]|. The global variable |line| contains the line number in the topmost open file, for use in error messages. If we are not reading from the terminal, |line_stack[index]| holds the line number for the enclosing level, so that |line| can be restored when the current file has been read. If more information about the input state is needed, it can be included in small arrays like those shown here. For example, the current page or segment number in the input file might be put into a variable |@!page|, maintained for enclosing levels in `\ignorespaces|@!page_stack:array[1..max_in_open] of integer|\unskip' by analogy with |line_stack|. @^system dependencies@> @d terminal_input==(name=0) {are we reading from the terminal?} @d cur_file==input_file[index] {the current |alpha_file| variable} @= @!in_open : 0..max_in_open; {the number of lines in the buffer, less one} @!input_file : array[1..max_in_open] of alpha_file; @!line : integer; {current line number in the current source file} @!line_stack : array[1..max_in_open] of integer; @ However, all this discussion about input state really applies only to the case that we are inputting from a file. There is another important case, namely when we are currently getting input from a token list. In this case |index>max_in_open|, and the conventions about the other state variables are different: \yskip\hang|loc| is a pointer to the current node in the token list, i.e., the node that will be read next. If |loc=null|, the token list has been fully read. \yskip\hang|start| points to the first node of the token list; this node may or may not contain a reference count, depending on the type of token list involved. \yskip\hang|token_type|, which takes the place of |index| in the discussion above, is a code number that explains what kind of token list is being scanned. \yskip\hang|name| points to the |eqtb| address of the control sequence being expanded, if the current token list is a macro not defined by \&{vardef}. Macros defined by \&{vardef} have |name=null|; their name can be deduced by looking at their first two parameters. \yskip\hang|param_start|, which takes the place of |limit|, tells where the parameters of the current macro or loop text begin in the |param_stack|. \yskip\noindent The |token_type| can take several values, depending on where the current token list came from: \yskip \indent|forever_text|, if the token list being scanned is the body of a \&{forever} loop; \indent|loop_text|, if the token list being scanned is the body of a \&{for} or \&{forsuffixes} loop; \indent|parameter|, if a \&{text} or \&{suffix} parameter is being scanned; \indent|backed_up|, if the token list being scanned has been inserted as `to be read again'. \indent|inserted|, if the token list being scanned has been inserted as part of error recovery; \indent|macro|, if the expansion of a user-defined symbolic token is being scanned. \yskip\noindent The token list begins with a reference count if and only if |token_type= macro|. @^reference counts@> @d token_type==index {type of current token list} @d token_state==(index>max_in_open) {are we scanning a token list?} @d file_state==(index<=max_in_open) {are we scanning a file line?} @d param_start==limit {base of macro parameters in |param_stack|} @d forever_text=max_in_open+1 {|token_type| code for loop texts} @d loop_text=max_in_open+2 {|token_type| code for loop texts} @d parameter=max_in_open+3 {|token_type| code for parameter texts} @d backed_up=max_in_open+4 {|token_type| code for texts to be reread} @d inserted=max_in_open+5 {|token_type| code for inserted texts} @d macro=max_in_open+6 {|token_type| code for macro replacement texts} @ The |param_stack| is an auxiliary array used to hold pointers to the token lists for parameters at the current level and subsidiary levels of input. This stack grows at a different rate from the others. @= @!param_stack:array [0..param_size] of pointer; {token list pointers for parameters} @!param_ptr:0..param_size; {first unused entry in |param_stack|} @!max_param_stack:integer; {largest value of |param_ptr|} @ Thus, the ``current input state'' can be very complicated indeed; there can be many levels and each level can arise in a variety of ways. The |show_context| procedure, which is used by \MF's error-reporting routine to print out the current input state on all levels down to the most recent line of characters from an input file, illustrates most of these conventions. The global variable |file_ptr| contains the lowest level that was displayed by this procedure. @= @!file_ptr:0..stack_size; {shallowest level shown by |show_context|} @ The status at each level is indicated by printing two lines, where the first line indicates what was read so far and the second line shows what remains to be read. The context is cropped, if necessary, so that the first line contains at most |half_error_line| characters, and the second contains at most |error_line|. Non-current input levels whose |token_type| is `|backed_up|' are shown only if they have not been fully read. @p procedure show_context; {prints where the scanner is} label done; var @!old_setting:0..max_selector; {saved |selector| setting} @@/ begin file_ptr:=input_ptr; input_stack[file_ptr]:=cur_input; {store current state} loop@+begin cur_input:=input_stack[file_ptr]; {enter into the context} @; if file_state then if (name>2) or (file_ptr=0) then goto done; decr(file_ptr); end; done: cur_input:=input_stack[input_ptr]; {restore original state} end; @ @= if (file_ptr=input_ptr) or file_state or (token_type<>backed_up) or (loc<>null) then {we omit backed-up token lists that have already been read} begin tally:=0; {get ready to count characters} old_setting:=selector; if file_state then begin @; @; end else begin @; @; end; selector:=old_setting; {stop pseudoprinting} @; end @ This routine should be changed, if necessary, to give the best possible indication of where the current line resides in the input file. For example, on some systems it is best to print both a page and line number. @^system dependencies@> @= if name<=1 then if terminal_input and(file_ptr=0) then print_nl("<*>") else print_nl("") else if name=2 then print_nl("") else begin print_nl("l."); print_int(line); end; print_char(" ") @ @= case token_type of forever_text: print_nl(" "); loop_text: @; parameter: print_nl(" "); backed_up: if loc=null then print_nl(" ") else print_nl(" "); inserted: print_nl(" "); macro: begin print_ln; if name<>null then print(text(name)) else @; print("->"); end; othercases print_nl("?") {this should never happen} @.?\relax@> endcases @ The parameter that corresponds to a loop text is either a token list (in the case of \&{forsuffixes}) or a ``capsule'' (in the case of \&{for}). We'll discuss capsules later; for now, all we need to know is that the |link| field in a capsule parameter is |void| and that |print_exp(p,0)| displays the value of capsule~|p| in abbreviated form. @= begin print_nl("null then if link(p)=void then print_exp(p,0) {we're in a \&{for} loop} else show_token_list(p,null,20,tally); print(")> "); end @ The first two parameters of a macro defined by \&{vardef} will be token lists representing the macro's prefix and ``at point.'' By putting these together, we get the macro's full name. @= begin p:=param_stack[param_start]; if p=null then show_token_list(param_stack[param_start+1],null,20,tally) else begin q:=p; while link(q)<>null do q:=link(q); link(q):=param_stack[param_start+1]; show_token_list(p,null,20,tally); link(q):=null; end; end @ Now it is necessary to explain a little trick. We don't want to store a long string that corresponds to a token list, because that string might take up lots of memory; and we are printing during a time when an error message is being given, so we dare not do anything that might overflow one of \MF's tables. So `pseudoprinting' is the answer: We enter a mode of printing that stores characters into a buffer of length |error_line|, where character $k+1$ is placed into \hbox{|trick_buf[k mod error_line]|} if |k(error_line, tally+1+error_line-half_error_line)|. At the end of the pseudoprinting, the values of |first_count|, |tally|, and |trick_count| give us all the information we need to print the two lines, and all of the necessary text is in |trick_buf|. Namely, let |l| be the length of the descriptive information that appears on the first line. The length of the context information gathered for that line is |k=first_count|, and the length of the context information gathered for line~2 is $m=\min(|tally|, |trick_count|)-k$. If |l+k<=h|, where |h=half_error_line|, we print |trick_buf[0..k-1]| after the descriptive information on line~1, and set |n:=l+k|; here |n| is the length of line~1. If $l+k>h$, some cropping is necessary, so we set |n:=h| and print `\.{...}' followed by $$\hbox{|trick_buf[(l+k-h+3)..k-1]|,}$$ where subscripts of |trick_buf| are circular modulo |error_line|. The second line consists of |n|~spaces followed by |trick_buf[k..(k+m-1)]|, unless |n+m>error_line|; in the latter case, further cropping is done. This is easier to program than to explain. @= @!i:0..buf_size; {index into |buffer|} @!l:integer; {length of descriptive information on line 1} @!m:integer; {context information gathered for line 2} @!n:0..error_line; {length of line 1} @!p: integer; {starting or ending place in |trick_buf|} @!q: integer; {temporary index} @ The following code sets up the print routines so that they will gather the desired information. @d begin_pseudoprint== begin l:=tally; tally:=0; selector:=pseudo; trick_count:=1000000; end @d set_trick_count== begin first_count:=tally; trick_count:=tally+1+error_line-half_error_line; if trick_count= if trick_count=1000000 then set_trick_count; {|set_trick_count| must be performed} if tallyerror_line then print("...") @ But the trick is distracting us from our current goal, which is to understand the input state. So let's concentrate on the data structures that are being pseudoprinted as we finish up the |show_context| procedure. @= begin_pseudoprint; if limit>0 then for i:=start to limit-1 do begin if i=loc then set_trick_count; print(buffer[i]); end @ @= begin_pseudoprint; if token_type<>macro then show_token_list(start,loc,100000,0) else show_macro(start,loc,100000) @ Here is the missing piece of |show_token_list| that is activated when the token beginning line~2 is about to be shown: @=set_trick_count @* \[32] Maintaining the input stacks. The following subroutines change the input status in commonly needed ways. First comes |push_input|, which stores the current state and creates a new level (having, initially, the same properties as the old). @d push_input==@t@> {enter a new input level, save the old} begin if input_ptr>max_in_stack then begin max_in_stack:=input_ptr; if input_ptr=stack_size then overflow("input stack size",stack_size); @:METAFONT capacity exceeded input stack size}{\quad input stack size@> end; input_stack[input_ptr]:=cur_input; {stack the record} incr(input_ptr); end @ And of course what goes up must come down. @d pop_input==@t@> {leave an input level, re-enter the old} begin decr(input_ptr); cur_input:=input_stack[input_ptr]; end @ Here is a procedure that starts a new level of token-list input, given a token list |p| and its type |t|. If |t=macro|, the calling routine should set |name|, reset~|loc|, and increase the macro's reference count. @d back_list(#)==begin_token_list(#,backed_up) {backs up a simple token list} @p procedure begin_token_list(@!p:pointer;@!t:quarterword); begin push_input; start:=p; token_type:=t; param_start:=param_ptr; loc:=p; end; @ When a token list has been fully scanned, the following computations should be done as we leave that level of input. @^inner loop@> @p procedure end_token_list; {leave a token-list input level} label done; var @!p:pointer; {temporary register} begin if token_type>=backed_up then {token list to be deleted} if token_type<=inserted then begin flush_token_list(start); goto done; end else delete_mac_ref(start); {update reference count} while param_ptr>param_start do {parameters must be flushed} begin decr(param_ptr); p:=param_stack[param_ptr]; if p<>null then if link(p)=void then {it's an \&{expr} parameter} begin recycle_value(p); free_node(p,value_node_size); end else flush_token_list(p); {it's a \&{suffix} or \&{text} parameter} end; done: pop_input; check_interrupt; end; @ The contents of |cur_cmd,cur_mod,cur_sym| are placed into an equivalent token by the |cur_tok| routine. @^inner loop@> @p @t\4@>@@;@/ function cur_tok:pointer; var @!p:pointer; {a new token node} @!save_type:small_number; {|cur_type| to be restored} @!save_exp:integer; {|cur_exp| to be restored} begin if cur_sym=0 then if cur_cmd=capsule_token then begin save_type:=cur_type; save_exp:=cur_exp; make_exp_copy(cur_mod); p:=stash_cur_exp; link(p):=null; cur_type:=save_type; cur_exp:=save_exp; end else begin p:=get_node(token_node_size); value(p):=cur_mod; name_type(p):=token; if cur_cmd=numeric_token then type(p):=known else type(p):=string_type; end else begin fast_get_avail(p); info(p):=cur_sym; end; cur_tok:=p; end; @ Sometimes \MF\ has read too far and wants to ``unscan'' what it has seen. The |back_input| procedure takes care of this by putting the token just scanned back into the input stream, ready to be read again. If |cur_sym<>0|, the values of |cur_cmd| and |cur_mod| are irrelevant. @p procedure back_input; {undoes one token of input} var @!p:pointer; {a token list of length one} begin p:=cur_tok; while token_state and(loc=null) do end_token_list; {conserve stack space} back_list(p); end; @ The |back_error| routine is used when we want to restore or replace an offending token just before issuing an error message. We disable interrupts during the call of |back_input| so that the help message won't be lost. @p procedure back_error; {back up one token and call |error|} begin OK_to_interrupt:=false; back_input; OK_to_interrupt:=true; error; end; @# procedure ins_error; {back up one inserted token and call |error|} begin OK_to_interrupt:=false; back_input; token_type:=inserted; OK_to_interrupt:=true; error; end; @ The |begin_file_reading| procedure starts a new level of input for lines of characters to be read from a file, or as an insertion from the terminal. It does not take care of opening the file, nor does it set |loc| or |limit| or |line|. @^system dependencies@> @p procedure begin_file_reading; begin if in_open=max_in_open then overflow("text input levels",max_in_open); @:METAFONT capacity exceeded text input levels}{\quad text input levels@> if first=buf_size then overflow("buffer size",buf_size); @:METAFONT capacity exceeded buffer size}{\quad buffer size@> incr(in_open); push_input; index:=in_open; line_stack[index]:=line; start:=first; name:=0; {|terminal_input| is now |true|} end; @ Conversely, the variables must be downdated when such a level of input is finished: @p procedure end_file_reading; begin first:=start; line:=line_stack[index]; if index<>in_open then confusion("endinput"); @:this can't happen endinput}{\quad endinput@> if name>2 then a_close(cur_file); {forget it} pop_input; decr(in_open); end; @ In order to keep the stack from overflowing during a long sequence of inserted `\.{show}' commands, the following routine removes completed error-inserted lines from memory. @p procedure clear_for_error_prompt; begin while file_state and terminal_input and@| (input_ptr>0)and(loc=limit) do end_file_reading; print_ln; clear_terminal; end; @ To get \MF's whole input mechanism going, we perform the following actions. @= begin input_ptr:=0; max_in_stack:=0; in_open:=0; max_buf_stack:=0; param_ptr:=0; max_param_stack:=0; first:=1; start:=1; index:=0; line:=0; name:=0; force_eof:=false; if not init_terminal then goto final_end; limit:=last; first:=last+1; {|init_terminal| has set |loc| and |last|} end; @* \[33] Getting the next token. The heart of \MF's input mechanism is the |get_next| procedure, which we shall develop in the next few sections of the program. Perhaps we shouldn't actually call it the ``heart,'' however; it really acts as \MF's eyes and mouth, reading the source files and gobbling them up. And it also helps \MF\ to regurgitate stored token lists that are to be processed again. The main duty of |get_next| is to input one token and to set |cur_cmd| and |cur_mod| to that token's command code and modifier. Furthermore, if the input token is a symbolic token, that token's |hash| address is stored in |cur_sym|; otherwise |cur_sym| is set to zero. Underlying this simple description is a certain amount of complexity because of all the cases that need to be handled, as explained above. However, the inner loop of |get_next| is reasonably short and fast. @ Before getting into |get_next|, we need to consider a mechanism by which \MF\ helps keep errors from propagating too far. Whenever the program goes into a mode where it keeps calling |get_next| repeatedly until a certain condition is met, it sets |scanner_status| to some value other than |normal|. Then if an input file ends, or if an `\&{outer}' symbol appears, an appropriate error recovery will be possible. The global variable |warning_info| helps in this error recovery by providing additional information. For example, |warning_info| might indicate the name of a macro whose replacement text is being scanned. @d normal=0 {|scanner_status| at ``quiet times''} @d skipping=1 {|scanner_status| when false conditional text is being skipped} @d flushing=2 {|scanner_status| when junk after a statement is being ignored} @d absorbing=3 {|scanner_status| when a \&{text} parameter is being scanned} @d var_defining=4 {|scanner_status| when a \&{vardef} is being scanned} @d op_defining=5 {|scanner_status| when a macro \&{def} is being scanned} @d loop_defining=6 {|scanner_status| when a \&{for} loop is being scanned} @= @!scanner_status:normal..loop_defining; {are we scanning at high speed?} @!warning_info:integer; {if so, what else do we need to know, in case an error occurs during the scanning?} @ @= scanner_status:=normal; @ The following subroutine is called when an `\&{outer}' symbolic token has been scanned or when the end of a file has been reached. These two cases are distinguished by |cur_sym|, which is zero at the end of a file. @p function check_outer_validity:boolean; var @!p:pointer; {points to inserted token list} begin if scanner_status=normal then check_outer_validity:=true else begin deletions_allowed:=false; @; if scanner_status>skipping then @ else begin print_err("Incomplete if; all text was ignored after line "); @.Incomplete if...@> print_int(warning_info);@/ help3("A forbidden `outer' token occurred in skipped text.")@/ ("This kind of error happens when you say `if...' and forget")@/ ("the matching `fi'. I've inserted a `fi'; this might work."); if cur_sym=0 then help_line[2]:=@| "The file ended while I was skipping conditional text."; cur_sym:=frozen_fi; ins_error; end; deletions_allowed:=true; check_outer_validity:=false; end; end; @ @= if cur_sym<>0 then begin p:=get_avail; info(p):=cur_sym; back_list(p); {prepare to read the symbolic token again} end @ @= begin runaway; {print the definition-so-far} if cur_sym=0 then print_err("File ended") @.File ended while scanning...@> else begin print_err("Forbidden token found"); @.Forbidden token found...@> end; print(" while scanning "); help4("I suspect you have forgotten an `enddef',")@/ ("causing me to read past where you wanted me to stop.")@/ ("I'll try to recover; but if the error is serious,")@/ ("you'd better type `E' or `X' now and fix your file.");@/ case scanner_status of @t\4@>@@; end; {there are no other cases} ins_error; end @ As we consider various kinds of errors, it is also appropriate to change the first line of the help message just given; |help_line[3]| points to the string that might be changed. @= flushing: begin print("to the end of the statement"); help_line[3]:="A previous error seems to have propagated,"; cur_sym:=frozen_semicolon; end; absorbing: begin print("a text argument"); help_line[3]:="It seems that a right delimiter was left out,"; if warning_info=0 then cur_sym:=frozen_end_group else begin cur_sym:=frozen_right_delimiter; equiv(frozen_right_delimiter):=warning_info; end; end; var_defining, op_defining: begin print("the definition of "); if scanner_status=op_defining then print(text(warning_info)) else print_variable_name(warning_info); cur_sym:=frozen_end_def; end; loop_defining: begin print("the text of a "); print(text(warning_info)); print(" loop"); help_line[3]:="I suspect you have forgotten an `endfor',"; cur_sym:=frozen_end_for; end; @ The |runaway| procedure displays the first part of the text that occurred when \MF\ began its special |scanner_status|, if that text has been saved. @= procedure runaway; begin if scanner_status>flushing then begin print_nl("Runaway "); case scanner_status of absorbing: print("text?"); var_defining,op_defining: print("definition?"); loop_defining: print("loop?"); end; {there are no other cases} print_ln; show_token_list(link(hold_head),null,error_line-10,0); end; end; @ We need to mention a procedure that may be called by |get_next|. @p procedure@?firm_up_the_line; forward; @ And now we're ready to take the plunge into |get_next| itself. @d switch=25 {a label in |get_next|} @d start_numeric_token=85 {another} @d start_decimal_token=86 {and another} @d fin_numeric_token=87 {and still another, although |goto| is considered harmful} @p procedure get_next; {sets |cur_cmd|, |cur_mod|, |cur_sym| to next token} @^inner loop@> label restart, {go here to get the next input token} exit, {go here when the next input token has been got} found, {go here when the end of a symbolic token has been found} switch, {go here to branch on the class of an input character} start_numeric_token,start_decimal_token,fin_numeric_token,done; {go here at crucial stages when scanning a number} var @!k:0..buf_size; {an index into |buffer|} @!c:ASCII_code; {the current character in the buffer} @!class:ASCII_code; {its class number} @!n,@!f:integer; {registers for decimal-to-binary conversion} begin restart: cur_sym:=0; if file_state then @ else @; @; exit:end; @ When a symbolic token is declared to be `\&{outer}', its command code is increased by |outer_tag|. @^inner loop@> @= cur_cmd:=eq_type(cur_sym); cur_mod:=equiv(cur_sym); if cur_cmd>=outer_tag then if check_outer_validity then cur_cmd:=cur_cmd-outer_tag else goto restart @ A percent sign appears in |buffer[limit]|; this makes it unnecessary to have a special test for end-of-line. @^inner loop@> @= begin switch: c:=buffer[loc]; incr(loc); class:=char_class[c]; case class of digit_class: goto start_numeric_token; period_class: begin class:=char_class[buffer[loc]]; if class>period_class then goto switch else if class end; space_class: goto switch; percent_class: begin @; check_interrupt; goto switch; end; string_class: @; isolated_classes: begin k:=loc-1; goto found; end; invalid_class: @; othercases do_nothing {letters, etc.} endcases;@/ k:=loc-1; while char_class[buffer[loc]]=class do incr(loc); goto found; start_numeric_token:@; start_decimal_token:@; fin_numeric_token:@; found: cur_sym:=id_lookup(k,loc-k); end @ We go to |restart| instead of to |switch|, because |state| might equal |token_list| after the error has been dealt with (cf.\ |clear_for_error_prompt|). @= begin print_err("Text line contains an invalid character"); @.Text line contains...@> help2("A funny symbol that I can't read has just been input.")@/ ("Continue, and I'll forget that it ever happened.");@/ deletions_allowed:=false; error; deletions_allowed:=true; goto restart; end @ @= begin if buffer[loc]="""" then cur_mod:="" else begin k:=loc; buffer[limit+1]:=""""; repeat incr(loc); until buffer[loc]=""""; if loc>limit then @; if loc=k+1 then cur_mod:=buffer[k] else begin str_room(loc-k); repeat append_char(buffer[k]); incr(k); until k=loc; cur_mod:=make_string; end; end; incr(loc); cur_cmd:=string_token; return; end @ We go to |restart| after this error message, instead of to |switch|, because the |clear_for_error_prompt| routine might have reinstated |token_state| after |error| has finished. @= begin loc:=limit; {the next character to be read on this line will be |"%"|} print_err("Incomplete string token has been flushed"); @.Incomplete string token...@> help3("Strings should finish on the same line as they began.")@/ ("I've deleted the partial string; you might want to")@/ ("insert another by typing, e.g., `I""new string""'.");@/ deletions_allowed:=false; error; deletions_allowed:=true; goto restart; end @ @= n:=c-"0"; while char_class[buffer[loc]]=digit_class do begin if n<4096 then n:=10*n+buffer[loc]-"0"; incr(loc); end; if buffer[loc]="." then if char_class[buffer[loc+1]]=digit_class then goto done; f:=0; goto fin_numeric_token; done: incr(loc) @ @= k:=0; repeat if k<17 then {digits for |k>=17| cannot affect the result} begin dig[k]:=buffer[loc]-"0"; incr(k); end; incr(loc); until char_class[buffer[loc]]<>digit_class; f:=round_decimals(k); if f=unity then begin incr(n); f:=0; end @ @= if n<4096 then cur_mod:=n*unity+f else begin print_err("Enormous number has been reduced"); @.Enormous number...@> help2("I can't handle numbers bigger than about 4095.99998;")@/ ("so I've changed your constant to that maximum amount.");@/ deletions_allowed:=false; error; deletions_allowed:=true; cur_mod:=@'1777777777; end; cur_cmd:=numeric_token; return @ Let's consider now what happens when |get_next| is looking at a token list. @^inner loop@> @= if loc>=hi_mem_min then {one-word token} begin cur_sym:=info(loc); loc:=link(loc); {move to next} if cur_sym>=expr_base then if cur_sym>=suffix_base then @ else begin cur_cmd:=capsule_token; cur_mod:=param_stack[param_start+cur_sym-(expr_base)]; cur_sym:=0; return; end; end else if loc>null then @ else begin {we are done with this token list} end_token_list; goto restart; {resume previous level} end @ @= begin if cur_sym>=text_base then cur_sym:=cur_sym-param_size; {|param_size=text_base-suffix_base|} begin_token_list(param_stack[param_start+cur_sym-(suffix_base)],parameter); goto restart; end @ @= begin if name_type(loc)=token then begin cur_mod:=value(loc); if type(loc)=known then cur_cmd:=numeric_token else begin cur_cmd:=string_token; add_str_ref(cur_mod); end; end else begin cur_mod:=loc; cur_cmd:=capsule_token; end; loc:=link(loc); return; end @ All of the easy branches of |get_next| have now been taken care of. There is one more branch. @= if name>2 then @ else begin if input_ptr>0 then {text was inserted during error recovery or by \&{scantokens}} begin end_file_reading; goto restart; {resume previous level} end; if selectornonstop_mode then begin if limit=start then {previous line was empty} print_nl("(Please type a command or say `end')"); @.Please type...@> print_ln; first:=start; prompt_input("*"); {input on-line into |buffer|} @.*\relax@> limit:=last; buffer[limit]:="%"; first:=limit+1; loc:=start; end else fatal_error("*** (job aborted, no legal end found)"); @.job aborted@> {nonstop mode, which is intended for overnight batch processing, never waits for on-line input} end @ The global variable |force_eof| is normally |false|; it is set |true| by an \&{endinput} command. @= @!force_eof:boolean; {should the next \&{input} be aborted early?} @ @= begin incr(line); first:=start; if not force_eof then begin if input_ln(cur_file,true) then {not end of file} firm_up_the_line {this sets |limit|} else force_eof:=true; end; if force_eof then begin print_char(")"); force_eof:=false; update_terminal; {show user that file has been read} end_file_reading; {resume previous level} if check_outer_validity then goto restart@+else goto restart; end; buffer[limit]:="%"; first:=limit+1; loc:=start; {ready to read} end @ If the user has set the |pausing| parameter to some positive value, and if nonstop mode has not been selected, each line of input is displayed on the terminal and the transcript file, followed by `\.{=>}'. \MF\ waits for a response. If the response is null (i.e., if nothing is typed except perhaps a few blank spaces), the original line is accepted as it stands; otherwise the line typed is used instead of the line in the file. @p procedure firm_up_the_line; var @!k:0..buf_size; {an index into |buffer|} begin limit:=last; if internal[pausing]>0 then if interaction>nonstop_mode then begin wake_up_terminal; print_ln; if start"); {wait for user response} @.=>@> if last>first then begin for k:=first to last-1 do {move line down in buffer} buffer[k+start-first]:=buffer[k]; limit:=start+last-first; end; end; end; @* \[34] Scanning macro definitions. \MF\ has a variety of ways to tuck tokens away into token lists for later use: Macros can be defined with \&{def}, \&{vardef}, \&{primarydef}, etc.; repeatable code can be defined with \&{for}, \&{forever}, \&{forsuffixes}. All such operations are handled by the routines in this part of the program. The modifier part of each command code is zero for the ``ending delimiters'' like \&{enddef} and \&{endfor}. @d start_def=1 {command modifier for \&{def}} @d var_def=2 {command modifier for \&{vardef}} @d end_def=0 {command modifier for \&{enddef}} @d start_forever=1 {command modifier for \&{forever}} @d end_for=0 {command modifier for \&{endfor}} @= primitive("def",macro_def,start_def);@/ @!@:def_}{\&{def} primitive@> primitive("vardef",macro_def,var_def);@/ @!@:var_def_}{\&{vardef} primitive@> primitive("primarydef",macro_def,secondary_primary_macro);@/ @!@:primary_def_}{\&{primarydef} primitive@> primitive("secondarydef",macro_def,tertiary_secondary_macro);@/ @!@:secondary_def_}{\&{secondarydef} primitive@> primitive("tertiarydef",macro_def,expression_tertiary_macro);@/ @!@:tertiary_def_}{\&{tertiarydef} primitive@> primitive("enddef",macro_def,end_def); eqtb[frozen_end_def]:=eqtb[cur_sym];@/ @!@:end_def_}{\&{enddef} primitive@> @# primitive("for",iteration,expr_base);@/ @!@:for_}{\&{for} primitive@> primitive("forsuffixes",iteration,suffix_base);@/ @!@:for_suffixes_}{\&{forsuffixes} primitive@> primitive("forever",iteration,start_forever);@/ @!@:forever_}{\&{forever} primitive@> primitive("endfor",iteration,end_for); eqtb[frozen_end_for]:=eqtb[cur_sym];@/ @!@:end_for_}{\&{endfor} primitive@> @ @= macro_def:if m<=var_def then if m=start_def then print("def") else if m0 then begin @; if cur_cmd=terminator then @ else if cur_cmd=macro_special then @; end; link(p):=cur_tok; p:=link(p); end; done: link(p):=tail_end; flush_node_list(subst_list); scan_toks:=link(hold_head); end; @ @= begin q:=subst_list; while q<>null do begin if info(q)=cur_sym then begin cur_sym:=value(q); cur_cmd:=relax; goto found; end; q:=link(q); end; found:end @ @= if cur_mod>0 then incr(balance) else begin decr(balance); if balance=0 then goto done; end @ Four commands are intended to be used only within macro texts: \&{quote}, \.{\#\AT!}, \.{\AT!}, and \.{\AT!\#}. They are variants of a single command code called |macro_special|. @d quote=0 {|macro_special| modifier for \&{quote}} @d macro_prefix=1 {|macro_special| modifier for \.{\#\AT!}} @d macro_at=2 {|macro_special| modifier for \.{\AT!}} @d macro_suffix=3 {|macro_special| modifier for \.{\AT!\#}} @= primitive("quote",macro_special,quote);@/ @!@:quote_}{\&{quote} primitive@> primitive("#@@",macro_special,macro_prefix);@/ @!@:]]]\#\AT!_}{\.{\#\AT!} primitive@> primitive("@@",macro_special,macro_at);@/ @!@:]]]\AT!_}{\.{\AT!} primitive@> primitive("@@#",macro_special,macro_suffix);@/ @!@:]]]\AT!\#_}{\.{\AT!\#} primitive@> @ @= macro_special: case m of macro_prefix: print("#@@"); macro_at: print_char("@@"); macro_suffix: print("@@#"); othercases print("quote") endcases; @ @= begin if cur_mod=quote then get_next else if cur_mod<=suffix_count then cur_sym:=suffix_base-1+cur_mod; end @ Here is a routine that's used whenever a token will be redefined. If the user's token is unredefinable, the `|frozen_inaccessible|' token is substituted; the latter is redefinable but essentially impossible to use, hence \MF's tables won't get fouled up. @p procedure get_symbol; {sets |cur_sym| to a safe symbol} label restart; begin restart: get_next; if (cur_sym=0)or(cur_sym>frozen_inaccessible) then begin print_err("Missing symbolic token inserted"); @.Missing symbolic token...@> help3("Sorry: You can't redefine a number, string, or expr.")@/ ("I've inserted an inaccessible symbol so that your")@/ ("definition will be completed without mixing me up too badly."); if cur_sym>0 then help_line[2]:="Sorry: You can't redefine my error-recovery tokens." else if cur_cmd=string_token then delete_str_ref(cur_mod); cur_sym:=frozen_inaccessible; ins_error; goto restart; end; end; @ Before we actually redefine a symbolic token, we need to clear away its former value, if it was a variable. The following stronger version of |get_symbol| does that. @p procedure get_clear_symbol; begin get_symbol; clear_symbol(cur_sym,false); end; @ Here's another little subroutine; it checks that an equals sign or assignment sign comes along at the proper place in a macro definition. @p procedure check_equals; begin if cur_cmd<>equals then if cur_cmd<>assignment then begin missing_err("=");@/ @.Missing `='@> help5("The next thing in this `def' should have been `=',")@/ ("because I've already looked at the definition heading.")@/ ("But don't worry; I'll pretend that an equals sign")@/ ("was present. Everything from here to `enddef'")@/ ("will be the replacement text of this macro."); back_error; end; end; @ A \&{primarydef}, \&{secondarydef}, or \&{tertiarydef} is rather easily handled now that we have |scan_toks|. In this case there are two parameters, which will be \.{EXPR0} and \.{EXPR1} (i.e., |expr_base| and |expr_base+1|). @p procedure make_op_def; var @!m:command_code; {the type of definition} @!p,@!q,@!r:pointer; {for list manipulation} begin m:=cur_mod;@/ get_symbol; q:=get_node(token_node_size); info(q):=cur_sym; value(q):=expr_base;@/ get_clear_symbol; warning_info:=cur_sym;@/ get_symbol; p:=get_node(token_node_size); info(p):=cur_sym; value(p):=expr_base+1; link(p):=q;@/ get_next; check_equals;@/ scanner_status:=op_defining; q:=get_avail; ref_count(q):=null; r:=get_avail; link(q):=r; info(r):=general_macro; link(r):=scan_toks(macro_def,p,null,0); scanner_status:=normal; eq_type(warning_info):=m; equiv(warning_info):=q; get_x_next; end; @ Parameters to macros are introduced by the keywords \&{expr}, \&{suffix}, \&{text}, \&{primary}, \&{secondary}, and \&{tertiary}. @= primitive("expr",param_type,expr_base);@/ @!@:expr_}{\&{expr} primitive@> primitive("suffix",param_type,suffix_base);@/ @!@:suffix_}{\&{suffix} primitive@> primitive("text",param_type,text_base);@/ @!@:text_}{\&{text} primitive@> primitive("primary",param_type,primary_macro);@/ @!@:primary_}{\&{primary} primitive@> primitive("secondary",param_type,secondary_macro);@/ @!@:secondary_}{\&{secondary} primitive@> primitive("tertiary",param_type,tertiary_macro);@/ @!@:tertiary_}{\&{tertiary} primitive@> @ @= param_type:if m>=expr_base then if m=expr_base then print("expr") else if m=suffix_base then print("suffix") else print("text") else if m@@; @t\4@>@@; procedure scan_def; var @!m:start_def..var_def; {the type of definition} @!n:0..3; {the number of special suffix parameters} @!k:0..param_size; {the total number of parameters} @!c:general_macro..text_macro; {the kind of macro we're defining} @!r:pointer; {parameter-substitution list} @!q:pointer; {tail of the macro token list} @!p:pointer; {temporary storage} @!base:halfword; {|expr_base|, |suffix_base|, or |text_base|} @!l_delim,@!r_delim:pointer; {matching delimiters} begin m:=cur_mod; c:=general_macro; link(hold_head):=null;@/ q:=get_avail; ref_count(q):=null; r:=null;@/ @; k:=n; if cur_cmd=left_delimiter then @; if cur_cmd=param_type then @; check_equals; p:=get_avail; info(p):=c; link(q):=p; @; scanner_status:=normal; get_x_next; end; @ We don't put `|frozen_end_group|' into the replacement text of a \&{vardef}, because the user may want to redefine `\.{endgroup}'. @= if m=start_def then link(p):=scan_toks(macro_def,r,null,n) else begin q:=get_avail; info(q):=bg_loc; link(p):=q; p:=get_avail; info(p):=eg_loc; link(q):=scan_toks(macro_def,r,p,n); end; if warning_info=bad_vardef then flush_token_list(value(bad_vardef)) @ @= @!bg_loc,@!eg_loc:1..hash_end; {hash addresses of `\.{begingroup}' and `\.{endgroup}'} @ @= if m=start_def then begin get_clear_symbol; warning_info:=cur_sym; get_next; scanner_status:=op_defining; n:=0; eq_type(warning_info):=defined_macro; equiv(warning_info):=q; end else begin p:=scan_declared_variable; flush_variable(equiv(info(p)),link(p),true); warning_info:=find_variable(p); flush_list(p); if warning_info=null then @; scanner_status:=var_defining; n:=2; if cur_cmd=macro_special then if cur_mod=macro_suffix then {\.{\AT!\#}} begin n:=3; get_next; end; type(warning_info):=unsuffixed_macro-2+n; value(warning_info):=q; end {|suffixed_macro=unsuffixed_macro+1|} @ @= begin print_err("This variable already starts with a macro"); @.This variable already...@> help2("After `vardef a' you can't say `vardef a.b'.")@/ ("So I'll have to discard this definition."); error; warning_info:=bad_vardef; end @ @= name_type(bad_vardef):=root; link(bad_vardef):=frozen_bad_vardef; equiv(frozen_bad_vardef):=bad_vardef; eq_type(frozen_bad_vardef):=tag_token; @ @= repeat l_delim:=cur_sym; r_delim:=cur_mod; get_next; if (cur_cmd=param_type)and(cur_mod>=expr_base) then base:=cur_mod else begin print_err("Missing parameter type; `expr' will be assumed"); @.Missing parameter type@> help1("You should've had `expr' or `suffix' or `text' here."); back_error; base:=expr_base; end; @; check_delimiter(l_delim,r_delim); get_next; until cur_cmd<>left_delimiter @ @= repeat link(q):=get_avail; q:=link(q); info(q):=base+k;@/ get_symbol; p:=get_node(token_node_size); value(p):=base+k; info(p):=cur_sym; if k=param_size then overflow("parameter stack size",param_size); @:METAFONT capacity exceeded parameter stack size}{\quad parameter stack size@> incr(k); link(p):=r; r:=p; get_next; until cur_cmd<>comma @ @= begin p:=get_node(token_node_size); if cur_mod=min_command|. In other words, |get_x_next| expands macros and removes conditionals or iterations or input instructions that might be present. It follows that |get_x_next| might invoke itself recursively. In fact, there is massive recursion, since macro expansion can involve the scanning of arbitrarily complex expressions, which in turn involve macro expansion and conditionals, etc. @^recursion@> Therefore it's necessary to declare a whole bunch of |forward| procedures at this point, and to insert some other procedures that will be invoked by |get_x_next|. @p procedure@?scan_primary; forward;@t\2@> procedure@?scan_secondary; forward;@t\2@> procedure@?scan_tertiary; forward;@t\2@> procedure@?scan_expression; forward;@t\2@> procedure@?scan_suffix; forward;@t\2@>@/ @t\4@>@@;@/ procedure@?get_boolean; forward;@t\2@> procedure@?pass_text; forward;@t\2@> procedure@?conditional; forward;@t\2@> procedure@?start_input; forward;@t\2@> procedure@?begin_iteration; forward;@t\2@> procedure@?resume_iteration; forward;@t\2@> procedure@?stop_iteration; forward;@t\2@> @ An auxiliary subroutine called |expand| is used by |get_x_next| when it has to do exotic expansion commands. @p procedure expand; var @!p:pointer; {for list manipulation} @!k:integer; {something that we hope is |<=buf_size|} @!j:pool_pointer; {index into |str_pool|} begin if internal[tracing_commands]>unity then if cur_cmd<>defined_macro then show_cur_cmd_mod; case cur_cmd of if_test:conditional; {this procedure is discussed in Part 36 below} fi_or_else:@; input:@; iteration:if cur_mod=end_for then @ else begin_iteration; {this procedure is discussed in Part 37 below} repeat_loop: @; exit_test: @; relax: do_nothing; expand_after: @; scan_tokens: @; defined_macro:macro_call(cur_mod,null,cur_sym); end; {there are no other cases} end; @ @= begin print_err("Extra `endfor'"); @.Extra `endfor'@> help2("I'm not currently working on a for loop,")@/ ("so I had better not try to end anything.");@/ error; end @ The processing of \&{input} involves the |start_input| subroutine, which will be declared later; the processing of \&{endinput} is trivial. @= primitive("input",input,0);@/ @!@:input_}{\&{input} primitive@> primitive("endinput",input,1);@/ @!@:end_input_}{\&{endinput} primitive@> @ @= input: if m=0 then print("input")@+else print("endinput"); @ @= if cur_mod>0 then force_eof:=true else start_input @ We'll discuss the complicated parts of loop operations later. For now it suffices to know that there's a global variable called |loop_ptr| that will be |null| if no loop is in progress. @= begin while token_state and(loc=null) do end_token_list; {conserve stack space} if loop_ptr=null then begin print_err("Lost loop"); @.Lost loop@> help2("I'm confused; after exiting from a loop, I still seem")@/ ("to want to repeat it. I'll try to forget the problem.");@/ error; end else resume_iteration; {this procedure is in Part 37 below} end @ @= begin get_boolean; if internal[tracing_commands]>unity then show_cmd_mod(nullary,cur_exp); if cur_exp=true_code then if loop_ptr=null then begin print_err("No loop is in progress"); @.No loop is in progress@> help1("Why say `exitif' when there's nothing to exit from?"); if cur_cmd=semicolon then error@+else back_error; end else @ else if cur_cmd<>semicolon then begin missing_err(";");@/ @.Missing `;'@> help2("After `exitif ' I expect to see a semicolon.")@/ ("I shall pretend that one was there."); back_error; end; end @ Here we use the fact that |forever_text| is the only |token_type| that is less than |loop_text|. @= begin p:=null; repeat if file_state then end_file_reading else begin if token_type<=loop_text then p:=start; end_token_list; end; until p<>null; if p<>info(loop_ptr) then fatal_error("*** (loop confusion)"); @.loop confusion@> stop_iteration; {this procedure is in Part 37 below} end @ @= begin get_next; p:=cur_tok; get_next; if cur_cmd= begin get_x_next; scan_primary; if cur_type<>string_type then begin disp_err(null,"Not a string"); @.Not a string@> help2("I'm going to flush this expression, since")@/ ("scantokens should be followed by a known string."); put_get_flush_error(0); end else begin back_input; if length(cur_exp)>0 then @; end; end @ @= begin begin_file_reading; name:=2; k:=first+length(cur_exp); if k>=max_buf_stack then begin if k>=buf_size then begin max_buf_stack:=buf_size; overflow("buffer size",buf_size); @:METAFONT capacity exceeded buffer size}{\quad buffer size@> end; max_buf_stack:=k+1; end; j:=str_start[cur_exp]; limit:=k; while first @p procedure get_x_next; var @!save_exp:pointer; {a capsule to save |cur_type| and |cur_exp|} begin get_next; if cur_cmd=min_command; unstash_cur_exp(save_exp); {that restores |cur_type| and |cur_exp|} end; end; @ Now let's consider the |macro_call| procedure, which is used to start up all user-defined macros. Since the arguments to a macro might be expressions, |macro_call| is recursive. @^recursion@> The first parameter to |macro_call| points to the reference count of the token list that defines the macro. The second parameter contains any arguments that have already been parsed (see below). The third parameter points to the symbolic token that names the macro. If the third parameter is |null|, the macro was defined by \&{vardef}, so its name can be reconstructed from the prefix and ``at'' arguments found within the second parameter. What is this second parameter? It's simply a linked list of one-word items, whose |info| fields point to the arguments. In other words, if |arg_list=null|, no arguments have been scanned yet; otherwise |info(arg_list)| points to the first scanned argument, and |link(arg_list)| points to the list of further arguments (if any). Arguments of type \&{expr} are so-called capsules, which we will discuss later when we concentrate on expressions; they can be recognized easily because their |link| field is |void|. Arguments of type \&{suffix} and \&{text} are token lists without reference counts. @ After argument scanning is complete, the arguments are moved to the |param_stack|. (They can't be put on that stack any sooner, because the stack is growing and shrinking in unpredictable ways as more arguments are being acquired.) Then the macro body is fed to the scanner; i.e., the replacement text of the macro is placed at the top of the \MF's input stack, so that |get_next| will proceed to read it next. @= @t\4@>@@; @t\4@>@@; @t\4@>@@; procedure macro_call(@!def_ref,@!arg_list,@!macro_name:pointer); {invokes a user-defined control sequence} label found; var @!r:pointer; {current node in the macro's token list} @!p,@!q:pointer; {for list manipulation} @!n:integer; {the number of arguments} @!l_delim,@!r_delim:pointer; {a delimiter pair} @!tail:pointer; {tail of the argument list} begin r:=link(def_ref); add_mac_ref(def_ref); if arg_list=null then n:=0 else @; if internal[tracing_macros]>0 then @; @; @; end; @ @= begin begin_diagnostic; print_ln; print_macro_name(arg_list,macro_name); if n=3 then print("@@#"); {indicate a suffixed macro} show_macro(def_ref,null,100000); if arg_list<>null then begin n:=0; p:=arg_list; repeat q:=info(p); print_arg(q,n,0); incr(n); p:=link(p); until p=null; end; end_diagnostic(false); end @ @= procedure print_macro_name(@!a,@!n:pointer); var @!p,@!q:pointer; {they traverse the first part of |a|} begin if n<>null then print(text(n)) else begin p:=info(a); if p=null then print(text(info(info(link(a))))) else begin q:=p; while link(q)<>null do q:=link(q); link(q):=info(link(a)); show_token_list(p,null,1000,0); link(q):=null; end; end; end; @ @= procedure print_arg(@!q:pointer;@!n:integer;@!b:pointer); begin if link(q)=void then print_nl("(EXPR") else if (btext_macro) then print_nl("(SUFFIX") else print_nl("(TEXT"); print_int(n); print(")<-"); if link(q)=void then print_exp(q,1) else show_token_list(q,null,1000,0); end; @ @= begin n:=1; tail:=arg_list; while link(tail)<>null do begin incr(n); tail:=link(tail); end; end @ @= cur_cmd:=comma+1; {anything |<>comma| will do} while info(r)>=expr_base do begin @; r:=link(r); end; if cur_cmd=comma then begin print_err("Too many arguments to "); @.Too many arguments...@> print_macro_name(arg_list,macro_name); print_char(";"); print_nl(" Missing `"); print(text(r_delim)); @.Missing `)'...@> print("' has been inserted"); help3("I'm going to assume that the comma I just read was a")@/ ("right delimiter, and then I'll begin expanding the macro.")@/ ("You might want to delete some tokens before continuing."); error; end; if info(r)<>general_macro then @; r:=link(r) @ At this point, the reader will find it advisable to review the explanation of token list format that was presented earlier, paying special attention to the conventions that apply only at the beginning of a macro's token list. On the other hand, the reader will have to take the expression-parsing aspects of the following program on faith; we will explain |cur_type| and |cur_exp| later. (Several things in this program depend on each other, and it's necessary to jump into the circle somewhere.) @= if cur_cmd<>comma then begin get_x_next; if cur_cmd<>left_delimiter then begin print_err("Missing argument to "); @.Missing argument...@> print_macro_name(arg_list,macro_name); help3("That macro has more parameters than you thought.")@/ ("I'll continue by pretending that each missing argument")@/ ("is either zero or null."); if info(r)>=suffix_base then begin cur_exp:=null; cur_type:=token_list; end else begin cur_exp:=0; cur_type:=known; end; back_error; cur_cmd:=right_delimiter; goto found; end; l_delim:=cur_sym; r_delim:=cur_mod; end; @; if cur_cmd<>comma then @; found: @ @ @= if (cur_cmd<>right_delimiter)or(cur_mod<>l_delim) then if info(link(r))>=expr_base then begin missing_err(","); @.Missing `,'@> help3("I've finished reading a macro argument and am about to")@/ ("read another; the arguments weren't delimited correctly.")@/ ("You might want to delete some tokens before continuing."); back_error; cur_cmd:=comma; end else begin missing_err(text(r_delim)); @.Missing `)'@> help2("I've gotten to the end of the macro parameter list.")@/ ("You might want to delete some tokens before continuing."); back_error; end @ A \&{suffix} or \&{text} parameter will be have been scanned as a token list pointed to by |cur_exp|, in which case we will have |cur_type=token_list|. @= begin p:=get_avail; if cur_type=token_list then info(p):=cur_exp else info(p):=stash_cur_exp; if internal[tracing_macros]>0 then begin begin_diagnostic; print_arg(info(p),n,info(r)); end_diagnostic(false); end; if arg_list=null then arg_list:=p else link(tail):=p; tail:=p; incr(n); end @ @= if info(r)>=text_base then scan_text_arg(l_delim,r_delim) else begin get_x_next; if info(r)>=suffix_base then scan_suffix else scan_expression; end @ The parameters to |scan_text_arg| are either a pair of delimiters or zero; the latter case is for undelimited text arguments, which end with the first semicolon or \&{endgroup} or \&{end} that is not contained in a group. @= procedure scan_text_arg(@!l_delim,@!r_delim:pointer); label done; var @!balance:integer; {excess of |l_delim| over |r_delim|} @!p:pointer; {list tail} begin warning_info:=l_delim; scanner_status:=absorbing; p:=hold_head; balance:=1; link(hold_head):=null; loop@+ begin get_next; if l_delim=0 then @ else @; link(p):=cur_tok; p:=link(p); end; done: cur_exp:=link(hold_head); cur_type:=token_list; scanner_status:=normal; end; @ @= begin if cur_cmd=right_delimiter then begin if cur_mod=l_delim then begin decr(balance); if balance=0 then goto done; end; end else if cur_cmd=left_delimiter then if cur_mod=r_delim then incr(balance); end @ @= begin if end_of_statement then {|cur_cmd=semicolon|, |end_group|, or |stop|} begin if balance=1 then goto done else if cur_cmd=end_group then decr(balance); end else if cur_cmd=begin_group then incr(balance); end @ @= begin if info(r)suffix_macro then if (cur_cmd=equals)or(cur_cmd=assignment) then get_x_next; end; case info(r) of primary_macro:scan_primary; secondary_macro:scan_secondary; tertiary_macro:scan_tertiary; expr_macro:scan_expression; of_macro:@; suffix_macro:@; text_macro:scan_text_arg(0,0); end; {there are no other cases} back_input; @; end @ @= begin scan_expression; p:=get_avail; info(p):=stash_cur_exp; if internal[tracing_macros]>0 then begin begin_diagnostic; print_arg(info(p),n,0); end_diagnostic(false); end; if arg_list=null then arg_list:=p@+else link(tail):=p; tail:=p;incr(n); if cur_cmd<>of_token then begin missing_err("of"); print(" for "); @.Missing `of'@> print_macro_name(arg_list,macro_name); help1("I've got the first argument; will look now for the other."); back_error; end; get_x_next; scan_primary; end @ @= begin if cur_cmd<>left_delimiter then l_delim:=null else begin l_delim:=cur_sym; r_delim:=cur_mod; get_x_next; end; scan_suffix; if l_delim<>null then begin if(cur_cmd<>right_delimiter)or(cur_mod<>l_delim) then begin missing_err(text(r_delim)); @.Missing `)'@> help2("I've gotten to the end of the macro parameter list.")@/ ("You might want to delete some tokens before continuing."); back_error; end; get_x_next; end; end @ Before we put a new token list on the input stack, it is wise to clean off all token lists that have recently been depleted. Then a user macro that ends with a call to itself will not require unbounded stack space. @= while token_state and(loc=null) do end_token_list; {conserve stack space} if param_ptr+n>max_param_stack then begin max_param_stack:=param_ptr+n; if max_param_stack>param_size then overflow("parameter stack size",param_size); @:METAFONT capacity exceeded parameter stack size}{\quad parameter stack size@> end; begin_token_list(def_ref,macro); name:=macro_name; loc:=r; if n>0 then begin p:=arg_list; repeat param_stack[param_ptr]:=info(p); incr(param_ptr); p:=link(p); until p=null; flush_list(arg_list); end @ It's sometimes necessary to put a single argument onto |param_stack|. The |stack_argument| subroutine does this. @p procedure stack_argument(@!p:pointer); begin if param_ptr=max_param_stack then begin incr(max_param_stack); if max_param_stack>param_size then overflow("parameter stack size",param_size); @:METAFONT capacity exceeded parameter stack size}{\quad parameter stack size@> end; param_stack[param_ptr]:=p; incr(param_ptr); end; @* \[36] Conditional processing. Let's consider now the way \&{if} commands are handled. Conditions can be inside conditions, and this nesting has a stack that is independent of other stacks. Four global variables represent the top of the condition stack: |cond_ptr| points to pushed-down entries, if~any; |cur_if| tells whether we are processing \&{if} or \&{elseif}; |if_limit| specifies the largest code of a |fi_or_else| command that is syntactically legal; and |if_line| is the line number at which the current conditional began. If no conditions are currently in progress, the condition stack has the special state |cond_ptr=null|, |if_limit=normal|, |cur_if=0|, |if_line=0|. Otherwise |cond_ptr| points to a two-word node; the |type|, |name_type|, and |link| fields of the first word contain |if_limit|, |cur_if|, and |cond_ptr| at the next level, and the second word contains the corresponding |if_line|. @d if_node_size=2 {number of words in stack entry for conditionals} @d if_line_field(#)==mem[#+1].int @d if_code=1 {code for \&{if} being evaluated} @d fi_code=2 {code for \&{fi}} @d else_code=3 {code for \&{else}} @d else_if_code=4 {code for \&{elseif}} @= @!cond_ptr:pointer; {top of the condition stack} @!if_limit:normal..else_if_code; {upper bound on |fi_or_else| codes} @!cur_if:small_number; {type of conditional being worked on} @!if_line:integer; {line where that conditional began} @ @= cond_ptr:=null; if_limit:=normal; cur_if:=0; if_line:=0; @ @= primitive("if",if_test,if_code);@/ @!@:if_}{\&{if} primitive@> primitive("fi",fi_or_else,fi_code); eqtb[frozen_fi]:=eqtb[cur_sym];@/ @!@:fi_}{\&{fi} primitive@> primitive("else",fi_or_else,else_code);@/ @!@:else_}{\&{else} primitive@> primitive("elseif",fi_or_else,else_if_code);@/ @!@:else_if_}{\&{elseif} primitive@> @ @= if_test,fi_or_else: case m of if_code:print("if"); fi_code:print("fi"); else_code:print("else"); othercases print("elseif") endcases; @ Here is a procedure that ignores text until coming to an \&{elseif}, \&{else}, or \&{fi} at level zero of $\&{if}\ldots\&{fi}$ nesting. After it has acted, |cur_mod| will indicate the token that was found. \MF's smallest two command codes are |if_test| and |fi_or_else|; this makes the skipping process a bit simpler. @p procedure pass_text; label done; var l:integer; begin scanner_status:=skipping; l:=0; warning_info:=line; loop@+ begin get_next; if cur_cmd<=fi_or_else then if cur_cmd; end; done: scanner_status:=normal; end; @ @= if cur_cmd=string_token then delete_str_ref(cur_mod) @ When we begin to process a new \&{if}, we set |if_limit:=if_code|; then if \&{elseif} or \&{else} or \&{fi} occurs before the current \&{if} condition has been evaluated, a colon will be inserted. A construction like `\.{if fi}' would otherwise get \MF\ confused. @= begin p:=get_node(if_node_size); link(p):=cond_ptr; type(p):=if_limit; name_type(p):=cur_if; if_line_field(p):=if_line; cond_ptr:=p; if_limit:=if_code; if_line:=line; cur_if:=if_code; end @ @= begin p:=cond_ptr; if_line:=if_line_field(p); cur_if:=name_type(p); if_limit:=type(p); cond_ptr:=link(p); free_node(p,if_node_size); end @ Here's a procedure that changes the |if_limit| code corresponding to a given value of |cond_ptr|. @p procedure change_if_limit(@!l:small_number;@!p:pointer); label exit; var q:pointer; begin if p=cond_ptr then if_limit:=l {that's the easy case} else begin q:=cond_ptr; loop@+ begin if q=null then confusion("if"); @:this can't happen if}{\quad if@> if link(q)=p then begin type(q):=l; return; end; q:=link(q); end; end; exit:end; @ The user is supposed to put colons into the proper parts of conditional statements. Therefore, \MF\ has to check for their presence. @p procedure check_colon; begin if cur_cmd<>colon then begin missing_err(":");@/ @.Missing `:'@> help2("There should've been a colon after the condition.")@/ ("I shall pretend that one was there.");@; back_error; end; end; @ A condition is started when the |get_x_next| procedure encounters an |if_test| command; in that case |get_x_next| calls |conditional|, which is a recursive procedure. @^recursion@> @p procedure conditional; label exit,done,reswitch,found; var @!save_cond_ptr:pointer; {|cond_ptr| corresponding to this conditional} @!new_if_limit:fi_code..else_if_code; {future value of |if_limit|} @!p:pointer; {temporary register} begin @;@+save_cond_ptr:=cond_ptr; reswitch: get_boolean; new_if_limit:=else_if_code; if internal[tracing_commands]>unity then @; found: check_colon; if cur_exp=true_code then begin change_if_limit(new_if_limit,save_cond_ptr); return; {wait for \&{elseif}, \&{else}, or \&{fi}} end; @; done: cur_if:=cur_mod; if_line:=line; if cur_mod=fi_code then @ else if cur_mod=else_if_code then goto reswitch else begin cur_exp:=true_code; new_if_limit:=fi_code; get_x_next; goto found; end; exit:end; @ In a construction like `\&{if} \&{if} \&{true}: $0=1$: \\{foo} \&{else}: \\{bar} \&{fi}', the first \&{else} that we come to after learning that the \&{if} is false is not the \&{else} we're looking for. Hence the following curious logic is needed. @= loop@+ begin pass_text; if cond_ptr=save_cond_ptr then goto done else if cur_mod=fi_code then @; end @ @= begin begin_diagnostic; if cur_exp=true_code then print("{true}")@+else print("{false}"); end_diagnostic(false); end @ The processing of conditionals is complete except for the following code, which is actually part of |get_x_next|. It comes into play when \&{elseif}, \&{else}, or \&{fi} is scanned. @= if cur_mod>if_limit then if if_limit=if_code then {condition not yet evaluated} begin missing_err(":"); @.Missing `:'@> back_input; cur_sym:=frozen_colon; ins_error; end else begin print_err("Extra "); print_cmd_mod(fi_or_else,cur_mod); @.Extra else@> @.Extra elseif@> @.Extra fi@> help1("I'm ignoring this; it doesn't match any if."); error; end else begin while cur_mod<>fi_code do pass_text; {skip to \&{fi}} @; end @* \[37] Iterations. To bring our treatment of |get_x_next| to a close, we need to consider what \MF\ does when it sees \&{for}, \&{forsuffixes}, and \&{forever}. There's a global variable |loop_ptr| that keeps track of the \&{for} loops that are currently active. If |loop_ptr=null|, no loops are in progress; otherwise |info(loop_ptr)| points to the iterative text of the current (innermost) loop, and |link(loop_ptr)| points to the data for any other loops that enclose the current one. A loop-control node also has two other fields, called |loop_type| and |loop_list|, whose contents depend on the type of loop: \yskip\indent|loop_type(loop_ptr)=null| means that |loop_list(loop_ptr)| points to a list of one-word nodes whose |info| fields point to the remaining argument values of a suffix list and expression list. \yskip\indent|loop_type(loop_ptr)=void| means that the current loop is `\&{forever}'. \yskip\indent|loop_type(loop_ptr)=p>void| means that |value(p)|, |step_size(p)|, and |final_value(p)| contain the data for an arithmetic progression. \yskip\noindent In the latter case, |p| points to a ``progression node'' whose first word is not used. (No value could be stored there because the link field of words in the dynamic memory area cannot be arbitrary.) @d loop_list_loc(#)==#+1 {where the |loop_list| field resides} @d loop_type(#)==info(loop_list_loc(#)) {the type of \&{for} loop} @d loop_list(#)==link(loop_list_loc(#)) {the remaining list elements} @d loop_node_size=2 {the number of words in a loop control node} @d progression_node_size=4 {the number of words in a progression node} @d step_size(#)==mem[#+2].sc {the step size in an arithmetic progression} @d final_value(#)==mem[#+3].sc {the final value in an arithmetic progression} @= @!loop_ptr:pointer; {top of the loop-control-node stack} @ @= loop_ptr:=null; @ If the expressions that define an arithmetic progression in a \&{for} loop don't have known numeric values, the |bad_for| subroutine screams at the user. @p procedure bad_for(@!s:str_number); begin disp_err(null,"Improper "); {show the bad expression above the message} @.Improper...replaced by 0@> print(s); print(" has been replaced by 0"); help4("When you say `for x=a step b until c',")@/ ("the initial value `a' and the step size `b'")@/ ("and the final value `c' must have known numeric values.")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); put_get_flush_error(0); end; @ Here's what \MF\ does when \&{for}, \&{forsuffixes}, or \&{forever} has just been scanned. (This code requires slight familiarity with expression-parsing routines that we have not yet discussed; but it seems to belong in the present part of the program, even though the author didn't write it until later. The reader may wish to come back to it.) @p procedure begin_iteration; label continue,done,found; var @!m:halfword; {|expr_base| (\&{for}) or |suffix_base| (\&{forsuffixes})} @!n:halfword; {hash address of the current symbol} @!p,@!q,@!s,@!pp:pointer; {link manipulation registers} begin m:=cur_mod; n:=cur_sym; s:=get_node(loop_node_size); if m=start_forever then begin loop_type(s):=void; p:=null; get_x_next; goto found; end; get_symbol; p:=get_node(token_node_size); info(p):=cur_sym; value(p):=m;@/ get_x_next; if (cur_cmd<>equals)and(cur_cmd<>assignment) then begin missing_err("=");@/ @.Missing `='@> help3("The next thing in this loop should have been `=' or `:='.")@/ ("But don't worry; I'll pretend that an equals sign")@/ ("was present, and I'll look for the values next.");@/ back_error; end; @; found:@; @; resume_iteration; end; @ @= if cur_cmd<>colon then begin missing_err(":");@/ @.Missing `:'@> help3("The next thing in this loop should have been a `:'.")@/ ("So I'll pretend that a colon was present;")@/ ("everything from here to `endfor' will be iterated."); back_error; end @ We append a special |frozen_repeat_loop| token in place of the `\&{endfor}' at the end of the loop. This will come through \MF's scanner at the proper time to cause the loop to be repeated. (If the user tries some shenanigan like `\&{for} $\ldots$ \&{let} \&{endfor}', he will be foiled by the |get_symbol| routine, which keeps frozen tokens unchanged. Furthermore the |frozen_repeat_loop| is an \&{outer} token, so it won't be lost accidentally.) @ @= q:=get_avail; info(q):=frozen_repeat_loop; scanner_status:=loop_defining; warning_info:=n; info(s):=scan_toks(iteration,p,q,0); scanner_status:=normal;@/ link(s):=loop_ptr; loop_ptr:=s @ @= eq_type(frozen_repeat_loop):=repeat_loop+outer_tag; text(frozen_repeat_loop):=" ENDFOR"; @ The loop text is inserted into \MF's scanning apparatus by the |resume_iteration| routine. @p procedure resume_iteration; label not_found,exit; var @!p,@!q:pointer; {link registers} begin p:=loop_type(loop_ptr); if p>void then {|p| points to a progression node} begin cur_exp:=value(p); if @ then goto not_found; cur_type:=known; q:=stash_cur_exp; {make |q| an \&{expr} argument} value(p):=cur_exp+step_size(p); {set |value(p)| for the next iteration} end else if punity then @; return; not_found:stop_iteration; exit:end; @ @= ((step_size(p)>0)and(cur_exp>final_value(p)))or@| ((step_size(p)<0)and(cur_exp= begin begin_diagnostic; print_nl("{loop value="); @.loop value=n@> if (q<>null)and(link(q)=void) then print_exp(q,1) else show_token_list(q,null,50,0); print_char("}"); end_diagnostic(false); end @ A level of loop control disappears when |resume_iteration| has decided not to resume, or when an \&{exitif} construction has removed the loop text from the input stack. @p procedure stop_iteration; var @!p,@!q:pointer; {the usual} begin p:=loop_type(loop_ptr); if p>void then free_node(p,progression_node_size) else if pnull do begin p:=info(q); if p<>null then if link(p)=void then {it's an \&{expr} parameter} begin recycle_value(p); free_node(p,value_node_size); end else flush_token_list(p); {it's a \&{suffix} or \&{text} parameter} p:=q; q:=link(q); free_avail(p); end; end; p:=loop_ptr; loop_ptr:=link(p); flush_token_list(info(p)); free_node(p,loop_node_size); end; @ Now that we know all about loop control, we can finish up the missing portion of |begin_iteration| and we'll be done. The following code is performed after the `\.=' has been scanned in a \&{for} construction (if |m=expr_base|) or a \&{forsuffixes} construction (if |m=suffix_base|). @= loop_type(s):=null; q:=loop_list_loc(s); link(q):=null; {|link(q)=loop_list(s)|} repeat get_x_next; if m<>expr_base then scan_suffix else begin if cur_cmd>=colon then if cur_cmd<=comma then goto continue; scan_expression; if cur_cmd=step_token then if q=loop_list_loc(s) then @; cur_exp:=stash_cur_exp; end; link(q):=get_avail; q:=link(q); info(q):=cur_exp; cur_type:=vacuous; continue: until cur_cmd<>comma; done: @ @= begin if cur_type<>known then bad_for("initial value"); pp:=get_node(progression_node_size); value(pp):=cur_exp;@/ get_x_next; scan_expression; if cur_type<>known then bad_for("step size"); step_size(pp):=cur_exp; if cur_cmd<>until_token then begin missing_err("until");@/ @.Missing `until'@> help2("I assume you meant to say `until' after `step'.")@/ ("So I'll look for the final value and colon next."); back_error; end; get_x_next; scan_expression; if cur_type<>known then bad_for("final value"); final_value(pp):=cur_exp; loop_type(s):=pp; goto done; end @* \[38] File names. It's time now to fret about file names. Besides the fact that different operating systems treat files in different ways, we must cope with the fact that completely different naming conventions are used by different groups of people. The following programs show what is required for one particular operating system; similar routines for other systems are not difficult to devise. @^system dependencies@> \MF\ assumes that a file name has three parts: the name proper; its ``extension''; and a ``file area'' where it is found in an external file system. The extension of an input file is assumed to be `\.{.mf}' unless otherwise specified; it is `\.{.log}' on the transcript file that records each run of \MF; it is `\.{.tfm}' on the font metric files that describe characters in the fonts \MF\ creates; it is `\.{.gf}' on the output files that specify generic font information; and it is `\.{.base}' on the base files written by \.{INIMF} to initialize \MF. The file area can be arbitrary on input files, but files are usually output to the user's current area. If an input file cannot be found on the specified area, \MF\ will look for it on a special system area; this special area is intended for commonly used input files. Simple uses of \MF\ refer only to file names that have no explicit extension or area. For example, a person usually says `\.{input} \.{cmr10}' instead of `\.{input} \.{cmr10.new}'. Simple file names are best, because they make the \MF\ source files portable; whenever a file name consists entirely of letters and digits, it should be treated in the same way by all implementations of \MF. However, users need the ability to refer to other files in their environment, especially when responding to error messages concerning unopenable files; therefore we want to let them use the syntax that appears in their favorite operating system. @ \MF\ uses the same conventions that have proved to be satisfactory for \TeX. In order to isolate the system-dependent aspects of file names, the @^system dependencies@> system-independent parts of \MF\ make use of three system-dependent procedures that are called |begin_name|, |more_name|, and |end_name|. In essence, if the user-specified characters of the file name are $c_1\ldots c_n$, the system-independent driver program does the operations $$|begin_name|;\,|more_name|(c_1);\,\ldots\,;|more_name|(c_n); \,|end_name|.$$ These three procedures communicate with each other via global variables. Afterwards the file name will appear in the string pool as three strings called |cur_name|\penalty10000\hskip-.05em, |cur_area|, and |cur_ext|; the latter two are null (i.e., |""|), unless they were explicitly specified by the user. Actually the situation is slightly more complicated, because \MF\ needs to know when the file name ends. The |more_name| routine is a function (with side effects) that returns |true| on the calls |more_name|$(c_1)$, \dots, |more_name|$(c_{n-1})$. The final call |more_name|$(c_n)$ returns |false|; or, it returns |true| and $c_n$ is the last character on the current input line. In other words, |more_name| is supposed to return |true| unless it is sure that the file name has been completely scanned; and |end_name| is supposed to be able to finish the assembly of |cur_name|, |cur_area|, and |cur_ext| regardless of whether $|more_name|(c_n)$ returned |true| or |false|. @= @!cur_name:str_number; {name of file just scanned} @!cur_area:str_number; {file area just scanned, or \.{""}} @!cur_ext:str_number; {file extension just scanned, or \.{""}} @ The file names we shall deal with for illustrative purposes have the following structure: If the name contains `\.>' or `\.:', the file area consists of all characters up to and including the final such character; otherwise the file area is null. If the remaining file name contains `\..', the file extension consists of all such characters from the first remaining `\..' to the end, otherwise the file extension is null. @^system dependencies@> We can scan such file names easily by using two global variables that keep track of the occurrences of area and extension delimiters: @= @!area_delimiter:pool_pointer; {the most recent `\.>' or `\.:', if any} @!ext_delimiter:pool_pointer; {the relevant `\..', if any} @ Input files that can't be found in the user's area may appear in a standard system area called |MF_area|. This system area name will, of course, vary from place to place. @^system dependencies@> @d MF_area=="MFinputs:" @.MFinputs@> @ Here now is the first of the system-dependent routines for file name scanning. @^system dependencies@> @p procedure begin_name; begin area_delimiter:=0; ext_delimiter:=0; end; @ And here's the second. @^system dependencies@> @p function more_name(@!c:ASCII_code):boolean; begin if c=" " then more_name:=false else begin if (c=">")or(c=":") then begin area_delimiter:=pool_ptr; ext_delimiter:=0; end else if (c=".")and(ext_delimiter=0) then ext_delimiter:=pool_ptr; str_room(1); append_char(c); {contribute |c| to the current string} more_name:=true; end; end; @ The third. @^system dependencies@> @p procedure end_name; begin if str_ptr+3>max_str_ptr then begin if str_ptr+3>max_strings then overflow("number of strings",max_strings-init_str_ptr); @:METAFONT capacity exceeded number of strings}{\quad number of strings@> max_str_ptr:=str_ptr+3; end; if area_delimiter=0 then cur_area:="" else begin cur_area:=str_ptr; incr(str_ptr); str_start[str_ptr]:=area_delimiter+1; end; if ext_delimiter=0 then begin cur_ext:=""; cur_name:=make_string; end else begin cur_name:=str_ptr; incr(str_ptr); str_start[str_ptr]:=ext_delimiter; cur_ext:=make_string; end; end; @ Conversely, here is a routine that takes three strings and prints a file name that might have produced them. (The routine is system dependent, because some operating systems put the file area last instead of first.) @^system dependencies@> @= procedure print_file_name(@!n,@!a,@!e:integer); begin print(a); print(n); print(e); end; @ Another system-dependent routine is needed to convert three \MF\ strings to the |name_of_file| value that is used to open files. The present code allows both lowercase and uppercase letters in the file name. @^system dependencies@> @d append_to_name(#)==begin c:=#; incr(k); if k<=file_name_size then name_of_file[k]:=xchr[c]; end @p procedure pack_file_name(@!n,@!a,@!e:str_number); var @!k:integer; {number of positions filled in |name_of_file|} @!c: ASCII_code; {character being packed} @!j:pool_pointer; {index into |str_pool|} begin k:=0; for j:=str_start[a] to str_start[a+1]-1 do append_to_name(str_pool[j]); for j:=str_start[n] to str_start[n+1]-1 do append_to_name(str_pool[j]); for j:=str_start[e] to str_start[e+1]-1 do append_to_name(str_pool[j]); if k<=file_name_size then name_length:=k@+else name_length:=file_name_size; for k:=name_length+1 to file_name_size do name_of_file[k]:=' '; end; @ A messier routine is also needed, since base file names must be scanned before \MF's string mechanism has been initialized. We shall use the global variable |MF_base_default| to supply the text for default system areas and extensions related to base files. @^system dependencies@> @d base_default_length=18 {length of the |MF_base_default| string} @d base_area_length=8 {length of its area part} @d base_ext_length=5 {length of its `\.{.base}' part} @= @!MF_base_default:packed array[1..base_default_length] of char; @ @= MF_base_default:='MFbases:plain.base'; @.MFbases@> @.plain@> @^system dependencies@> @ @= if base_default_length>file_name_size then bad:=41; @ Here is the messy routine that was just mentioned. It sets |name_of_file| from the first |n| characters of |MF_base_default|, followed by |buffer[a..b]|, followed by the last |base_ext_length| characters of |MF_base_default|. We dare not give error messages here, since \MF\ calls this routine before the |error| routine is ready to roll. Instead, we simply drop excess characters, since the error will be detected in another way when a strange file name isn't found. @^system dependencies@> @p procedure pack_buffered_name(@!n:small_number;@!a,@!b:integer); var @!k:integer; {number of positions filled in |name_of_file|} @!c: ASCII_code; {character being packed} @!j:integer; {index into |buffer| or |MF_base_default|} begin if n+b-a+1+base_ext_length>file_name_size then b:=a+file_name_size-n-1-base_ext_length; k:=0; for j:=1 to n do append_to_name(xord[MF_base_default[j]]); for j:=a to b do append_to_name(buffer[j]); for j:=base_default_length-base_ext_length+1 to base_default_length do append_to_name(xord[MF_base_default[j]]); if k<=file_name_size then name_length:=k@+else name_length:=file_name_size; for k:=name_length+1 to file_name_size do name_of_file[k]:=' '; end; @ Here is the only place we use |pack_buffered_name|. This part of the program becomes active when a ``virgin'' \MF\ is trying to get going, just after the preliminary initialization, or when the user is substituting another base file by typing `\.\&' after the initial `\.{**}' prompt. The buffer contains the first line of input in |buffer[loc..(last-1)]|, where |loc" "|. @= function open_base_file:boolean; label found,exit; var @!j:0..buf_size; {the first space after the file name} begin j:=loc; if buffer[loc]="&" then begin incr(loc); j:=loc; buffer[last]:=" "; while buffer[j]<>" " do incr(j); pack_buffered_name(0,loc,j-1); {try first without the system file area} if w_open_in(base_file) then goto found; pack_buffered_name(base_area_length,loc,j-1); {now try the system base file area} if w_open_in(base_file) then goto found; wake_up_terminal; wterm_ln('Sorry, I can''t find that base;',' will try PLAIN.'); @.Sorry, I can't find...@> update_terminal; end; {now pull out all the stops: try for the system \.{plain} file} pack_buffered_name(base_default_length-base_ext_length,1,0); if not w_open_in(base_file) then begin wake_up_terminal; wterm_ln('I can''t find the PLAIN base file!'); @.I can't find PLAIN...@> @.plain@> open_base_file:=false; return; end; found:loc:=j; open_base_file:=true; exit:end; @ Operating systems often make it possible to determine the exact name (and possible version number) of a file that has been opened. The following routine, which simply makes a \MF\ string from the value of |name_of_file|, should ideally be changed to deduce the full name of file~|f|, which is the file most recently opened, if it is possible to do this in a \PASCAL\ program. @^system dependencies@> This routine might be called after string memory has overflowed, hence we dare not use `|str_room|'. @p function make_name_string:str_number; var @!k:1..file_name_size; {index into |name_of_file|} begin if (pool_ptr+name_length>pool_size)or(str_ptr=max_strings) then make_name_string:="?" else begin for k:=1 to name_length do append_char(xord[name_of_file[k]]); make_name_string:=make_string; end; end; function a_make_name_string(var @!f:alpha_file):str_number; begin a_make_name_string:=make_name_string; end; function b_make_name_string(var @!f:byte_file):str_number; begin b_make_name_string:=make_name_string; end; function w_make_name_string(var @!f:word_file):str_number; begin w_make_name_string:=make_name_string; end; @ Now let's consider the routines by which \MF\ deals with file names in a system-independent manner. First comes a procedure that looks for a file name in the input by taking the information from the input buffer. (We can't use |get_next|, because the conversion to tokens would destroy necessary information.) This procedure doesn't allow semicolons or percent signs to be part of file names, because of other conventions of \MF. The manual doesn't use semicolons or percents immediately after file names, but some users no doubt will find it natural to do so; therefore system-dependent changes to allow such characters in file names should probably be made with reluctance, and only when an entire file name that includes special characters is ``quoted'' somehow. @^system dependencies@> @p procedure scan_file_name; label done; begin begin_name; while buffer[loc]=" " do incr(loc); loop@+begin if (buffer[loc]=";")or(buffer[loc]="%") then goto done; if not more_name(buffer[loc]) then goto done; incr(loc); end; done: end_name; end; @ The global variable |job_name| contains the file name that was first \&{input} by the user. This name is extended by `\.{log}' and `\.{gf}' and `\.{base}' and `\.{tfm}' in order to make the names of \MF's output files. @= @!job_name:str_number; {principal file name} @!log_opened:boolean; {has the transcript file been opened?} @!log_name:str_number; {full name of the log file} @ Initially |job_name=0|; it becomes nonzero as soon as the true name is known. We have |job_name=0| if and only if the `\.{log}' file has not been opened, except of course for a short time just after |job_name| has become nonzero. @=job_name:=0; log_opened:=false; @ Here is a routine that manufactures the output file names, assuming that |job_name<>0|. It ignores and changes the current settings of |cur_area| and |cur_ext|. @d pack_cur_name==pack_file_name(cur_name,cur_area,cur_ext) @p procedure pack_job_name(@!s:str_number); {|s = ".log"|, |".gf"|, or |".base"|} begin cur_area:=""; cur_ext:=s; cur_name:=job_name; pack_cur_name; end; @ Actually the main output file extension is usually something like |".300gf"| instead of just |".gf"|; the additional number indicates the resolution in pixels per inch, based on the setting of |hppp| when the file is opened. @= @!gf_ext:str_number; {default extension for the output file} @ If some trouble arises when \MF\ tries to open a file, the following routine calls upon the user to supply another file name. Parameter~|s| is used in the error message to identify the type of file; parameter~|e| is the default extension if none is given. Upon exit from the routine, variables |cur_name|, |cur_area|, |cur_ext|, and |name_of_file| are ready for another attempt at file opening. @p procedure prompt_file_name(@!s,@!e:str_number); label done; var @!k:0..buf_size; {index into |buffer|} begin if interaction=scroll_mode then wake_up_terminal; if s="input file name" then print_err("I can't find file `") @.I can't find file x@> else print_err("I can't write on file `"); @.I can't write on file x@> print_file_name(cur_name,cur_area,cur_ext); print("'."); if e=".mf" then show_context; print_nl("Please type another "); print(s); @.Please type...@> if interaction clear_terminal; prompt_input(": "); @; if cur_ext="" then cur_ext:=e; pack_cur_name; end; @ @= begin begin_name; k:=first; while (buffer[k]=" ")and(k; log_name:=a_make_name_string(log_file); selector:=log_only; log_opened:=true; @; input_stack[input_ptr]:=cur_input; {make sure bottom level is in memory} print_nl("**"); @.**@> l:=input_stack[0].limit_field-1; {last position of first line} for k:=1 to l do print(buffer[k]); print_ln; {now the transcript file contains the first line of input} selector:=old_setting+2; {|log_only| or |term_and_log|} end; @ Sometimes |open_log_file| is called at awkward moments when \MF\ is unable to print error messages or even to |show_context|. The |prompt_file_name| routine can result in a |fatal_error|, but the |error| routine will not be invoked because |log_opened| will be false. The normal idea of |batch_mode| is that nothing at all should be written on the terminal. However, in the unusual case that no log file could be opened, we make an exception and allow an explanatory message to be seen. Incidentally, the program always refers to the log file as a `\.{transcript file}', because some systems cannot use the extension `\.{.log}' for this file. @= begin selector:=term_only; prompt_file_name("transcript file name",".log"); end @ @= begin wlog(banner); print(base_ident); print(" "); print_int(round_unscaled(internal[day])); print_char(" "); months:='JANFEBMARAPRMAYJUNJULAUGSEPOCTNOVDEC'; m:=round_unscaled(internal[month]); for k:=3*m-2 to 3*m do wlog(months[k]); print_char(" "); print_int(round_unscaled(internal[year])); print_char(" "); m:=round_unscaled(internal[time]); print_dd(m div 60); print_char(":"); print_dd(m mod 60); end @ Here's an example of how these conventions are used. We shall use the macro |set_output_file_name| when it is time to crank up the output file. @d set_output_file_name== begin if job_name=0 then open_log_file; pack_job_name(gf_ext); while not b_open_out(gf_file) do prompt_file_name("file name for output",gf_ext); output_file_name:=b_make_name_string(gf_file); end @= @!gf_file: byte_file; {the generic font output goes here} @!output_file_name: str_number; {full name of the output file} @ @=output_file_name:=0; @ Let's turn now to the procedure that is used to initiate file reading when an `\.{input}' command is being processed. @p procedure start_input; {\MF\ will \.{input} something} label done; begin @; if cur_ext="" then cur_ext:=".mf"; pack_cur_name; loop@+ begin begin_file_reading; {set up |cur_file| and new level of input} if a_open_in(cur_file) then goto done; pack_file_name(cur_name,MF_area,cur_ext); if a_open_in(cur_file) then goto done; end_file_reading; {remove the level that didn't work} prompt_file_name("input file name",".mf"); end; done: name:=a_make_name_string(cur_file); str_ref[cur_name]:=max_str_ref; if job_name=0 then begin job_name:=cur_name; open_log_file; end; {|open_log_file| doesn't |show_context|, so |limit| and |loc| needn't be set to meaningful values yet} if term_offset+length(name)>max_print_line-2 then print_ln else if (term_offset>0)or(file_offset>0) then print_char(" "); print_char("("); print(name); update_terminal; if name=str_ptr-1 then {we can conserve string pool space now} begin flush_string(name); name:=cur_name; end; @; end; @ Here we have to remember to tell the |input_ln| routine not to start with a |get|. If the file is empty, it is considered to contain a single blank line. @^system dependencies@> @= begin if not input_ln(cur_file,false) then do_nothing; firm_up_the_line; buffer[limit]:="%"; first:=limit+1; loc:=start; line:=1; end @ @= while token_state and(loc=null) do end_token_list; if token_state then begin print_err("File names can't appear within macros"); @.File names can't...@> help3("Sorry...I've converted what follows to tokens,")@/ ("possibly garbaging the name you gave.")@/ ("Please delete the tokens and insert the name again.");@/ error; end; if file_state then scan_file_name else begin cur_name:=""; cur_ext:=""; cur_area:=""; end @* \[39] Introduction to the parsing routines. We come now to the central nervous system that sparks many of \MF's activities. By evaluating expressions, from their primary constituents to ever larger subexpressions, \MF\ builds the structures that ultimately define fonts. Four mutually recursive subroutines are involved in this process: We call them $$\hbox{|scan_primary|, |scan_secondary|, |scan_tertiary|, and |scan_expression|.}$$ @^recursion@> Each of them is parameterless and begins with the first token to be scanned already represented in |cur_cmd|, |cur_mod|, and |cur_sym|. After execution, the value of the primary or secondary or tertiary or expression that was found will appear in the global variables |cur_type| and |cur_exp|. The token following the expression will be represented in |cur_cmd|, |cur_mod|, and |cur_sym|. Technically speaking, the parsing algorithms are ``LL(1),'' more or less; backup mechanisms have been added in order to provide reasonable error recovery. @= @!cur_type:small_number; {the type of the expression just found} @!cur_exp:integer; {the value of the expression just found} @ @= cur_exp:=0; @ Many different kinds of expressions are possible, so it is wise to have precise descriptions of what |cur_type| and |cur_exp| mean in all cases: \smallskip\hang |cur_type=vacuous| means that this expression didn't turn out to have a value at all, because it arose from a \&{begingroup}$\,\ldots\,$\&{endgroup} construction in which there was no expression before the \&{endgroup}. In this case |cur_exp| has some irrelevant value. \smallskip\hang |cur_type=boolean_type| means that |cur_exp| is either |true_code| or |false_code|. \smallskip\hang |cur_type=unknown_boolean| means that |cur_exp| points to a capsule node that is in the ring of variables equivalent to at least one undefined boolean variable. \smallskip\hang |cur_type=string_type| means that |cur_exp| is a string number (i.e., an integer in the range |0<=cur_exp= function stash_cur_exp:pointer; var @!p:pointer; {the capsule that will be returned} begin case cur_type of unknown_types,transform_type,pair_type,dependent,proto_dependent, independent:p:=cur_exp; othercases begin p:=get_node(value_node_size); name_type(p):=capsule; type(p):=cur_type; value(p):=cur_exp; end endcases;@/ cur_type:=vacuous; link(p):=void; stash_cur_exp:=p; end; @ The inverse of |stash_cur_exp| is the following procedure, which deletes an unnecessary capsule and puts its contents into |cur_type| and |cur_exp|. The program steps of \MF\ can be divided into two categories: those in which |cur_type| and |cur_exp| are ``alive'' and those in which they are ``dead,'' in the sense that |cur_type| and |cur_exp| contain relevant information or not. It's important not to ignore them when they're alive, and it's important not to pay attention to them when they're dead. There's also an intermediate category: If |cur_type=vacuous|, then |cur_exp| is irrelevant, hence we can proceed without caring if |cur_type| and |cur_exp| are alive or dead. In such cases we say that |cur_type| and |cur_exp| are {\sl dormant}. It is permissible to call |get_x_next| only when they are alive or dormant. The \\{stash} procedure above assumes that |cur_type| and |cur_exp| are alive or dormant. The \\{unstash} procedure assumes that they are dead or dormant; it resuscitates them. @= procedure unstash_cur_exp(@!p:pointer); begin cur_type:=type(p); case cur_type of unknown_types,transform_type,pair_type,dependent,proto_dependent, independent: cur_exp:=p; othercases begin cur_exp:=value(p); free_node(p,value_node_size); end endcases;@/ end; @ The following procedure prints the values of expressions in an abbreviated format. If its first parameter |p| is null, the value of |(cur_type,cur_exp)| is displayed; otherwise |p| should be a capsule containing the desired value. The second parameter controls the amount of output. If it is~0, dependency lists will be abbreviated to `\.{linearform}' unless they consist of a single term. If it is greater than~1, complicated structures (pens, pictures, and paths) will be displayed in full. @= @t\4@>@@; @t\4@>@@; procedure print_exp(@!p:pointer;@!verbosity:small_number); var @!restore_cur_exp:boolean; {should |cur_exp| be restored?} @!t:small_number; {the type of the expression} @!v:integer; {the value of the expression} @!q:pointer; {a big node being displayed} begin if p<>null then restore_cur_exp:=false else begin p:=stash_cur_exp; restore_cur_exp:=true; end; t:=type(p); if t; if restore_cur_exp then unstash_cur_exp(p); end; @ @= case t of vacuous:print("vacuous"); boolean_type:if v=true_code then print("true")@+else print("false"); unknown_types,numeric_type:@; string_type:begin print_char(""""); slow_print(v); print_char(""""); end; pen_type,future_pen,path_type,picture_type:@; transform_type,pair_type:if v=null then print_type(t) else @; known:print_scaled(v); dependent,proto_dependent:print_dp(t,v,verbosity); independent:print_variable_name(p); othercases confusion("exp") @:this can't happen exp}{\quad exp@> endcases @ @= begin print_char("("); q:=v+big_node_size[t]; repeat if type(v)=known then print_scaled(value(v)) else if type(v)=independent then print_variable_name(v) else print_dp(type(v),dep_list(v),verbosity); v:=v+2; if v<>q then print_char(","); until v=q; print_char(")"); end @ Values of type \&{picture}, \&{path}, and \&{pen} are displayed verbosely in the log file only, unless the user has given a positive value to \\{tracingonline}. @= if verbosity<=1 then print_type(t) else begin if selector=term_and_log then if internal[tracing_online]<=0 then begin selector:=term_only; print_type(t); print(" (see the transcript file)"); selector:=term_and_log; end; case t of pen_type:print_pen(v,"",false); future_pen:print_path(v," (future pen)",false); path_type:print_path(v,"",false); picture_type:begin cur_edges:=v; print_edges("",false,0,0); end; end; {there are no other cases} end @ @= procedure print_dp(@!t:small_number;@!p:pointer;@!verbosity:small_number); var @!q:pointer; {the node following |p|} begin q:=link(p); if (info(q)=null) or (verbosity>0) then print_dependency(p,t) else print("linearform"); end; @ The displayed name of a variable in a ring will not be a capsule unless the ring consists entirely of capsules. @= begin print_type(t); if v<>null then begin print_char(" "); while (name_type(v)=capsule) and (v<>p) do v:=value(v); print_variable_name(v); end; end @ When errors are detected during parsing, it is often helpful to display an expression just above the error message, using |exp_err| or |disp_err| instead of |print_err|. @d exp_err(#)==disp_err(null,#) {displays the current expression} @= procedure disp_err(@!p:pointer;@!s:str_number); begin if interaction=error_stop_mode then wake_up_terminal; print_nl(">> "); @.>>@> print_exp(p,1); {``medium verbose'' printing of the expression} if s<>"" then begin print_nl("! "); print(s); @.!\relax@> end; end; @ If |cur_type| and |cur_exp| contain relevant information that should be recycled, we will use the following procedure, which changes |cur_type| to |known| and stores a given value in |cur_exp|. We can think of |cur_type| and |cur_exp| as either alive or dormant after this has been done, because |cur_exp| will not contain a pointer value. @= procedure flush_cur_exp(@!v:scaled); begin case cur_type of unknown_types,transform_type,pair_type,@|dependent,proto_dependent,independent: begin recycle_value(cur_exp); free_node(cur_exp,value_node_size); end; pen_type: delete_pen_ref(cur_exp); string_type:delete_str_ref(cur_exp); future_pen,path_type: toss_knot_list(cur_exp); picture_type:toss_edges(cur_exp); othercases do_nothing endcases;@/ cur_type:=known; cur_exp:=v; end; @ There's a much more general procedure that is capable of releasing the storage associated with any two-word value packet. @= procedure recycle_value(@!p:pointer); label done; var @!t:small_number; {a type code} @!v:integer; {a value} @!vv:integer; {another value} @!q,@!r,@!s,@!pp:pointer; {link manipulation registers} begin t:=type(p); if t; dependent,proto_dependent:@; independent:@; token_list,structured:confusion("recycle"); @:this can't happen recycle}{\quad recycle@> unsuffixed_macro,suffixed_macro:delete_mac_ref(value(p)); end; {there are no other cases} type(p):=undefined; end; @ @= if v<>null then begin q:=v+big_node_size[t]; repeat q:=q-2; recycle_value(q); until q=v; free_node(v,big_node_size[t]); end @ @= begin q:=dep_list(p); while info(q)<>null do q:=link(q); link(prev_dep(p)):=link(q); prev_dep(link(q)):=prev_dep(p); link(q):=null; flush_node_list(dep_list(p)); end @ When an independent variable disappears, it simply fades away, unless something depends on it. In the latter case, a dependent variable whose coefficient of dependence is maximal will take its place. The relevant algorithm is due to Ignacio~A. Zabala, who implemented it as part of his Ph.D. thesis (Stanford University, December 1982). @^Zabala Salelles, Ignacio Andres@> For example, suppose that variable $x$ is being recycled, and that the only variables depending on~$x$ are $y=2x+a$ and $z=x+b$. In this case we want to make $y$ independent and $z=.5y-.5a+b$; no other variables will depend on~$y$. If $\\{tracingequations}>0$ in this situation, we will print `\.{\#\#\# -2x=-y+a}'. There's a slight complication, however: An independent variable $x$ can occur both in dependency lists and in proto-dependency lists. This makes it necessary to be careful when deciding which coefficient is maximal. Furthermore, this complication is not so slight when a proto-dependent variable is chosen to become independent. For example, suppose that $y=2x+100a$ is proto-dependent while $z=x+b$ is dependent; then we must change $z=.5y-50a+b$ to a proto-dependency, because of the large coefficient `50'. In order to deal with these complications without wasting too much time, we shall link together the occurrences of~$x$ among all the linear dependencies, maintaining separate lists for the dependent and proto-dependent cases. @= begin max_c[dependent]:=0; max_c[proto_dependent]:=0;@/ max_link[dependent]:=null; max_link[proto_dependent]:=null;@/ q:=link(dep_head); while q<>dep_head do begin s:=value_loc(q); {now |link(s)=dep_list(q)|} loop@+ begin r:=link(s); if info(r)=null then goto done; if info(r)<>p then s:=r else begin t:=type(q); link(s):=link(r); info(r):=q; if abs(value(r))>max_c[t] then @ else begin link(r):=max_link[t]; max_link[t]:=r; end; end; end; done: q:=link(r); end; if (max_c[dependent]>0)or(max_c[proto_dependent]>0) then @; end @ The code for independency removal makes use of three two-word arrays. @= @!max_c:array[dependent..proto_dependent] of integer; {max coefficient magnitude} @!max_ptr:array[dependent..proto_dependent] of pointer; {where |p| occurs with |max_c|} @!max_link:array[dependent..proto_dependent] of pointer; {other occurrences of |p|} @ @= begin if max_c[t]>0 then begin link(max_ptr[t]):=max_link[t]; max_link[t]:=max_ptr[t]; end; max_c[t]:=abs(value(r)); max_ptr[t]:=r; end @ @= begin if (max_c[dependent]>=fraction_one)or@| (max_c[dependent] div @'10000 >= max_c[proto_dependent]) then t:=dependent else t:=proto_dependent; @; t:=dependent+proto_dependent-t; {complement |t|} if max_c[t]>0 then {we need to pick up an unchosen dependency} begin link(max_ptr[t]):=max_link[t]; max_link[t]:=max_ptr[t]; end; if t<>dependent then @ else @; flush_node_list(s); if fix_needed then fix_dependencies; check_arith; end @ Let |s=max_ptr[t]|. At this point we have $|value|(s)=\pm|max_c|[t]$, and |info(s)| points to the dependent variable~|pp| of type~|t| from whose dependency list we have removed node~|s|. We must reinsert node~|s| into the dependency list, with coefficient $-1.0$, and with |pp| as the new independent variable. Since |pp| will have a larger serial number than any other variable, we can put node |s| at the head of the list. @= s:=max_ptr[t]; pp:=info(s); v:=value(s); if t=dependent then value(s):=-fraction_one@+else value(s):=-unity; r:=dep_list(pp); link(s):=r; while info(r)<>null do r:=link(r); q:=link(r); link(r):=null; prev_dep(q):=prev_dep(pp); link(prev_dep(pp)):=q; new_indep(pp); if cur_exp=pp then if cur_type=t then cur_type:=independent; if internal[tracing_equations]>0 then @ @ Now $(-v)$ times the formerly independent variable~|p| is being replaced by the dependency list~|s|. @= if interesting(p) then begin begin_diagnostic; print_nl("### "); @:]]]\#\#\#_}{\.{\#\#\#}@> if v>0 then print_char("-"); if t=dependent then vv:=round_fraction(max_c[dependent]) else vv:=max_c[proto_dependent]; if vv<>unity then print_scaled(vv); print_variable_name(p); while value(p) mod s_scale>0 do begin print("*4"); value(p):=value(p)-2; end; if t=dependent then print_char("=")@+else print(" = "); print_dependency(s,t); end_diagnostic(false); end @ Finally, there are dependent and proto-dependent variables whose dependency lists must be brought up to date. @= for t:=dependent to proto_dependent do begin r:=max_link[t]; while r<>null do begin q:=info(r); dep_list(q):=p_plus_fq(dep_list(q),@| make_fraction(value(r),-v),s,t,dependent); if dep_list(q)=dep_final then make_known(q,dep_final); q:=r; r:=link(r); free_node(q,dep_node_size); end; end @ @= for t:=dependent to proto_dependent do begin r:=max_link[t]; while r<>null do begin q:=info(r); if t=dependent then {for safety's sake, we change |q| to |proto_dependent|} begin if cur_exp=q then if cur_type=dependent then cur_type:=proto_dependent; dep_list(q):=p_over_v(dep_list(q),unity,dependent,proto_dependent); type(q):=proto_dependent; value(r):=round_fraction(value(r)); end; dep_list(q):=p_plus_fq(dep_list(q),@| make_scaled(value(r),-v),s,proto_dependent,proto_dependent); if dep_list(q)=dep_final then make_known(q,dep_final); q:=r; r:=link(r); free_node(q,dep_node_size); end; end @ Here are some routines that provide handy combinations of actions that are often needed during error recovery. For example, `|flush_error|' flushes the current expression, replaces it by a given value, and calls |error|. Errors often are detected after an extra token has already been scanned. The `\\{put\_get}' routines put that token back before calling |error|; then they get it back again. (Or perhaps they get another token, if the user has changed things.) @= procedure flush_error(@!v:scaled);@+begin error; flush_cur_exp(v);@+end; @# procedure@?back_error; forward;@t\2@>@/ procedure@?get_x_next; forward;@t\2@>@/ @# procedure put_get_error;@+begin back_error; get_x_next;@+end; @# procedure put_get_flush_error(@!v:scaled);@+begin put_get_error; flush_cur_exp(v);@+end; @ A global variable called |var_flag| is set to a special command code just before \MF\ calls |scan_expression|, if the expression should be treated as a variable when this command code immediately follows. For example, |var_flag| is set to |assignment| at the beginning of a statement, because we want to know the {\sl location\/} of a variable at the left of `\.{:=}', not the {\sl value\/} of that variable. The |scan_expression| subroutine calls |scan_tertiary|, which calls |scan_secondary|, which calls |scan_primary|, which sets |var_flag:=0|. In this way each of the scanning routines ``knows'' when it has been called with a special |var_flag|, but |var_flag| is usually zero. A variable preceding a command that equals |var_flag| is converted to a token list rather than a value. Furthermore, an `\.{=}' sign following an expression with |var_flag=assignment| is not considered to be a relation that produces boolean expressions. @= @!var_flag:0..max_command_code; {command that wants a variable} @ @= var_flag:=0; @* \[40] Parsing primary expressions. The first parsing routine, |scan_primary|, is also the most complicated one, since it involves so many different cases. But each case---with one exception---is fairly simple by itself. When |scan_primary| begins, the first token of the primary to be scanned should already appear in |cur_cmd|, |cur_mod|, and |cur_sym|. The values of |cur_type| and |cur_exp| should be either dead or dormant, as explained earlier. If |cur_cmd| is not between |min_primary_command| and |max_primary_command|, inclusive, a syntax error will be signalled. @= procedure scan_primary; label restart, done, done1, done2; var @!p,@!q,@!r:pointer; {for list manipulation} @!c:quarterword; {a primitive operation code} @!my_var_flag:0..max_command_code; {initial value of |my_var_flag|} @!l_delim,@!r_delim:pointer; {hash addresses of a delimiter pair} @@; begin my_var_flag:=var_flag; var_flag:=0; restart:check_arith; @; case cur_cmd of left_delimiter:@; begin_group:@; string_token:@; numeric_token:@; nullary:@; unary,type_name,cycle,plus_or_minus:@; primary_binary:@; str_op:@; internal_quantity:@; capsule_token:make_exp_copy(cur_mod); tag_token:@; othercases begin bad_exp("A primary"); goto restart; @.A primary expression...@> end endcases;@/ get_x_next; {the routines |goto done| if they don't want this} done: if cur_cmd=left_bracket then if cur_type>=known then @; end; @ Errors at the beginning of expressions are flagged by |bad_exp|. @p procedure bad_exp(@!s:str_number); var save_flag:0..max_command_code; begin print_err(s); print(" expression can't begin with `"); print_cmd_mod(cur_cmd,cur_mod); print_char("'"); help4("I'm afraid I need some sort of value in order to continue,")@/ ("so I've tentatively inserted `0'. You may want to")@/ ("delete this zero and insert something else;")@/ ("see Chapter 27 of The METAFONTbook for an example."); @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> back_input; cur_sym:=0; cur_cmd:=numeric_token; cur_mod:=0; ins_error;@/ save_flag:=var_flag; var_flag:=0; get_x_next; var_flag:=save_flag; end; @ @= debug if panicking then check_mem(false);@+gubed@;@/ if interrupt<>0 then if OK_to_interrupt then begin back_input; check_interrupt; get_x_next; end @ @= begin l_delim:=cur_sym; r_delim:=cur_mod; get_x_next; scan_expression; if (cur_cmd=comma) and (cur_type>=known) then @ else check_delimiter(l_delim,r_delim); end @ The |stash_in| subroutine puts the current (numeric) expression into a field within a ``big node.'' @p procedure stash_in(@!p:pointer); var @!q:pointer; {temporary register} begin type(p):=cur_type; if cur_type=known then value(p):=cur_exp else begin if cur_type=independent then @ else begin mem[value_loc(p)]:=mem[value_loc(cur_exp)]; {|dep_list(p):=dep_list(cur_exp)| and |prev_dep(p):=prev_dep(cur_exp)|} link(prev_dep(p)):=p; end; free_node(cur_exp,value_node_size); end; cur_type:=vacuous; end; @ In rare cases the current expression can become |independent|. There may be many dependency lists pointing to such an independent capsule, so we can't simply move it into place within a big node. Instead, we copy it, then recycle it. @ @= begin q:=single_dependency(cur_exp); if q=dep_final then begin type(p):=known; value(p):=0; free_node(q,dep_node_size); end else begin type(p):=dependent; new_dep(p,q); end; recycle_value(cur_exp); end @ @= begin p:=get_node(value_node_size); type(p):=pair_type; name_type(p):=capsule; init_big_node(p); q:=value(p); stash_in(x_part_loc(q));@/ get_x_next; scan_expression; if cur_type help4("I thought you were giving me a pair `(x,y)'; but")@/ ("after finding a nice xpart `x' I found a ypart `y'")@/ ("that isn't of numeric type. So I've changed y to zero.")@/ ("(The y that I didn't like appears above the error message.)"); put_get_flush_error(0); end; stash_in(y_part_loc(q)); check_delimiter(l_delim,r_delim); cur_type:=pair_type; cur_exp:=p; end @ The local variable |group_line| keeps track of the line where a \&{begingroup} command occurred; this will be useful in an error message if the group doesn't actually end. @= @!group_line:integer; {where a group began} @ @= begin group_line:=line; if internal[tracing_commands]>0 then show_cur_cmd_mod; save_boundary_item(p); repeat do_statement; {ends with |cur_cmd>=semicolon|} until cur_cmd<>semicolon; if cur_cmd<>end_group then begin print_err("A group begun on line "); @.A group...never ended@> print_int(group_line); print(" never ended"); help2("I saw a `begingroup' back there that hasn't been matched")@/ ("by `endgroup'. So I've inserted `endgroup' now."); back_error; cur_cmd:=end_group; end; unsave; {this might change |cur_type|, if independent variables are recycled} if internal[tracing_commands]>0 then show_cur_cmd_mod; end @ @= begin cur_type:=string_type; cur_exp:=cur_mod; end @ Later we'll come to procedures that perform actual operations like addition, square root, and so on; our purpose now is to do the parsing. But we might as well mention those future procedures now, so that the suspense won't be too bad: \smallskip |do_nullary(c)| does primitive operations that have no operands (e.g., `\&{true}' or `\&{pencircle}'); \smallskip |do_unary(c)| applies a primitive operation to the current expression; \smallskip |do_binary(p,c)| applies a primitive operation to the capsule~|p| and the current expression. @=do_nullary(cur_mod) @ @= begin c:=cur_mod; get_x_next; scan_primary; do_unary(c); goto done; end @ A numeric token might be a primary by itself, or it might be the numerator of a fraction composed solely of numeric tokens, or it might multiply the primary that follows (provided that the primary doesn't begin with a plus sign or a minus sign). The code here uses the facts that |max_primary_command=plus_or_minus| and |max_primary_command-1=numeric_token|. If a fraction is found that is less than unity, we try to retain higher precision when we use it in scalar multiplication. @= @!num,@!denom:scaled; {for primaries that are fractions, like `1/2'} @ @= begin cur_exp:=cur_mod; cur_type:=known; get_x_next; if cur_cmd<>slash then begin num:=0; denom:=0; end else begin get_x_next; if cur_cmd<>numeric_token then begin back_input; cur_cmd:=slash; cur_mod:=over; cur_sym:=frozen_slash; goto done; end; num:=cur_exp; denom:=cur_mod; if denom=0 then @ else cur_exp:=make_scaled(num,denom); check_arith; get_x_next; end; if cur_cmd>=min_primary_command then if cur_cmdplus_or_minus|} begin p:=stash_cur_exp; scan_primary; if (abs(num)>=abs(denom))or(cur_type= begin print_err("Division by zero"); @.Division by zero@> help1("I'll pretend that you meant to divide by 1."); error; end @ @= begin c:=cur_mod; get_x_next; scan_expression; if cur_cmd<>of_token then begin missing_err("of"); print(" for "); print_cmd_mod(primary_binary,c); @.Missing `of'@> help1("I've got the first argument; will look now for the other."); back_error; end; p:=stash_cur_exp; get_x_next; scan_primary; do_binary(p,c); goto done; end @ @= begin get_x_next; scan_suffix; old_setting:=selector; selector:=new_string; show_token_list(cur_exp,null,100000,0); flush_token_list(cur_exp); cur_exp:=make_string; selector:=old_setting; cur_type:=string_type; goto done; end @ If an internal quantity appears all by itself on the left of an assignment, we return a token list of length one, containing the address of the internal quantity plus |hash_end|. (This accords with the conventions of the save stack, as described earlier.) @= begin q:=cur_mod; if my_var_flag=assignment then begin get_x_next; if cur_cmd=assignment then begin cur_exp:=get_avail; info(cur_exp):=q+hash_end; cur_type:=token_list; goto done; end; back_input; end; cur_type:=known; cur_exp:=internal[q]; end @ The most difficult part of |scan_primary| has been saved for last, since it was necessary to build up some confidence first. We can now face the task of scanning a variable. As we scan a variable, we build a token list containing the relevant names and subscript values, simultaneously following along in the ``collective'' structure to see if we are actually dealing with a macro instead of a value. The local variables |pre_head| and |post_head| will point to the beginning of the prefix and suffix lists; |tail| will point to the end of the list that is currently growing. Another local variable, |tt|, contains partial information about the declared type of the variable-so-far. If |tt>=unsuffixed_macro|, the relation |tt=type(q)| will always hold. If |tt=undefined|, the routine doesn't bother to update its information about type. And if |undefined= @!pre_head,@!post_head,@!tail:pointer; {prefix and suffix list variables} @!tt:small_number; {approximation to the type of the variable-so-far} @!t:pointer; {a token} @!macro_ref:pointer; {reference count for a suffixed macro} @ @= begin fast_get_avail(pre_head); tail:=pre_head; post_head:=null; tt:=vacuous; loop@+ begin t:=cur_tok; link(tail):=t; if tt<>undefined then begin @; if tt>=unsuffixed_macro then @; end; get_x_next; tail:=t; if cur_cmd=left_bracket then @; if cur_cmd>max_suffix_token then goto done1; if cur_cmd; end @ @= begin link(tail):=null; if tt>unsuffixed_macro then {|tt=suffixed_macro|} begin post_head:=get_avail; tail:=post_head; link(tail):=t;@/ tt:=undefined; macro_ref:=value(q); add_mac_ref(macro_ref); end else @; end @ @= begin get_x_next; scan_expression; if cur_cmd<>right_bracket then @ else begin if cur_type<>known then bad_subscript; cur_cmd:=numeric_token; cur_mod:=cur_exp; cur_sym:=0; end; end @ The left bracket that we thought was introducing a subscript might have actually been the left bracket in a mediation construction like `\.{x[a,b]}'. So we don't issue an error message at this point; but we do want to back up so as to avoid any embarrassment about our incorrect assumption. @= begin back_input; {that was the token following the current expression} back_expr; cur_cmd:=left_bracket; cur_mod:=0; cur_sym:=frozen_left_bracket; end @ Here's a routine that puts the current expression back to be read again. @p procedure back_expr; var @!p:pointer; {capsule token} begin p:=stash_cur_exp; link(p):=null; back_list(p); end; @ Unknown subscripts lead to the following error message. @p procedure bad_subscript; begin exp_err("Improper subscript has been replaced by zero"); @.Improper subscript...@> help3("A bracketed subscript must have a known numeric value;")@/ ("unfortunately, what I found was the value that appears just")@/ ("above this error message. So I'll try a zero subscript."); flush_error(0); end; @ Every time we call |get_x_next|, there's a chance that the variable we've been looking at will disappear. Thus, we cannot safely keep |q| pointing into the variable structure; we need to start searching from the root each time. @= @^inner loop@> begin p:=link(pre_head); q:=info(p); tt:=undefined; if eq_type(q) mod outer_tag=tag_token then begin q:=equiv(q); if q=null then goto done2; loop@+ begin p:=link(p); if p=null then begin tt:=type(q); goto done2; end; if type(q)<>structured then goto done2; q:=link(attr_head(q)); {the |collective_subscript| attribute} if p>=hi_mem_min then {it's not a subscript} begin repeat q:=link(q); until attr_loc(q)>=info(p); if attr_loc(q)>info(p) then goto done2; end; end; end; done2:end @ How do things stand now? Well, we have scanned an entire variable name, including possible subscripts and/or attributes; |cur_cmd|, |cur_mod|, and |cur_sym| represent the token that follows. If |post_head=null|, a token list for this variable name starts at |link(pre_head)|, with all subscripts evaluated. But if |post_head<>null|, the variable turned out to be a suffixed macro; |pre_head| is the head of the prefix list, while |post_head| is the head of a token list containing both `\.{\AT!}' and the suffix. Our immediate problem is to see if this variable still exists. (Variable structures can change drastically whenever we call |get_x_next|; users aren't supposed to do this, but the fact that it is possible means that we must be cautious.) The following procedure prints an error message when a variable unexpectedly disappears. Its help message isn't quite right for our present purposes, but we'll be able to fix that up. @p procedure obliterated(@!q:pointer); begin print_err("Variable "); show_token_list(q,null,1000,0); print(" has been obliterated"); @.Variable...obliterated@> help5("It seems you did a nasty thing---probably by accident,")@/ ("but nevertheless you nearly hornswoggled me...")@/ ("While I was evaluating the right-hand side of this")@/ ("command, something happened, and the left-hand side")@/ ("is no longer a variable! So I won't change anything."); end; @ If the variable does exist, we also need to check for a few other special cases before deciding that a plain old ordinary variable has, indeed, been scanned. @= if post_head<>null then @; q:=link(pre_head); free_avail(pre_head); if cur_cmd=my_var_flag then begin cur_type:=token_list; cur_exp:=q; goto done; end; p:=find_variable(q); if p<>null then make_exp_copy(p) else begin obliterated(q);@/ help_line[2]:="While I was evaluating the suffix of this variable,"; help_line[1]:="something was redefined, and it's no longer a variable!"; help_line[0]:="In order to get back on my feet, I've inserted `0' instead."; put_get_flush_error(0); end; flush_node_list(q); goto done @ The only complication associated with macro calling is that the prefix and ``at'' parameters must be packaged in an appropriate list of lists. @= begin p:=get_avail; info(pre_head):=link(pre_head); link(pre_head):=p; info(p):=t; macro_call(value(q),pre_head,null); get_x_next; goto restart; end @ If the ``variable'' that turned out to be a suffixed macro no longer exists, we don't care, because we have reserved a pointer (|macro_ref|) to its token list. @= begin back_input; p:=get_avail; q:=link(post_head); info(pre_head):=link(pre_head); link(pre_head):=post_head; info(post_head):=q; link(post_head):=p; info(p):=link(q); link(q):=null; macro_call(macro_ref,pre_head,null); decr(ref_count(macro_ref)); get_x_next; goto restart; end @ Our remaining job is simply to make a copy of the value that has been found. Some cases are harder than others, but complexity arises solely because of the multiplicity of possible cases. @= @t\4@>@@; procedure make_exp_copy(@!p:pointer); label restart; var @!q,@!r,@!t:pointer; {registers for list manipulation} begin restart: cur_type:=type(p); case cur_type of vacuous,boolean_type,known:cur_exp:=value(p); unknown_types:cur_exp:=new_ring_entry(p); string_type:begin cur_exp:=value(p); add_str_ref(cur_exp); end; pen_type:begin cur_exp:=value(p); add_pen_ref(cur_exp); end; picture_type:cur_exp:=copy_edges(value(p)); path_type,future_pen:cur_exp:=copy_path(value(p)); transform_type,pair_type:@; dependent,proto_dependent:encapsulate(copy_dep_list(dep_list(p))); numeric_type:begin new_indep(p); goto restart; end; independent: begin q:=single_dependency(p); if q=dep_final then begin cur_type:=known; cur_exp:=0; free_node(q,value_node_size); end else begin cur_type:=dependent; encapsulate(q); end; end; othercases confusion("copy") @:this can't happen copy}{\quad copy@> endcases; end; @ The |encapsulate| subroutine assumes that |dep_final| is the tail of dependency list~|p|. @= procedure encapsulate(@!p:pointer); begin cur_exp:=get_node(value_node_size); type(cur_exp):=cur_type; name_type(cur_exp):=capsule; new_dep(cur_exp,p); end; @ The most tedious case arises when the user refers to a \&{pair} or \&{transform} variable; we must copy several fields, each of which can be |independent|, |dependent|, |proto_dependent|, or |known|. @= begin if value(p)=null then init_big_node(p); t:=get_node(value_node_size); name_type(t):=capsule; type(t):=cur_type; init_big_node(t);@/ q:=value(p)+big_node_size[cur_type]; r:=value(t)+big_node_size[cur_type]; repeat q:=q-2; r:=r-2; install(r,q); until q=value(p); cur_exp:=t; end @ The |install| procedure copies a numeric field~|q| into field~|r| of a big node that will be part of a capsule. @= procedure install(@!r,@!q:pointer); var p:pointer; {temporary register} begin if type(q)=known then begin value(r):=value(q); type(r):=known; end else if type(q)=independent then begin p:=single_dependency(q); if p=dep_final then begin type(r):=known; value(r):=0; free_node(p,value_node_size); end else begin type(r):=dependent; new_dep(r,p); end; end else begin type(r):=type(q); new_dep(r,copy_dep_list(dep_list(q))); end; end; @ Expressions of the form `\.{a[b,c]}' are converted into `\.{b+a*(c-b)}', without checking the types of \.b~or~\.c. @= begin p:=stash_cur_exp; get_x_next; scan_expression; if cur_cmd<>comma then begin @; unstash_cur_exp(p); end else begin q:=stash_cur_exp; get_x_next; scan_expression; if cur_cmd<>right_bracket then begin missing_err("]");@/ @.Missing `]'@> help3("I've scanned an expression of the form `a[b,c',")@/ ("so a right bracket should have come next.")@/ ("I shall pretend that one was there.");@/ back_error; end; r:=stash_cur_exp; make_exp_copy(q);@/ do_binary(r,minus); do_binary(p,times); do_binary(q,plus); get_x_next; end; end @ Here is a comparatively simple routine that is used to scan the \&{suffix} parameters of a macro. @= procedure scan_suffix; label done; var @!h,@!t:pointer; {head and tail of the list being built} @!p:pointer; {temporary register} begin h:=get_avail; t:=h; loop@+ begin if cur_cmd=left_bracket then @; if cur_cmd=numeric_token then p:=new_num_tok(cur_mod) else if (cur_cmd=tag_token)or(cur_cmd=internal_quantity) then begin p:=get_avail; info(p):=cur_sym; end else goto done; link(t):=p; t:=p; get_x_next; end; done: cur_exp:=link(h); free_avail(h); cur_type:=token_list; end; @ @= begin get_x_next; scan_expression; if cur_type<>known then bad_subscript; if cur_cmd<>right_bracket then begin missing_err("]");@/ @.Missing `]'@> help3("I've seen a `[' and a subscript value, in a suffix,")@/ ("so a right bracket should have come next.")@/ ("I shall pretend that one was there.");@/ back_error; end; cur_cmd:=numeric_token; cur_mod:=cur_exp; end @* \[41] Parsing secondary and higher expressions. After the complications of |scan_primary|, the |scan_secondary| routine is refreshingly simple. It's not trivial, but the operations are relatively straightforward; the main difficulty is, again, that expressions and data structures might change drastically every time we call |get_x_next|, so a cautious approach is mandatory. For example, a macro defined by \&{primarydef} might have disappeared by the time its second argument has been scanned; we solve this by increasing the reference count of its token list, so that the macro can be called even after it has been clobbered. @= procedure scan_secondary; label restart,continue; var @!p:pointer; {for list manipulation} @!c,@!d:halfword; {operation codes or modifiers} @!mac_name:pointer; {token defined with \&{primarydef}} begin restart:if(cur_cmdmax_primary_command) then bad_exp("A secondary"); @.A secondary expression...@> scan_primary; continue: if cur_cmd<=max_secondary_command then if cur_cmd>=min_secondary_command then begin p:=stash_cur_exp; c:=cur_mod; d:=cur_cmd; if d=secondary_primary_macro then begin mac_name:=cur_sym; add_mac_ref(c); end; get_x_next; scan_primary; if d<>secondary_primary_macro then do_binary(p,c) else begin back_input; binary_mac(p,c,mac_name); decr(ref_count(c)); get_x_next; goto restart; end; goto continue; end; end; @ The following procedure calls a macro that has two parameters, |p| and |cur_exp|. @p procedure binary_mac(@!p,@!c,@!n:pointer); var @!q,@!r:pointer; {nodes in the parameter list} begin q:=get_avail; r:=get_avail; link(q):=r;@/ info(q):=p; info(r):=stash_cur_exp;@/ macro_call(c,q,n); end; @ The next procedure, |scan_tertiary|, is pretty much the same deal. @= procedure scan_tertiary; label restart,continue; var @!p:pointer; {for list manipulation} @!c,@!d:halfword; {operation codes or modifiers} @!mac_name:pointer; {token defined with \&{secondarydef}} begin restart:if(cur_cmdmax_primary_command) then bad_exp("A tertiary"); @.A tertiary expression...@> scan_secondary; if cur_type=future_pen then materialize_pen; continue: if cur_cmd<=max_tertiary_command then if cur_cmd>=min_tertiary_command then begin p:=stash_cur_exp; c:=cur_mod; d:=cur_cmd; if d=tertiary_secondary_macro then begin mac_name:=cur_sym; add_mac_ref(c); end; get_x_next; scan_secondary; if d<>tertiary_secondary_macro then do_binary(p,c) else begin back_input; binary_mac(p,c,mac_name); decr(ref_count(c)); get_x_next; goto restart; end; goto continue; end; end; @ A |future_pen| becomes a full-fledged pen here. @p procedure materialize_pen; label common_ending; var @!a_minus_b,@!a_plus_b,@!major_axis,@!minor_axis:scaled; {ellipse variables} @!theta:angle; {amount by which the ellipse has been rotated} @!p:pointer; {path traverser} @!q:pointer; {the knot list to be made into a pen} begin q:=cur_exp; if left_type(q)=endpoint then begin print_err("Pen path must be a cycle"); @.Pen path must be a cycle@> help2("I can't make a pen from the given path.")@/ ("So I've replaced it by the trivial path `(0,0)..cycle'."); put_get_error; cur_exp:=null_pen; goto common_ending; end else if left_type(q)=open then @; cur_exp:=make_pen(q); common_ending: toss_knot_list(q); cur_type:=pen_type; end; @ We placed the three points $(0,0)$, $(1,0)$, $(0,1)$ into a \&{pencircle}, and they have now been transformed to $(u,v)$, $(A+u,B+v)$, $(C+u,D+v)$; this gives us enough information to deduce the transformation $(x,y)\mapsto(Ax+Cy+u,Bx+Dy+v)$. Given ($A,B,C,D)$ we can always find $(a,b,\theta,\phi)$ such that $$\eqalign{A&=a\cos\phi\cos\theta-b\sin\phi\sin\theta;\cr B&=a\cos\phi\sin\theta+b\sin\phi\cos\theta;\cr C&=-a\sin\phi\cos\theta-b\cos\phi\sin\theta;\cr D&=-a\sin\phi\sin\theta+b\cos\phi\cos\theta.\cr}$$ In this notation, the unit circle $(\cos t,\sin t)$ is transformed into $$\bigl(a\cos(\phi+t)\cos\theta-b\sin(\phi+t)\sin\theta,\; a\cos(\phi+t)\sin\theta+b\sin(\phi+t)\cos\theta\bigr)\;+\;(u,v),$$ which is an ellipse with semi-axes~$(a,b)$, rotated by~$\theta$ and shifted by~$(u,v)$. To solve the stated equations, we note that it is necessary and sufficient to solve $$\eqalign{A-D&=(a-b)\cos(\theta-\phi),\cr B+C&=(a-b)\sin(\theta-\phi),\cr} \qquad \eqalign{A+D&=(a+b)\cos(\theta+\phi),\cr B-C&=(a+b)\sin(\theta+\phi);\cr}$$ and it is easy to find $a-b$, $a+b$, $\theta-\phi$, and $\theta+\phi$ from these formulas. The code below uses |(txx,tyx,txy,tyy,tx,ty)| to stand for $(A,B,C,D,u,v)$. @= begin tx:=x_coord(q); ty:=y_coord(q); txx:=left_x(q)-tx; tyx:=left_y(q)-ty; txy:=right_x(q)-tx; tyy:=right_y(q)-ty; a_minus_b:=pyth_add(txx-tyy,tyx+txy); a_plus_b:=pyth_add(txx+tyy,tyx-txy); major_axis:=half(a_minus_b+a_plus_b); minor_axis:=half(abs(a_plus_b-a_minus_b)); if major_axis=minor_axis then theta:=0 {circle} else theta:=half(n_arg(txx-tyy,tyx+txy)+n_arg(txx+tyy,tyx-txy)); free_node(q,knot_node_size); q:=make_ellipse(major_axis,minor_axis,theta); if (tx<>0)or(ty<>0) then @; end @ @= begin p:=q; repeat x_coord(p):=x_coord(p)+tx; y_coord(p):=y_coord(p)+ty; p:=link(p); until p=q; end @ Finally we reach the deepest level in our quartet of parsing routines. This one is much like the others; but it has an extra complication from the construction of paths, which materialize here. @d continue_path=25 {a label inside of |scan_expression|} @d finish_path=26 {another} @= procedure scan_expression; label restart,done,continue,continue_path,finish_path,exit; var @!p,@!q,@!r,@!pp,@!qq:pointer; {for list manipulation} @!c,@!d:halfword; {operation codes or modifiers} @!my_var_flag:0..max_command_code; {initial value of |var_flag|} @!mac_name:pointer; {token defined with \&{tertiarydef}} @!cycle_hit:boolean; {did a path expression just end with `\&{cycle}'?} @!x,@!y:scaled; {explicit coordinates or tension at a path join} @!t:endpoint..open; {knot type following a path join} begin my_var_flag:=var_flag; restart:if(cur_cmdmax_primary_command) then bad_exp("An"); @.An expression...@> scan_tertiary; continue: if cur_cmd<=max_expression_command then if cur_cmd>=min_expression_command then if (cur_cmd<>equals)or(my_var_flag<>assignment) then begin p:=stash_cur_exp; c:=cur_mod; d:=cur_cmd; if d=expression_tertiary_macro then begin mac_name:=cur_sym; add_mac_ref(c); end; if (d else begin get_x_next; scan_tertiary; if d<>expression_tertiary_macro then do_binary(p,c) else begin back_input; binary_mac(p,c,mac_name); decr(ref_count(c)); get_x_next; goto restart; end; end; goto continue; end; exit:end; @ The reader should review the data structure conventions for paths before hoping to understand the next part of this code. @= begin cycle_hit:=false; @; continue_path: @; if cur_cmd=cycle then @ else begin scan_tertiary; @; end; @; if cur_cmd>=min_expression_command then if cur_cmd<=ampersand then if not cycle_hit then goto continue_path; finish_path: @; end @ @= begin unstash_cur_exp(p); if cur_type=pair_type then p:=new_knot else if cur_type=path_type then p:=cur_exp else return; q:=p; while link(q)<>p do q:=link(q); if left_type(p)<>endpoint then {open up a cycle} begin r:=copy_knot(p); link(q):=r; q:=r; end; left_type(p):=open; right_type(q):=open; end @ A pair of numeric values is changed into a knot node for a one-point path when \MF\ discovers that the pair is part of a path. @p@t\4@>@@; function new_knot:pointer; {convert a pair to a knot with two endpoints} var @!q:pointer; {the new node} begin q:=get_node(knot_node_size); left_type(q):=endpoint; right_type(q):=endpoint; link(q):=q;@/ known_pair; x_coord(q):=cur_x; y_coord(q):=cur_y; new_knot:=q; end; @ The |known_pair| subroutine sets |cur_x| and |cur_y| to the components of the current expression, assuming that the current expression is a pair of known numerics. Unknown components are zeroed, and the current expression is flushed. @= procedure known_pair; var @!p:pointer; {the pair node} begin if cur_type<>pair_type then begin exp_err("Undefined coordinates have been replaced by (0,0)"); @.Undefined coordinates...@> help5("I need x and y numbers for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> ("you might want to type `I ???' now.)"); put_get_flush_error(0); cur_x:=0; cur_y:=0; end else begin p:=value(cur_exp); @; flush_cur_exp(0); end; end; @ @= if type(x_part_loc(p))=known then cur_x:=value(x_part_loc(p)) else begin disp_err(x_part_loc(p), "Undefined x coordinate has been replaced by 0"); @.Undefined coordinates...@> help5("I need a `known' x value for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> ("you might want to type `I ???' now.)"); put_get_error; recycle_value(x_part_loc(p)); cur_x:=0; end; if type(y_part_loc(p))=known then cur_y:=value(y_part_loc(p)) else begin disp_err(y_part_loc(p), "Undefined y coordinate has been replaced by 0"); help5("I need a `known' y value for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ ("you might want to type `I ???' now.)"); put_get_error; recycle_value(y_part_loc(p)); cur_y:=0; end @ At this point |cur_cmd| is either |ampersand|, |left_brace|, or |path_join|. @= if cur_cmd=left_brace then @; d:=cur_cmd; if d=path_join then @ else if d<>ampersand then goto finish_path; get_x_next; if cur_cmd=left_brace then @ else if right_type(q)<>explicit then begin t:=open; x:=0; end @ The |scan_direction| subroutine looks at the directional information that is enclosed in braces, and also scans ahead to the following character. A type code is returned, either |open| (if the direction was $(0,0)$), or |curl| (if the direction was a curl of known value |cur_exp|), or |given| (if the direction is given by the |angle| value that now appears in |cur_exp|). There's nothing difficult about this subroutine, but the program is rather lengthy because a variety of potential errors need to be nipped in the bud. @p function scan_direction:small_number; var @!t:given..open; {the type of information found} @!x:scaled; {an |x| coordinate} begin get_x_next; if cur_cmd=curl_command then @ else @; if cur_cmd<>right_brace then begin missing_err("}");@/ @.Missing `\char`\}'@> help3("I've scanned a direction spec for part of a path,")@/ ("so a right brace should have come next.")@/ ("I shall pretend that one was there.");@/ back_error; end; get_x_next; scan_direction:=t; end; @ @= begin get_x_next; scan_expression; if (cur_type<>known)or(cur_exp<0) then begin exp_err("Improper curl has been replaced by 1"); @.Improper curl@> help1("A curl must be a known, nonnegative number."); put_get_flush_error(unity); end; t:=curl; end @ @= begin scan_expression; if cur_type>pair_type then @ else known_pair; if (cur_x=0)and(cur_y=0) then t:=open else begin t:=given; cur_exp:=n_arg(cur_x,cur_y); end; end @ @= begin if cur_type<>known then begin exp_err("Undefined x coordinate has been replaced by 0"); @.Undefined coordinates...@> help5("I need a `known' x value for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> ("you might want to type `I ???' now.)"); put_get_flush_error(0); end; x:=cur_exp; if cur_cmd<>comma then begin missing_err(",");@/ @.Missing `,'@> help2("I've got the x coordinate of a path direction;")@/ ("will look for the y coordinate next."); back_error; end; get_x_next; scan_expression; if cur_type<>known then begin exp_err("Undefined y coordinate has been replaced by 0"); help5("I need a `known' y value for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ ("you might want to type `I ???' now.)"); put_get_flush_error(0); end; cur_y:=cur_exp; cur_x:=x; end @ At this point |right_type(q)| is usually |open|, but it may have been set to some other value by a previous splicing operation. We must maintain the value of |right_type(q)| in unusual cases such as `\.{..z1\{z2\}\&\{z3\}z1\{0,0\}..}'. @= begin t:=scan_direction; if t<>open then begin right_type(q):=t; right_given(q):=cur_exp; if left_type(q)=open then begin left_type(q):=t; left_given(q):=cur_exp; end; {note that |left_given(q)=left_curl(q)|} end; end @ Since |left_tension| and |left_y| share the same position in knot nodes, and since |left_given| is similarly equivalent to |left_x|, we use |x| and |y| to hold the given direction and tension information when there are no explicit control points. @= begin t:=scan_direction; if right_type(q)<>explicit then x:=cur_exp else t:=explicit; {the direction information is superfluous} end @ @= begin get_x_next; if cur_cmd=tension then @ else if cur_cmd=controls then @ else begin right_tension(q):=unity; y:=unity; back_input; {default tension} goto done; end; if cur_cmd<>path_join then begin missing_err("..");@/ @.Missing `..'@> help1("A path join command should end with two dots."); back_error; end; done:end @ @= begin get_x_next; y:=cur_cmd; if cur_cmd=at_least then get_x_next; scan_primary; @; if y=at_least then negate(cur_exp); right_tension(q):=cur_exp; if cur_cmd=and_command then begin get_x_next; y:=cur_cmd; if cur_cmd=at_least then get_x_next; scan_primary; @; if y=at_least then negate(cur_exp); end; y:=cur_exp; end @ @d min_tension==three_quarter_unit @= if (cur_type<>known)or(cur_exp help1("The expression above should have been a number > 3/4."); put_get_flush_error(unity); end @ @= begin right_type(q):=explicit; t:=explicit; get_x_next; scan_primary;@/ known_pair; right_x(q):=cur_x; right_y(q):=cur_y; if cur_cmd<>and_command then begin x:=right_x(q); y:=right_y(q); end else begin get_x_next; scan_primary;@/ known_pair; x:=cur_x; y:=cur_y; end; end @ @= begin if cur_type<>path_type then pp:=new_knot else pp:=cur_exp; qq:=pp; while link(qq)<>pp do qq:=link(qq); if left_type(pp)<>endpoint then {open up a cycle} begin r:=copy_knot(pp); link(qq):=r; qq:=r; end; left_type(pp):=open; right_type(qq):=open; end @ If a person tries to define an entire path by saying `\.{(x,y)\&cycle}', we silently change the specification to `\.{(x,y)..cycle}', since a cycle shouldn't have length zero. @= begin cycle_hit:=true; get_x_next; pp:=p; qq:=p; if d=ampersand then if p=q then begin d:=path_join; right_tension(q):=unity; y:=unity; end; end @ @= begin if d=ampersand then if (x_coord(q)<>x_coord(pp))or(y_coord(q)<>y_coord(pp)) then begin print_err("Paths don't touch; `&' will be changed to `..'"); @.Paths don't touch@> help3("When you join paths `p&q', the ending point of p")@/ ("must be exactly equal to the starting point of q.")@/ ("So I'm going to pretend that you said `p..q' instead."); put_get_error; d:=path_join; right_tension(q):=unity; y:=unity; end; @; if d=ampersand then @ else begin @; link(q):=pp; left_y(pp):=y; if t<>open then begin left_x(pp):=x; left_type(pp):=t; end; end; q:=qq; end @ @= if right_type(q)=open then if (left_type(q)=curl)or(left_type(q)=given) then begin right_type(q):=left_type(q); right_given(q):=left_given(q); end @ @= if right_type(pp)=open then if (t=curl)or(t=given) then begin right_type(pp):=t; right_given(pp):=x; end @ @= begin if left_type(q)=open then if right_type(q)=open then begin left_type(q):=curl; left_curl(q):=unity; end; if right_type(pp)=open then if t=open then begin right_type(pp):=curl; right_curl(pp):=unity; end; right_type(q):=right_type(pp); link(q):=link(pp);@/ right_x(q):=right_x(pp); right_y(q):=right_y(pp); free_node(pp,knot_node_size); if qq=pp then qq:=q; end @ @= if cycle_hit then begin if d=ampersand then p:=q; end else begin left_type(p):=endpoint; if right_type(p)=open then begin right_type(p):=curl; right_curl(p):=unity; end; right_type(q):=endpoint; if left_type(q)=open then begin left_type(q):=curl; left_curl(q):=unity; end; link(q):=p; end; make_choices(p); cur_type:=path_type; cur_exp:=p @ Finally, we sometimes need to scan an expression whose value is supposed to be either |true_code| or |false_code|. @= procedure get_boolean; begin get_x_next; scan_expression; if cur_type<>boolean_type then begin exp_err("Undefined condition will be treated as `false'"); @.Undefined condition...@> help2("The expression shown above should have had a definite")@/ ("true-or-false value. I'm changing it to `false'.");@/ put_get_flush_error(false_code); cur_type:=boolean_type; end; end; @* \[42] Doing the operations. The purpose of parsing is primarily to permit people to avoid piles of parentheses. But the real work is done after the structure of an expression has been recognized; that's when new expressions are generated. We turn now to the guts of \MF, which handles individual operators that have come through the parsing mechanism. We'll start with the easy ones that take no operands, then work our way up to operators with one and ultimately two arguments. In other words, we will write the three procedures |do_nullary|, |do_unary|, and |do_binary| that are invoked periodically by the expression scanners. First let's make sure that all of the primitive operators are in the hash table. Although |scan_primary| and its relatives made use of the \\{cmd} code for these operators, the \\{do} routines base everything on the \\{mod} code. For example, |do_binary| doesn't care whether the operation it performs is a |primary_binary| or |secondary_binary|, etc. @= primitive("true",nullary,true_code);@/ @!@:true_}{\&{true} primitive@> primitive("false",nullary,false_code);@/ @!@:false_}{\&{false} primitive@> primitive("nullpicture",nullary,null_picture_code);@/ @!@:null_picture_}{\&{nullpicture} primitive@> primitive("nullpen",nullary,null_pen_code);@/ @!@:null_pen_}{\&{nullpen} primitive@> primitive("jobname",nullary,job_name_op);@/ @!@:job_name_}{\&{jobname} primitive@> primitive("readstring",nullary,read_string_op);@/ @!@:read_string_}{\&{readstring} primitive@> primitive("pencircle",nullary,pen_circle);@/ @!@:pen_circle_}{\&{pencircle} primitive@> primitive("normaldeviate",nullary,normal_deviate);@/ @!@:normal_deviate_}{\&{normaldeviate} primitive@> primitive("odd",unary,odd_op);@/ @!@:odd_}{\&{odd} primitive@> primitive("known",unary,known_op);@/ @!@:known_}{\&{known} primitive@> primitive("unknown",unary,unknown_op);@/ @!@:unknown_}{\&{unknown} primitive@> primitive("not",unary,not_op);@/ @!@:not_}{\&{not} primitive@> primitive("decimal",unary,decimal);@/ @!@:decimal_}{\&{decimal} primitive@> primitive("reverse",unary,reverse);@/ @!@:reverse_}{\&{reverse} primitive@> primitive("makepath",unary,make_path_op);@/ @!@:make_path_}{\&{makepath} primitive@> primitive("makepen",unary,make_pen_op);@/ @!@:make_pen_}{\&{makepen} primitive@> primitive("totalweight",unary,total_weight_op);@/ @!@:total_weight_}{\&{totalweight} primitive@> primitive("oct",unary,oct_op);@/ @!@:oct_}{\&{oct} primitive@> primitive("hex",unary,hex_op);@/ @!@:hex_}{\&{hex} primitive@> primitive("ASCII",unary,ASCII_op);@/ @!@:ASCII_}{\&{ASCII} primitive@> primitive("char",unary,char_op);@/ @!@:char_}{\&{char} primitive@> primitive("length",unary,length_op);@/ @!@:length_}{\&{length} primitive@> primitive("turningnumber",unary,turning_op);@/ @!@:turning_number_}{\&{turningnumber} primitive@> primitive("xpart",unary,x_part);@/ @!@:x_part_}{\&{xpart} primitive@> primitive("ypart",unary,y_part);@/ @!@:y_part_}{\&{ypart} primitive@> primitive("xxpart",unary,xx_part);@/ @!@:xx_part_}{\&{xxpart} primitive@> primitive("xypart",unary,xy_part);@/ @!@:xy_part_}{\&{xypart} primitive@> primitive("yxpart",unary,yx_part);@/ @!@:yx_part_}{\&{yxpart} primitive@> primitive("yypart",unary,yy_part);@/ @!@:yy_part_}{\&{yypart} primitive@> primitive("sqrt",unary,sqrt_op);@/ @!@:sqrt_}{\&{sqrt} primitive@> primitive("mexp",unary,m_exp_op);@/ @!@:m_exp_}{\&{mexp} primitive@> primitive("mlog",unary,m_log_op);@/ @!@:m_log_}{\&{mlog} primitive@> primitive("sind",unary,sin_d_op);@/ @!@:sin_d_}{\&{sind} primitive@> primitive("cosd",unary,cos_d_op);@/ @!@:cos_d_}{\&{cosd} primitive@> primitive("floor",unary,floor_op);@/ @!@:floor_}{\&{floor} primitive@> primitive("uniformdeviate",unary,uniform_deviate);@/ @!@:uniform_deviate_}{\&{uniformdeviate} primitive@> primitive("charexists",unary,char_exists_op);@/ @!@:char_exists_}{\&{charexists} primitive@> primitive("angle",unary,angle_op);@/ @!@:angle_}{\&{angle} primitive@> primitive("cycle",cycle,cycle_op);@/ @!@:cycle_}{\&{cycle} primitive@> primitive("+",plus_or_minus,plus);@/ @!@:+ }{\.{+} primitive@> primitive("-",plus_or_minus,minus);@/ @!@:- }{\.{-} primitive@> primitive("*",secondary_binary,times);@/ @!@:* }{\.{*} primitive@> primitive("/",slash,over); eqtb[frozen_slash]:=eqtb[cur_sym];@/ @!@:/ }{\.{/} primitive@> primitive("++",tertiary_binary,pythag_add);@/ @!@:++_}{\.{++} primitive@> primitive("+-+",tertiary_binary,pythag_sub);@/ @!@:+-+_}{\.{+-+} primitive@> primitive("and",and_command,and_op);@/ @!@:and_}{\&{and} primitive@> primitive("or",tertiary_binary,or_op);@/ @!@:or_}{\&{or} primitive@> primitive("<",expression_binary,less_than);@/ @!@:< }{\.{<} primitive@> primitive("<=",expression_binary,less_or_equal);@/ @!@:<=_}{\.{<=} primitive@> primitive(">",expression_binary,greater_than);@/ @!@:> }{\.{>} primitive@> primitive(">=",expression_binary,greater_or_equal);@/ @!@:>=_}{\.{>=} primitive@> primitive("=",equals,equal_to);@/ @!@:= }{\.{=} primitive@> primitive("<>",expression_binary,unequal_to);@/ @!@:<>_}{\.{<>} primitive@> primitive("substring",primary_binary,substring_of);@/ @!@:substring_}{\&{substring} primitive@> primitive("subpath",primary_binary,subpath_of);@/ @!@:subpath_}{\&{subpath} primitive@> primitive("directiontime",primary_binary,direction_time_of);@/ @!@:direction_time_}{\&{directiontime} primitive@> primitive("point",primary_binary,point_of);@/ @!@:point_}{\&{point} primitive@> primitive("precontrol",primary_binary,precontrol_of);@/ @!@:precontrol_}{\&{precontrol} primitive@> primitive("postcontrol",primary_binary,postcontrol_of);@/ @!@:postcontrol_}{\&{postcontrol} primitive@> primitive("penoffset",primary_binary,pen_offset_of);@/ @!@:pen_offset_}{\&{penoffset} primitive@> primitive("&",ampersand,concatenate);@/ @!@:!!!}{\.{\&} primitive@> primitive("rotated",secondary_binary,rotated_by);@/ @!@:rotated_}{\&{rotated} primitive@> primitive("slanted",secondary_binary,slanted_by);@/ @!@:slanted_}{\&{slanted} primitive@> primitive("scaled",secondary_binary,scaled_by);@/ @!@:scaled_}{\&{scaled} primitive@> primitive("shifted",secondary_binary,shifted_by);@/ @!@:shifted_}{\&{shifted} primitive@> primitive("transformed",secondary_binary,transformed_by);@/ @!@:transformed_}{\&{transformed} primitive@> primitive("xscaled",secondary_binary,x_scaled);@/ @!@:x_scaled_}{\&{xscaled} primitive@> primitive("yscaled",secondary_binary,y_scaled);@/ @!@:y_scaled_}{\&{yscaled} primitive@> primitive("zscaled",secondary_binary,z_scaled);@/ @!@:z_scaled_}{\&{zscaled} primitive@> primitive("intersectiontimes",tertiary_binary,intersect);@/ @!@:intersection_times_}{\&{intersectiontimes} primitive@> @ @= nullary,unary,primary_binary,secondary_binary,tertiary_binary, expression_binary,cycle,plus_or_minus,slash,ampersand,equals,and_command: print_op(m); @ OK, let's look at the simplest \\{do} procedure first. @p procedure do_nullary(@!c:quarterword); var @!k:integer; {all-purpose loop index} begin check_arith; if internal[tracing_commands]>two then show_cmd_mod(nullary,c); case c of true_code,false_code:begin cur_type:=boolean_type; cur_exp:=c; end; null_picture_code:begin cur_type:=picture_type; cur_exp:=get_node(edge_header_size); init_edges(cur_exp); end; null_pen_code:begin cur_type:=pen_type; cur_exp:=null_pen; end; normal_deviate:begin cur_type:=known; cur_exp:=norm_rand; end; pen_circle:@; job_name_op: begin if job_name=0 then open_log_file; cur_type:=string_type; cur_exp:=job_name; end; read_string_op:@; end; {there are no other cases} check_arith; end; @ @= begin cur_type:=future_pen; cur_exp:=get_node(knot_node_size); left_type(cur_exp):=open; right_type(cur_exp):=open; link(cur_exp):=cur_exp;@/ x_coord(cur_exp):=0; y_coord(cur_exp):=0;@/ left_x(cur_exp):=unity; left_y(cur_exp):=0;@/ right_x(cur_exp):=0; right_y(cur_exp):=unity;@/ end @ @= begin if interaction<=nonstop_mode then fatal_error("*** (cannot readstring in nonstop modes)"); begin_file_reading; name:=1; prompt_input(""); str_room(last-start); for k:=start to last-1 do append_char(buffer[k]); end_file_reading; cur_type:=string_type; cur_exp:=make_string; end @ Things get a bit more interesting when there's an operand. The operand to |do_unary| appears in |cur_type| and |cur_exp|. @p @t\4@>@@; procedure do_unary(@!c:quarterword); var @!p,@!q:pointer; {for list manipulation} @!x:integer; {a temporary register} begin check_arith; if internal[tracing_commands]>two then @; case c of plus:if cur_typepicture_type then bad_unary(plus); minus:@; @t\4@>@@; end; {there are no other cases} check_arith; end; @ The |nice_pair| function returns |true| if both components of a pair are known. @= function nice_pair(@!p:integer;@!t:quarterword):boolean; label exit; begin if t=pair_type then begin p:=value(p); if type(x_part_loc(p))=known then if type(y_part_loc(p))=known then begin nice_pair:=true; return; end; end; nice_pair:=false; exit:end; @ @= procedure print_known_or_unknown_type(@!t:small_number;@!v:integer); begin print_char("("); if tpair_type then print_type(t) else if nice_pair(v,pair_type) then print("pair") else print("unknown pair") else print("unknown numeric"); print_char(")"); end; @ @= procedure bad_unary(@!c:quarterword); begin exp_err("Not implemented: "); print_op(c); @.Not implemented...@> print_known_or_unknown_type(cur_type,cur_exp); help3("I'm afraid I don't know how to apply that operation to that")@/ ("particular type. Continue, and I'll simply return the")@/ ("argument (shown above) as the result of the operation."); put_get_error; end; @ @= begin begin_diagnostic; print_nl("{"); print_op(c); print_char("(");@/ print_exp(null,0); {show the operand, but not verbosely} print(")}"); end_diagnostic(false); end @ Negation is easy except when the current expression is of type |independent|, or when it is a pair with one or more |independent| components. It is tempting to argue that the negative of an independent variable is an independent variable, hence we don't have to do anything when negating it. The fallacy is that other dependent variables pointing to the current expression must change the sign of their coefficients if we make no change to the current expression. Instead, we work around the problem by copying the current expression and recycling it afterwards (cf.~the |stash_in| routine). @= case cur_type of pair_type,independent: begin q:=cur_exp; make_exp_copy(q); if cur_type=dependent then negate_dep_list(dep_list(cur_exp)) else if cur_type=pair_type then begin p:=value(cur_exp); if type(x_part_loc(p))=known then negate(value(x_part_loc(p))) else negate_dep_list(dep_list(x_part_loc(p))); if type(y_part_loc(p))=known then negate(value(y_part_loc(p))) else negate_dep_list(dep_list(y_part_loc(p))); end; {if |cur_type=known| then |cur_exp=0|} recycle_value(q); free_node(q,value_node_size); end; dependent,proto_dependent:negate_dep_list(dep_list(cur_exp)); known:negate(cur_exp); picture_type:negate_edges(cur_exp); othercases bad_unary(minus) endcases @ @= procedure negate_dep_list(@!p:pointer); label exit; begin loop@+begin negate(value(p)); if info(p)=null then return; p:=link(p); end; exit:end; @ @= not_op: if cur_type<>boolean_type then bad_unary(not_op) else cur_exp:=true_code+false_code-cur_exp; @ @d three_sixty_units==23592960 {that's |360*unity|} @d boolean_reset(#)==if # then cur_exp:=true_code@+else cur_exp:=false_code @= sqrt_op,m_exp_op,m_log_op,sin_d_op,cos_d_op,floor_op, uniform_deviate,odd_op,char_exists_op:@t@>@;@/ if cur_type<>known then bad_unary(c) else case c of sqrt_op:cur_exp:=square_rt(cur_exp); m_exp_op:cur_exp:=m_exp(cur_exp); m_log_op:cur_exp:=m_log(cur_exp); sin_d_op,cos_d_op:begin n_sin_cos((cur_exp mod three_sixty_units)*16); if c=sin_d_op then cur_exp:=round_fraction(n_sin) else cur_exp:=round_fraction(n_cos); end; floor_op:cur_exp:=floor_scaled(cur_exp); uniform_deviate:cur_exp:=unif_rand(cur_exp); odd_op: begin boolean_reset(odd(round_unscaled(cur_exp))); cur_type:=boolean_type; end; char_exists_op:@; end; {there are no other cases} @ @= angle_op:if nice_pair(cur_exp,cur_type) then begin p:=value(cur_exp); x:=n_arg(value(x_part_loc(p)),value(y_part_loc(p))); if x>=0 then flush_cur_exp((x+8)div 16) else flush_cur_exp(-((-x+8)div 16)); end else bad_unary(angle_op); @ If the current expression is a pair, but the context wants it to be a path, we call |pair_to_path|. @= procedure pair_to_path; begin cur_exp:=new_knot; cur_type:=path_type; end; @ @= x_part,y_part:if (cur_type<=pair_type)and(cur_type>=transform_type) then take_part(c) else bad_unary(c); xx_part,xy_part,yx_part,yy_part: if cur_type=transform_type then take_part(c) else bad_unary(c); @ In the following procedure, |cur_exp| points to a capsule, which points to a big node. We want to delete all but one part of the big node. @= procedure take_part(@!c:quarterword); var @!p:pointer; {the big node} begin p:=value(cur_exp); value(temp_val):=p; type(temp_val):=cur_type; link(p):=temp_val; free_node(cur_exp,value_node_size); make_exp_copy(p+2*(c-x_part)); recycle_value(temp_val); end; @ @= name_type(temp_val):=capsule; @ @= char_op: if cur_type<>known then bad_unary(char_op) else begin cur_exp:=round_unscaled(cur_exp) mod 128; cur_type:=string_type; if cur_exp<0 then cur_exp:=cur_exp+128; if length(cur_exp)<>1 then begin str_room(1); append_char(cur_exp); cur_exp:=make_string; end; end; decimal: if cur_type<>known then bad_unary(decimal) else begin old_setting:=selector; selector:=new_string; print_scaled(cur_exp); cur_exp:=make_string; selector:=old_setting; cur_type:=string_type; end; oct_op,hex_op,ASCII_op: if cur_type<>string_type then bad_unary(c) else str_to_num(c); @ @= procedure str_to_num(@!c:quarterword); {converts a string to a number} var @!n:integer; {accumulator} @!m:ASCII_code; {current character} @!k:pool_pointer; {index into |str_pool|} @!b:8..16; {radix of conversion} @!bad_char:boolean; {did the string contain an invalid digit?} begin if c=ASCII_op then if length(cur_exp)=0 then n:=-1 else n:=str_pool[str_start[cur_exp]] else begin if c=oct_op then b:=8@+else b:=16; n:=0; bad_char:=false; for k:=str_start[cur_exp] to str_start[cur_exp+1]-1 do begin m:=str_pool[k]; if (m>="0")and(m<="9") then m:=m-"0" else if (m>="A")and(m<="F") then m:=m-"A"+10 else if (m>="a")and(m<="f") then m:=m-"a"+10 else begin bad_char:=true; m:=0; end; if m>=b then begin bad_char:=true; m:=0; end; if n<32768 div b then n:=n*b+m@+else n:=32767; end; @=4096|@>; end; flush_cur_exp(n*unity); end; @ @= if bad_char then begin exp_err("String contains illegal digits"); @.String contains illegal digits@> if c=oct_op then help1("I zeroed out characters that weren't in the range 0..7.") else help1("I zeroed out characters that weren't hex digits."); put_get_error; end; if n>4095 then begin print_err("Number too large ("); print_int(n); print_char(")"); @.Number too large@> help1("I have trouble with numbers greater than 4095; watch out."); put_get_error; end @ The length operation is somewhat unusual in that it applies to a variety of different types of operands. @= length_op: if cur_type=string_type then flush_cur_exp(length(cur_exp)*unity) else if cur_type=path_type then flush_cur_exp(path_length) else if cur_type=known then cur_exp:=abs(cur_exp) else if nice_pair(cur_exp,cur_type) then flush_cur_exp(pyth_add(value(x_part_loc(value(cur_exp))),@| value(y_part_loc(value(cur_exp))))) else bad_unary(c); @ @= function path_length:scaled; {computes the length of the current path} var @!n:scaled; {the path length so far} @!p:pointer; {traverser} begin p:=cur_exp; if left_type(p)=endpoint then n:=-unity@+else n:=0; repeat p:=link(p); n:=n+unity; until p=cur_exp; path_length:=n; end; @ The turning number is computed only with respect to null pens. A different pen might affect the turning number, in degenerate cases, because autorounding will produce a slightly different path, or because excessively large coordinates might be truncated. @= turning_op:if cur_type=pair_type then flush_cur_exp(0) else if cur_type<>path_type then bad_unary(turning_op) else if left_type(cur_exp)=endpoint then flush_cur_exp(0) {not a cyclic path} else begin cur_pen:=null_pen; cur_path_type:=contour_code; cur_exp:=make_spec(cur_exp, fraction_one-half_unit-1-el_gordo,0); flush_cur_exp(turning_number*unity); {convert to |scaled|} end; @ @d type_test_end== flush_cur_exp(true_code) else flush_cur_exp(false_code); cur_type:=boolean_type; end @d type_range_end(#)==(cur_type<=#) then type_test_end @d type_range(#)==begin if (cur_type>=#) and type_range_end @d type_test(#)==begin if cur_type=# then type_test_end @= boolean_type: type_range(boolean_type)(unknown_boolean); string_type: type_range(string_type)(unknown_string); pen_type: type_range(pen_type)(future_pen); path_type: type_range(path_type)(unknown_path); picture_type: type_range(picture_type)(unknown_picture); transform_type,pair_type: type_test(c); numeric_type: type_range(known)(independent); known_op,unknown_op: test_known(c); @ @= procedure test_known(@!c:quarterword); label done; var @!b:true_code..false_code; {is the current expression known?} @!p,@!q:pointer; {locations in a big node} begin b:=false_code; case cur_type of vacuous,boolean_type,string_type,pen_type,future_pen,path_type,picture_type, known: b:=true_code; transform_type,pair_type:begin p:=value(cur_exp); q:=p+big_node_size[cur_type]; repeat q:=q-2; if type(q)<>known then goto done; until q=p; b:=true_code; done: end; othercases do_nothing endcases; if c=known_op then flush_cur_exp(b) else flush_cur_exp(true_code+false_code-b); cur_type:=boolean_type; end; @ @= cycle_op: begin if cur_type<>path_type then flush_cur_exp(false_code) else if left_type(cur_exp)<>endpoint then flush_cur_exp(true_code) else flush_cur_exp(false_code); cur_type:=boolean_type; end; @ @= make_pen_op: begin if cur_type=pair_type then pair_to_path; if cur_type=path_type then cur_type:=future_pen else bad_unary(make_pen_op); end; make_path_op: begin if cur_type=future_pen then materialize_pen; if cur_type<>pen_type then bad_unary(make_path_op) else begin flush_cur_exp(make_path(cur_exp)); cur_type:=path_type; end; end; total_weight_op: if cur_type<>picture_type then bad_unary(total_weight_op) else flush_cur_exp(total_weight(cur_exp)); reverse: if cur_type=path_type then begin p:=htap_ypoc(cur_exp); if right_type(p)=endpoint then p:=link(p); toss_knot_list(cur_exp); cur_exp:=p; end else if cur_type=pair_type then pair_to_path else bad_unary(reverse); @ Finally, we have the operations that combine a capsule~|p| with the current expression. @p @t\4@>@@; procedure do_binary(@!p:pointer;@!c:quarterword); label done,done1,exit; var @!q,@!r,@!rr:pointer; {for list manipulation} @!old_p,@!old_exp:pointer; {capsules to recycle} @!v:integer; {for numeric manipulation} begin check_arith; if internal[tracing_commands]>two then @; @; @; case c of plus,minus:@; @t\4@>@@; end; {there are no other cases} recycle_value(p); free_node(p,value_node_size); {|return| to avoid this} exit:check_arith; @; end; @ @= procedure bad_binary(@!p:pointer;@!c:quarterword); begin disp_err(p,""); exp_err("Not implemented: "); @.Not implemented...@> if c>=min_of then print_op(c); print_known_or_unknown_type(type(p),p); if c>=min_of then print("of")@+else print_op(c); print_known_or_unknown_type(cur_type,cur_exp);@/ help3("I'm afraid I don't know how to apply that operation to that")@/ ("combination of types. Continue, and I'll return the second")@/ ("argument (see above) as the result of the operation."); put_get_error; end; @ @= begin begin_diagnostic; print_nl("{("); print_exp(p,0); {show the operand, but not verbosely} print_char(")"); print_op(c); print_char("(");@/ print_exp(null,0); print(")}"); end_diagnostic(false); end @ Several of the binary operations are potentially complicated by the fact that |independent| values can sneak into capsules. For example, we've seen an instance of this difficulty in the unary operation of negation. In order to reduce the number of cases that need to be handled, we first change the two operands (if necessary) to rid them of |independent| components. The original operands are put into capsules called |old_p| and |old_exp|, which will be recycled after the binary operation has been safely carried out. @= if old_p<>null then begin recycle_value(old_p); free_node(old_p,value_node_size); end; if old_exp<>null then begin recycle_value(old_exp); free_node(old_exp,value_node_size); end @ A big node is considered to be ``tarnished'' if it contains at least one independent component. We will define a simple function called `|tarnished|' that returns |null| if and only if its argument is not tarnished. @= case type(p) of transform_type,pair_type: old_p:=tarnished(p); independent: old_p:=void; othercases old_p:=null endcases; if old_p<>null then begin q:=stash_cur_exp; old_p:=p; make_exp_copy(old_p); p:=stash_cur_exp; unstash_cur_exp(q); end; @ @= case cur_type of transform_type,pair_type:old_exp:=tarnished(cur_exp); independent:old_exp:=void; othercases old_exp:=null endcases; if old_exp<>null then begin old_exp:=cur_exp; make_exp_copy(old_exp); end @ @= function tarnished(@!p:pointer):pointer; label exit; var @!q:pointer; {beginning of the big node} @!r:pointer; {current position in the big node} begin q:=value(p); r:=q+big_node_size[type(p)]; repeat r:=r-2; if type(r)=independent then begin tarnished:=void; return; end; until r=q; tarnished:=null; exit:end; @ @= if (cur_typepair_type then bad_binary(p,c) else begin q:=value(p); r:=value(cur_exp); add_or_subtract(x_part_loc(q),x_part_loc(r),c); add_or_subtract(y_part_loc(q),y_part_loc(r),c); end else if type(p)=pair_type then bad_binary(p,c) else add_or_subtract(p,null,c) @ The first argument to |add_or_subtract| is the location of a value node in a capsule or pair node that will soon be recycled. The second argument is either a location within a pair or transform node of |cur_exp|, or it is null (which means that |cur_exp| itself should be the second argument). The third argument is either |plus| or |minus|. The sum or difference of the numeric quantities will replace the second operand. Arithmetic overflow may go undetected; users aren't supposed to be monkeying around with really big values. @= @t\4@>@@; procedure add_or_subtract(@!p,@!q:pointer;@!c:quarterword); label done,exit; var @!s,@!t:small_number; {operand types} @!r:pointer; {list traverser} @!v:integer; {second operand value} begin if q=null then begin t:=cur_type; if t; end else begin if c=minus then negate_dep_list(v); @; end; exit:end; @ @= r:=dep_list(p); while info(r)<>null do r:=link(r); value(r):=slow_add(value(r),v); if q=null then begin q:=get_node(value_node_size); cur_exp:=q; cur_type:=type(p); name_type(q):=capsule; end; dep_list(q):=dep_list(p); type(q):=type(p); prev_dep(q):=prev_dep(p); link(prev_dep(p)):=q; type(p):=known; {this will keep the recycler from collecting non-garbage} @ We prefer |dependent| lists to |proto_dependent| ones, because it is nice to retain the extra accuracy of |fraction| coefficients. But we have to handle both kinds, and mixtures too. @= if type(p)=known then @ else begin s:=type(p); r:=dep_list(p); if t=dependent then begin if s=dependent then if max_coef(r)+max_coef(v); end @ @= begin while info(v)<>null do v:=link(v); value(v):=slow_add(value(p),value(v)); end @ @= if q<>null then dep_finish(v,q,t) else begin cur_type:=t; dep_finish(v,null,t); end @ Here's the current situation: The dependency list |v| of type |t| should either be put into the current expression (if |q=null|) or into location |q| within a pair node (otherwise). The destination (|cur_exp| or |q|) formerly held a dependency list with the same final pointer as the list |v|. @= procedure dep_finish(@!v,@!q:pointer;@!t:small_number); var @!p:pointer; {the destination} @!vv:scaled; {the value, if it is |known|} begin if q=null then p:=cur_exp@+else p:=q; dep_list(p):=v; type(p):=t; if info(v)=null then begin vv:=value(v); if q=null then flush_cur_exp(vv) else begin recycle_value(p); type(q):=known; value(q):=vv; end; end else if q=null then cur_type:=t; if fix_needed then fix_dependencies; end; @ Let's turn now to the six basic relations of comparison. @= less_than,less_or_equal,greater_than,greater_or_equal,equal_to,unequal_to: begin@t@>@; if (cur_type>pair_type)and(type(p)>pair_type) then add_or_subtract(p,null,minus) {|cur_exp:=(p)-cur_exp|} else if cur_type<>type(p) then begin bad_binary(p,c); goto done; end else if cur_type=string_type then flush_cur_exp(str_vs_str(value(p),cur_exp)) else if (cur_type=unknown_string)or(cur_type=unknown_boolean) then @ else if (cur_type=pair_type)or(cur_type=transform_type) then @ else if cur_type=boolean_type then flush_cur_exp(cur_exp-value(p)) else begin bad_binary(p,c); goto done; end; @; done: end; @ @= if cur_type<>known then begin if cur_type put_get_flush_error(false_code); end else case c of less_than: boolean_reset(cur_exp<0); less_or_equal: boolean_reset(cur_exp<=0); greater_than: boolean_reset(cur_exp>0); greater_or_equal: boolean_reset(cur_exp>=0); equal_to: boolean_reset(cur_exp=0); unequal_to: boolean_reset(cur_exp<>0); end; {there are no other cases} cur_type:=boolean_type @ When two unknown strings are in the same ring, we know that they are equal. Otherwise, we don't know whether they are equal or not, so we make no change. @= begin q:=value(cur_exp); while (q<>cur_exp)and(q<>p) do q:=value(q); if q=p then flush_cur_exp(0); end @ @= begin q:=value(p); r:=value(cur_exp); rr:=r+big_node_size[cur_type]-2; loop@+ begin add_or_subtract(q,r,minus); if type(r)<>known then goto done1; if value(r)<>0 then goto done1; if r=rr then goto done1; q:=q+2; r:=r+2; end; done1:take_part(x_part+half(r-value(cur_exp))); end @ Here we use the sneaky fact that |and_op-false_code=or_op-true_code|. @= and_op,or_op: if (type(p)<>boolean_type)or(cur_type<>boolean_type) then bad_binary(p,c) else if value(p)=c+false_code-and_op then cur_exp:=value(p); @ @= times: if (cur_type else if (nice_pair(p,type(p))and(cur_type>pair_type)) or(nice_pair(cur_exp,cur_type)and(type(p)>pair_type)) then begin hard_times(p); return; end else bad_binary(p,times); @ @= begin if type(p)=known then begin v:=value(p); free_node(p,value_node_size); end else begin v:=cur_exp; unstash_cur_exp(p); end; if cur_type=known then cur_exp:=take_scaled(cur_exp,v) else if cur_type=pair_type then begin p:=value(cur_exp); dep_mult(x_part_loc(p),v,true); dep_mult(y_part_loc(p),v,true); end else dep_mult(null,v,true); return; end @ @= procedure dep_mult(@!p:pointer;@!v:integer;@!v_is_scaled:boolean); label exit; var @!q:pointer; {the dependency list being multiplied by |v|} @!s,@!t:small_number; {its type, before and after} begin if p=null then q:=cur_exp else if type(p)<>known then q:=p else begin if v_is_scaled then value(p):=take_scaled(value(p),v) else value(p):=take_fraction(value(p),v); return; end; t:=type(q); q:=dep_list(q); s:=t; if t=dependent then if v_is_scaled then if ab_vs_cd(max_coef(q),abs(v),coef_bound-1,unity)>=0 then t:=proto_dependent; q:=p_times_v(q,v,s,t,v_is_scaled); dep_finish(q,p,t); exit:end; @ Here is a routine that is similar to |times|; but it is invoked only internally, when |v| is a |fraction| whose magnitude is at most~1, and when |cur_type>=pair_type|. @p procedure frac_mult(@!n,@!d:scaled); {multiplies |cur_exp| by |n/d|} var @!p:pointer; {a pair node} @!old_exp:pointer; {a capsule to recycle} @!v:fraction; {|n/d|} begin if internal[tracing_commands]>two then @; case cur_type of transform_type,pair_type:old_exp:=tarnished(cur_exp); independent:old_exp:=void; othercases old_exp:=null endcases; if old_exp<>null then begin old_exp:=cur_exp; make_exp_copy(old_exp); end; v:=make_fraction(n,d); if cur_type=known then cur_exp:=take_fraction(cur_exp,v) else if cur_type=pair_type then begin p:=value(cur_exp); dep_mult(x_part_loc(p),v,false); dep_mult(y_part_loc(p),v,false); end else dep_mult(null,v,false); if old_exp<>null then begin recycle_value(old_exp); free_node(old_exp,value_node_size); end end; @ @= begin begin_diagnostic; print_nl("{("); print_scaled(n); print_char("/"); print_scaled(d); print(")*("); print_exp(null,0); print(")}"); end_diagnostic(false); end @ The |hard_times| routine multiplies a nice pair by a dependency list. @= procedure hard_times(@!p:pointer); var @!q:pointer; {a copy of the dependent variable |p|} @!r:pointer; {the big node for the nice pair} @!u,@!v:scaled; {the known values of the nice pair} begin if type(p)=pair_type then begin q:=stash_cur_exp; unstash_cur_exp(p); p:=q; end; {now |cur_type=pair_type|} r:=value(cur_exp); u:=value(x_part_loc(r)); v:=value(y_part_loc(r)); @; dep_mult(x_part_loc(r),u,true); dep_mult(y_part_loc(r),v,true); end; @ @= type(y_part_loc(r)):=type(p); new_dep(y_part_loc(r),copy_dep_list(dep_list(p)));@/ type(x_part_loc(r)):=type(p); mem[value_loc(x_part_loc(r))]:=mem[value_loc(p)]; link(prev_dep(p)):=x_part_loc(r); free_node(p,value_node_size) @ @= over: if (cur_type<>known)or(type(p) else begin if cur_type=known then cur_exp:=make_scaled(cur_exp,v) else if cur_type=pair_type then begin p:=value(cur_exp); dep_div(x_part_loc(p),v); dep_div(y_part_loc(p),v); end else dep_div(null,v); end; return; end; @ @= procedure dep_div(@!p:pointer;@!v:scaled); label exit; var @!q:pointer; {the dependency list being divided by |v|} @!s,@!t:small_number; {its type, before and after} begin if p=null then q:=cur_exp else if type(p)<>known then q:=p else begin value(p):=make_scaled(value(p),v); return; end; t:=type(q); q:=dep_list(q); s:=t; if t=dependent then if ab_vs_cd(max_coef(q),unity,coef_bound-1,abs(v))>=0 then t:=proto_dependent; q:=p_over_v(q,v,s,t); dep_finish(q,p,t); exit:end; @ @= begin exp_err("Division by zero"); @.Division by zero@> help2("You're trying to divide the quantity shown above the error")@/ ("message by zero. I'm going to divide it by one instead."); put_get_error; end @ @= pythag_add,pythag_sub: if (cur_type=known)and(type(p)=known) then if c=pythag_add then cur_exp:=pyth_add(value(p),cur_exp) else cur_exp:=pyth_sub(value(p),cur_exp) else bad_binary(p,c); @ The next few sections of the program deal with affine transformations of coordinate data. @= rotated_by,slanted_by,scaled_by,shifted_by,transformed_by, x_scaled,y_scaled,z_scaled: @t@>@;@/ if (type(p)=path_type)or(type(p)=future_pen)or(type(p)=pen_type) then begin path_trans(p,c); return; end else if (type(p)=pair_type)or(type(p)=transform_type) then big_trans(p,c) else if type(p)=picture_type then begin edges_trans(p,c); return; end else bad_binary(p,c); @ Let |c| be one of the eight transform operators. The procedure call |set_up_trans(c)| first changes |cur_exp| to a transform that corresponds to |c| and the original value of |cur_exp|. (In particular, |cur_exp| doesn't change at all if |c=transformed_by|.) Then, if all components of the resulting transform are |known|, they are moved to the global variables |txx|, |txy|, |tyx|, |tyy|, |tx|, |ty|; and |cur_exp| is changed to the known value zero. @= procedure set_up_trans(@!c:quarterword); label done,exit; var @!p,@!q,@!r:pointer; {list manipulation registers} begin if (c<>transformed_by)or(cur_type<>transform_type) then @; @; exit:end; @ @= @!txx,@!txy,@!tyx,@!tyy,@!tx,@!ty:scaled; {current transform coefficients} @ @= begin p:=stash_cur_exp; cur_exp:=id_transform; cur_type:=transform_type; q:=value(cur_exp); case c of @@; end; {there are no other cases} disp_err(p,"Improper transformation argument"); @.Improper transformation argument@> help3("The expression shown above has the wrong type,")@/ ("so I can't transform anything using it.")@/ ("Proceed, and I'll omit the transformation."); put_get_error; done: recycle_value(p); free_node(p,value_node_size); end @ @= q:=value(cur_exp); r:=q+transform_node_size; repeat r:=r-2; if type(r)<>known then return; until r=q; txx:=value(xx_part_loc(q)); txy:=value(xy_part_loc(q)); tyx:=value(yx_part_loc(q)); tyy:=value(yy_part_loc(q)); tx:=value(x_part_loc(q)); ty:=value(y_part_loc(q)); flush_cur_exp(0) @ @= rotated_by:if type(p)=known then @; slanted_by:if type(p)>pair_type then begin install(xy_part_loc(q),p); goto done; end; scaled_by:if type(p)>pair_type then begin install(xx_part_loc(q),p); install(yy_part_loc(q),p); goto done; end; shifted_by:if type(p)=pair_type then begin r:=value(p); install(x_part_loc(q),x_part_loc(r)); install(y_part_loc(q),y_part_loc(r)); goto done; end; x_scaled:if type(p)>pair_type then begin install(xx_part_loc(q),p); goto done; end; y_scaled:if type(p)>pair_type then begin install(yy_part_loc(q),p); goto done; end; z_scaled:if type(p)=pair_type then @; transformed_by:do_nothing; @ @= begin n_sin_cos((value(p) mod three_sixty_units)*16); value(xx_part_loc(q)):=round_fraction(n_cos); value(yx_part_loc(q)):=round_fraction(n_sin); value(xy_part_loc(q)):=-value(yx_part_loc(q)); value(yy_part_loc(q)):=value(xx_part_loc(q)); goto done; end @ @= begin r:=value(p); install(xx_part_loc(q),x_part_loc(r)); install(yy_part_loc(q),x_part_loc(r)); install(yx_part_loc(q),y_part_loc(r)); if type(y_part_loc(r))=known then negate(value(y_part_loc(r))) else negate_dep_list(dep_list(y_part_loc(r))); install(xy_part_loc(q),y_part_loc(r)); goto done; end @ Procedure |set_up_known_trans| is like |set_up_trans|, but it insists that the transformation be entirely known. @= procedure set_up_known_trans(@!c:quarterword); begin set_up_trans(c); if cur_type<>known then begin exp_err("Transform components aren't all known"); @.Transform components...@> help3("I'm unable to apply a partially specified transformation")@/ ("except to a fully known pair or transform.")@/ ("Proceed, and I'll omit the transformation."); put_get_flush_error(0); txx:=unity; txy:=0; tyx:=0; tyy:=unity; tx:=0; ty:=0; end; end; @ Here's a procedure that applies the transform |txx..ty| to a pair of coordinates in locations |p| and~|q|. @= procedure trans(@!p,@!q:pointer); var @!v:scaled; {the new |x| value} begin v:=take_scaled(mem[p].sc,txx)+take_scaled(mem[q].sc,txy)+tx; mem[q].sc:=take_scaled(mem[p].sc,tyx)+take_scaled(mem[q].sc,tyy)+ty; mem[p].sc:=v; end; @ The simplest transformation procedure applies a transform to all coordinates of a path. The |null_pen| remains unchanged if it isn't being shifted. @= procedure path_trans(@!p:pointer;@!c:quarterword); label exit; var @!q:pointer; {list traversor} begin set_up_known_trans(c); unstash_cur_exp(p); if cur_type=pen_type then begin if max_offset(cur_exp)=0 then if tx=0 then if ty=0 then return; flush_cur_exp(make_path(cur_exp)); cur_type:=future_pen; end; q:=cur_exp; repeat if left_type(q)<>endpoint then trans(q+3,q+4); {that's |left_x| and |left_y|} trans(q+1,q+2); {that's |x_coord| and |y_coord|} if right_type(q)<>endpoint then trans(q+5,q+6); {that's |right_x| and |right_y|} q:=link(q); until q=cur_exp; exit:end; @ The next simplest transformation procedure applies to edges. It is simple primarily because \MF\ doesn't allow very general transformations to be made, and because the tricky subroutines for edge transformation have already been written. @= procedure edges_trans(@!p:pointer;@!c:quarterword); label exit; begin set_up_known_trans(c); unstash_cur_exp(p); cur_edges:=cur_exp; if empty_edges(cur_edges) then return; {the empty set is easy to transform} if txx=0 then if tyy=0 then if txy mod unity=0 then if tyx mod unity=0 then begin xy_swap_edges; txx:=txy; tyy:=tyx; txy:=0; tyx:=0; if empty_edges(cur_edges) then return; end; if txy=0 then if tyx=0 then if txx mod unity=0 then if tyy mod unity=0 then @; print_err("That transformation is too hard"); @.That transformation...@> help3("I can apply complicated transformations to paths,")@/ ("but I can only do integer operations on pictures.")@/ ("Proceed, and I'll omit the transformation."); put_get_error; exit:end; @ @= begin if (txx=0)or(tyy=0) then begin toss_edges(cur_edges); cur_exp:=get_node(edge_header_size); init_edges(cur_exp); end else begin if txx<0 then begin x_reflect_edges; txx:=-txx; end; if tyy<0 then begin y_reflect_edges; tyy:=-tyy; end; if txx<>unity then x_scale_edges(txx div unity); if tyy<>unity then y_scale_edges(tyy div unity); @; end; return; end @ @= tx:=round_unscaled(tx); ty:=round_unscaled(ty); if (m_min(cur_edges)+tx<=0)or(m_max(cur_edges)+tx>=8192)or@| (n_min(cur_edges)+ty<=0)or(n_max(cur_edges)+ty>=8191)or@| (abs(tx)>=4096)or(abs(ty)>=4096) then begin print_err("Too far to shift"); @.Too far to shift@> help3("I can't shift the picture as requested---it would")@/ ("make some coordinates too large or too small.")@/ ("Proceed, and I'll omit the transformation."); put_get_error; end else begin if tx<>0 then begin if not valid_range(m_offset(cur_edges)-tx) then fix_offset; m_min(cur_edges):=m_min(cur_edges)+tx; m_max(cur_edges):=m_max(cur_edges)+tx; m_offset(cur_edges):=m_offset(cur_edges)-tx; last_window_time(cur_edges):=0; end; if ty<>0 then begin n_min(cur_edges):=n_min(cur_edges)+ty; n_max(cur_edges):=n_max(cur_edges)+ty; n_pos(cur_edges):=n_pos(cur_edges)+ty; last_window_time(cur_edges):=0; end; end @ The hard cases of transformation occur when big nodes are involved, and when some of their components are unknown. @= @t\4@>@@; procedure big_trans(@!p:pointer;@!c:quarterword); label exit; var @!q,@!r,@!pp,@!qq:pointer; {list manipulation registers} @!s:small_number; {size of a big node} begin s:=big_node_size[type(p)]; q:=value(p); r:=q+s; repeat r:=r-2; if type(r)<>known then @; until r=q; @; exit:end; {node |p| will now be recycled by |do_binary|} @ @= begin set_up_known_trans(c); make_exp_copy(p); r:=value(cur_exp); if cur_type=transform_type then begin bilin1(yy_part_loc(r),tyy,xy_part_loc(q),tyx,0); bilin1(yx_part_loc(r),tyy,xx_part_loc(q),tyx,0); bilin1(xy_part_loc(r),txx,yy_part_loc(q),txy,0); bilin1(xx_part_loc(r),txx,yx_part_loc(q),txy,0); end; bilin1(y_part_loc(r),tyy,x_part_loc(q),tyx,ty); bilin1(x_part_loc(r),txx,y_part_loc(q),txy,tx); return; end @ Let |p| point to a two-word value field inside a big node of |cur_exp|, and let |q| point to a another value field. The |bilin1| procedure replaces |p| by $p\cdot t+q\cdot u+\delta$. @= procedure bilin1(@!p:pointer;@!t:scaled;@!q:pointer;@!u,@!delta:scaled); var @!r:pointer; {list traverser} begin if t<>unity then dep_mult(p,t,true); if u<>0 then if type(q)=known then delta:=delta+take_scaled(value(q),u) else begin @; dep_list(p):=p_plus_fq(dep_list(p),u,dep_list(q),proto_dependent,type(q)); end; if type(p)=known then value(p):=value(p)+delta else begin r:=dep_list(p); while info(r)<>null do r:=link(r); delta:=value(r)+delta; if r<>dep_list(p) then value(r):=delta else begin recycle_value(p); type(p):=known; value(p):=delta; end; end; if fix_needed then fix_dependencies; end; @ @= if type(p)<>proto_dependent then begin if type(p)=known then new_dep(p,const_dependency(value(p))) else dep_list(p):=p_times_v(dep_list(p),unity,dependent,proto_dependent,true); type(p):=proto_dependent; end @ @= set_up_trans(c); if cur_type=known then @ else begin pp:=stash_cur_exp; qq:=value(pp); make_exp_copy(p); r:=value(cur_exp); if cur_type=transform_type then begin bilin2(yy_part_loc(r),yy_part_loc(qq), value(xy_part_loc(q)),yx_part_loc(qq),null); bilin2(yx_part_loc(r),yy_part_loc(qq), value(xx_part_loc(q)),yx_part_loc(qq),null); bilin2(xy_part_loc(r),xx_part_loc(qq), value(yy_part_loc(q)),xy_part_loc(qq),null); bilin2(xx_part_loc(r),xx_part_loc(qq), value(yx_part_loc(q)),xy_part_loc(qq),null); end; bilin2(y_part_loc(r),yy_part_loc(qq), value(x_part_loc(q)),yx_part_loc(qq),y_part_loc(qq)); bilin2(x_part_loc(r),xx_part_loc(qq), value(y_part_loc(q)),xy_part_loc(qq),x_part_loc(qq)); recycle_value(pp); free_node(pp,value_node_size); end; @ Let |p| be a |proto_dependent| value whose dependency list ends at |dep_final|. The following procedure adds |v| times another numeric quantity to~|p|. @= procedure add_mult_dep(@!p:pointer;@!v:scaled;@!r:pointer); begin if type(r)=known then value(dep_final):=value(dep_final)+take_scaled(value(r),v) else begin dep_list(p):= p_plus_fq(dep_list(p),v,dep_list(r),proto_dependent,type(r)); if fix_needed then fix_dependencies; end; end; @ The |bilin2| procedure is something like |bilin1|, but with known and unknown quantities reversed. Parameter |p| points to a value field within the big node for |cur_exp|; and |type(p)=known|. Parameters |t| and~|u| point to value fields elsewhere; so does parameter~|q|, unless it is |null| (which stands for zero). Location~|p| will be replaced by $p\cdot t+v\cdot u+q$. @= procedure bilin2(@!p,@!t:pointer;@!v:scaled;@!u,@!q:pointer); var @!vv:scaled; {temporary storage for |value(p)|} begin vv:=value(p); type(p):=proto_dependent; new_dep(p,const_dependency(0)); {this sets |dep_final|} if vv<>0 then add_mult_dep(p,vv,t); {|dep_final| doesn't change} if v<>0 then add_mult_dep(p,v,u); if q<>null then add_mult_dep(p,unity,q); if dep_list(p)=dep_final then begin vv:=value(dep_final); recycle_value(p); type(p):=known; value(p):=vv; end; end; @ @= begin make_exp_copy(p); r:=value(cur_exp); if cur_type=transform_type then begin bilin3(yy_part_loc(r),tyy,value(xy_part_loc(q)),tyx,0); bilin3(yx_part_loc(r),tyy,value(xx_part_loc(q)),tyx,0); bilin3(xy_part_loc(r),txx,value(yy_part_loc(q)),txy,0); bilin3(xx_part_loc(r),txx,value(yx_part_loc(q)),txy,0); end; bilin3(y_part_loc(r),tyy,value(x_part_loc(q)),tyx,ty); bilin3(x_part_loc(r),txx,value(y_part_loc(q)),txy,tx); end @ Finally, in |bilin3| everything is |known|. @= procedure bilin3(@!p:pointer;@!t,@!v,@!u,@!delta:scaled); begin if t<>unity then delta:=delta+take_scaled(value(p),t) else delta:=delta+value(p); if u<>0 then value(p):=delta+take_scaled(v,u) else value(p):=delta; end; @ @= concatenate: if (cur_type=string_type)and(type(p)=string_type) then cat(p) else bad_binary(p,concatenate); substring_of: if nice_pair(p,type(p))and(cur_type=string_type) then chop_string(value(p)) else bad_binary(p,substring_of); subpath_of: begin if cur_type=pair_type then pair_to_path; if nice_pair(p,type(p))and(cur_type=path_type) then chop_path(value(p)) else bad_binary(p,subpath_of); end; @ @= procedure cat(@!p:pointer); var @!a,@!b:str_number; {the strings being concatenated} @!k:pool_pointer; {index into |str_pool|} begin a:=value(p); b:=cur_exp; str_room(length(a)+length(b)); for k:=str_start[a] to str_start[a+1]-1 do append_char(str_pool[k]); for k:=str_start[b] to str_start[b+1]-1 do append_char(str_pool[k]); cur_exp:=make_string; delete_str_ref(b); end; @ @= procedure chop_string(@!p:pointer); var @!a,@!b:integer; {start and stop points} @!l:integer; {length of the original string} @!k:integer; {runs from |a| to |b|} @!s:str_number; {the original string} @!reversed:boolean; {was |a>b|?} begin a:=round_unscaled(value(x_part_loc(p))); b:=round_unscaled(value(y_part_loc(p))); if a<=b then reversed:=false else begin reversed:=true; k:=a; a:=b; b:=k; end; s:=cur_exp; l:=length(s); if a<0 then begin a:=0; if b<0 then b:=0; end; if b>l then begin b:=l; if a>l then a:=l; end; str_room(b-a); if reversed then for k:=str_start[s]+b-1 downto str_start[s]+a do append_char(str_pool[k]) else for k:=str_start[s]+a to str_start[s]+b-1 do append_char(str_pool[k]); cur_exp:=make_string; delete_str_ref(s); end; @ @= procedure chop_path(@!p:pointer); var @!q:pointer; {a knot in the original path} @!pp,@!qq,@!rr,@!ss:pointer; {link variables for copies of path nodes} @!a,@!b,@!k,@!l:scaled; {indices for chopping} @!reversed:boolean; {was |a>b|?} begin l:=path_length; a:=value(x_part_loc(p)); b:=value(y_part_loc(p)); if a<=b then reversed:=false else begin reversed:=true; k:=a; a:=b; b:=k; end; @l|@>; q:=cur_exp; while a>=unity do begin q:=link(q); a:=a-unity; b:=b-unity; end; if b=a then @ else @; left_type(pp):=endpoint; right_type(qq):=endpoint; link(qq):=pp; toss_knot_list(cur_exp); if reversed then begin cur_exp:=link(htap_ypoc(pp)); toss_knot_list(pp); end else cur_exp:=pp; end; @ @l|@>= if a<0 then if left_type(cur_exp)=endpoint then begin a:=0; if b<0 then b:=0; end else repeat a:=a+l; b:=b+l; until a>=0; {a cycle always has length |l>0|} if b>l then if left_type(cur_exp)=endpoint then begin b:=l; if a>l then a:=l; end else while a>=l do begin a:=a-l; b:=b-l; end @ @= begin pp:=copy_knot(q); qq:=pp; repeat q:=link(q); rr:=qq; qq:=copy_knot(q); link(rr):=qq; b:=b-unity; until b<=0; if a>0 then begin ss:=pp; pp:=link(pp); split_cubic(ss,a*@'10000,x_coord(pp),y_coord(pp)); pp:=link(ss); free_node(ss,knot_node_size); if rr=ss then begin b:=make_scaled(b,unity-a); rr:=pp; end; end; if b<0 then begin split_cubic(rr,(b+unity)*@'10000,x_coord(qq),y_coord(qq)); free_node(qq,knot_node_size); qq:=link(rr); end; end @ @= begin if a>0 then begin qq:=link(q); split_cubic(q,a*@'10000,x_coord(qq),y_coord(qq)); q:=link(q); end; pp:=copy_knot(q); qq:=pp; end @ The |pair_value| routine changes the current expression to a given ordered pair of values. @= procedure pair_value(@!x,@!y:scaled); var @!p:pointer; {a pair node} begin p:=get_node(value_node_size); flush_cur_exp(p); cur_type:=pair_type; type(p):=pair_type; name_type(p):=capsule; init_big_node(p); p:=value(p);@/ type(x_part_loc(p)):=known; value(x_part_loc(p)):=x;@/ type(y_part_loc(p)):=known; value(y_part_loc(p)):=y;@/ end; @ @= point_of,precontrol_of,postcontrol_of: begin if cur_type=pair_type then pair_to_path; if (cur_type=path_type)and(type(p)=known) then find_point(value(p),c) else bad_binary(p,c); end; pen_offset_of: begin if cur_type=future_pen then materialize_pen; if (cur_type=pen_type)and nice_pair(p,type(p)) then set_up_offset(value(p)) else bad_binary(p,pen_offset_of); end; direction_time_of: begin if cur_type=pair_type then pair_to_path; if (cur_type=path_type)and nice_pair(p,type(p)) then set_up_direction_time(value(p)) else bad_binary(p,direction_time_of); end; @ @= procedure set_up_offset(@!p:pointer); begin find_offset(value(x_part_loc(p)),value(y_part_loc(p)),cur_exp); pair_value(cur_x,cur_y); end; @# procedure set_up_direction_time(@!p:pointer); begin flush_cur_exp(find_direction_time(value(x_part_loc(p)), value(y_part_loc(p)),cur_exp)); end; @ @= procedure find_point(@!v:scaled;@!c:quarterword); var @!p:pointer; {the path} @!n:scaled; {its length} @!q:pointer; {successor of |p|} begin p:=cur_exp;@/ if left_type(p)=endpoint then n:=-unity@+else n:=0; repeat p:=link(p); n:=n+unity; until p=cur_exp; if n=0 then v:=0 else if v<0 then if left_type(p)=endpoint then v:=0 else v:=n-1-((-v-1) mod n) else if v>n then if left_type(p)=endpoint then v:=n else v:=v mod n; p:=cur_exp; while v>=unity do begin p:=link(p); v:=v-unity; end; if v<>0 then @; @; end; @ @= begin q:=link(p); split_cubic(p,v*@'10000,x_coord(q),y_coord(q)); p:=link(p); end @ @= case c of point_of: pair_value(x_coord(p),y_coord(p)); precontrol_of: if left_type(p)=endpoint then pair_value(x_coord(p),y_coord(p)) else pair_value(left_x(p),left_y(p)); postcontrol_of: if right_type(p)=endpoint then pair_value(x_coord(p),y_coord(p)) else pair_value(right_x(p),right_y(p)); end {there are no other cases} @ @= intersect: begin if type(p)=pair_type then begin q:=stash_cur_exp; unstash_cur_exp(p); pair_to_path; p:=stash_cur_exp; unstash_cur_exp(q); end; if cur_type=pair_type then pair_to_path; if (cur_type=path_type)and(type(p)=path_type) then begin path_intersection(value(p),cur_exp); pair_value(cur_t,cur_tt); end else bad_binary(p,intersect); end; @* \[43] Statements and commands. The chief executive of \MF\ is the |do_statement| routine, which contains the master switch that causes all the various pieces of \MF\ to do their things, in the right order. In a sense, this is the grand climax of the program: It applies all of the tools that we have worked so hard to construct. In another sense, this is the messiest part of the program: It necessarily refers to other pieces of code all over the place, so that a person can't fully understand what is going on without paging back and forth to be reminded of conventions that are defined elsewhere. We are now at the hub of the web. The structure of |do_statement| itself is quite simple. The first token of the statement is fetched using |get_x_next|. If it can be the first token of an expression, we look for an equation, an assignment, or a title. Otherwise we use a \&{case} construction to branch at high speed to the appropriate routine for various and sundry other types of commands, each of which has an ``action procedure'' that does the necessary work. The program uses the fact that |min_primary_command=max_statement_command =type_name| to interpret a statement that starts with, e.g., `\&{string}', as a type declaration rather than a boolean expression. @p @t\4@>@@; @t\4@>@@; procedure do_statement; {governs \MF's activities} begin cur_type:=vacuous; get_x_next; if cur_cmd>max_primary_command then @ else if cur_cmd>max_statement_command then @ else @; if cur_cmd; error_count:=0; end; @ The only command codes |>max_primary_command| that can be present at the beginning of a statement are |semicolon| and higher; these occur when the statement is null. @= begin if cur_cmd print_cmd_mod(cur_cmd,cur_mod); print_char("'"); help5("I was looking for the beginning of a new statement.")@/ ("If you just proceed without changing anything, I'll ignore")@/ ("everything up to the next `;'. Please insert a semicolon")@/ ("now in front of anything that you don't want me to delete.")@/ ("(See Chapter 27 of The METAFONTbook for an example.)");@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> back_error; get_x_next; end; end @ The help message printed here says that everything is flushed up to a semicolon, but actually the commands |end_group| and |stop| will also terminate a statement. @= begin print_err("Extra tokens will be flushed"); @.Extra tokens will be flushed@> help6("I've just read as much of that statement as I could fathom,")@/ ("so a semicolon should have been next. It's very puzzling...")@/ ("but I'll try to get myself back together, by ignoring")@/ ("everything up to the next `;'. Please insert a semicolon")@/ ("now in front of anything that you don't want me to delete.")@/ ("(See Chapter 27 of The METAFONTbook for an example.)");@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> back_error; scanner_status:=flushing; repeat get_next; @; until end_of_statement; {|cur_cmd=semicolon|, |end_group|, or |stop|} scanner_status:=normal; end @ If |do_statement| ends with |cur_cmd=end_group|, we should have |cur_type=vacuous| unless the statement was simply an expression; in the latter case, |cur_type| and |cur_exp| should represent that expression. @= begin if internal[tracing_commands]>0 then show_cur_cmd_mod; case cur_cmd of type_name:do_type_declaration; macro_def:if cur_mod>var_def then make_op_def else if cur_mod>end_def then scan_def; @t\4@>@@; end; {there are no other cases} cur_type:=vacuous; end @ The most important statements begin with expressions. @= begin var_flag:=assignment; scan_expression; if cur_cmd else if cur_type<>vacuous then begin exp_err("Isolated expression"); @.Isolated expression@> help3("I couldn't find an `=' or `:=' after the")@/ ("expression that is shown above this error message,")@/ ("so I guess I'll just ignore it and carry on."); put_get_error; end; flush_cur_exp(0); cur_type:=vacuous; end; end @ @= begin if internal[tracing_titles]>0 then begin print_nl(""); slow_print(cur_exp); update_terminal; end; if internal[proofing]>0 then @; end @ Equations and assignments are performed by the pair of mutually recursive @^recursion@> routines |do_equation| and |do_assignment|. These routines are called when |cur_cmd=equals| and |cur_cmd=assignment|, respectively; the left-hand side is in |cur_type| and |cur_exp|, while the right-hand side is yet to be scanned. After the routines are finished, |cur_type| and |cur_exp| will be equal to the right-hand side (which will normally be equal to the left-hand side). @= @t\4@>@@; @t\4@>@@; procedure@?do_assignment; forward;@t\2@>@/ procedure do_equation; var @!lhs:pointer; {capsule for the left-hand side} @!p:pointer; {temporary register} begin lhs:=stash_cur_exp; get_x_next; var_flag:=assignment; scan_expression; if cur_cmd=equals then do_equation else if cur_cmd=assignment then do_assignment; if internal[tracing_commands]>two then @; if cur_type=unknown_path then if type(lhs)=pair_type then begin p:=stash_cur_exp; unstash_cur_exp(lhs); lhs:=p; end; {in this case |make_eq| will change the pair to a path} make_eq(lhs); {equate |lhs| to |(cur_type,cur_exp)|} end; @ And |do_assignment| is similar to |do_expression|: @= procedure do_assignment; var @!lhs:pointer; {token list for the left-hand side} @!p:pointer; {where the left-hand value is stored} @!q:pointer; {temporary capsule for the right-hand value} begin if cur_type<>token_list then begin exp_err("Improper `:=' will be changed to `='"); @.Improper `:='@> help2("I didn't find a variable name at the left of the `:=',")@/ ("so I'm going to pretend that you said `=' instead.");@/ error; do_equation; end else begin lhs:=cur_exp; cur_type:=vacuous;@/ get_x_next; var_flag:=assignment; scan_expression; if cur_cmd=equals then do_equation else if cur_cmd=assignment then do_assignment; if internal[tracing_commands]>two then @; if info(lhs)>hash_end then @ else @; flush_node_list(lhs); end; end; @ @= begin begin_diagnostic; print_nl("{("); print_exp(lhs,0); print(")=("); print_exp(null,0); print(")}"); end_diagnostic(false); end @ @= begin begin_diagnostic; print_nl("{"); if info(lhs)>hash_end then print(int_name[info(lhs)-(hash_end)]) else show_token_list(lhs,null,1000,0); print(":="); print_exp(null,0); print_char("}"); end_diagnostic(false); end @ @= if cur_type=known then internal[info(lhs)-(hash_end)]:=cur_exp else begin exp_err("Internal quantity `"); @.Internal quantity...@> print(int_name[info(lhs)-(hash_end)]); print("' must receive a known value"); help2("I can't set an internal quantity to anything but a known")@/ ("numeric value, so I'll have to ignore this assignment."); put_get_error; end @ @= begin p:=find_variable(lhs); if p<>null then begin q:=stash_cur_exp; cur_type:=und_type(p); recycle_value(p); type(p):=cur_type; value(p):=null; make_exp_copy(p); p:=stash_cur_exp; unstash_cur_exp(q); make_eq(p); end else begin obliterated(lhs); put_get_error; end; end @ And now we get to the nitty-gritty. The |make_eq| procedure is given a pointer to a capsule that is to be equated to the current expression. @= procedure make_eq(@!lhs:pointer); label restart,done, not_found; var @!t:small_number; {type of the left-hand side} @!v:integer; {value of the left-hand side} @!p,@!q:pointer; {pointers inside of big nodes} begin restart: t:=type(lhs); if t<=pair_type then v:=value(lhs); case t of @t\4@>@@; end; {all cases have been listed} @; done:check_arith; recycle_value(lhs); free_node(lhs,value_node_size); end; @ @= disp_err(lhs,""); exp_err("Equation cannot be performed ("); @.Equation cannot be performed@> if type(lhs)<=pair_type then print_type(type(lhs))@+else print("numeric"); print_char("="); if cur_type<=pair_type then print_type(cur_type)@+else print("numeric"); print_char(")");@/ help2("I'm sorry, but I don't know how to make such things equal.")@/ ("(See the two expressions just above the error message.)"); put_get_error @ @= boolean_type,string_type,pen_type,path_type,picture_type: if cur_type=t+unknown_tag then begin nonlinear_eq(v,cur_exp,false); goto done; end else if cur_type=t then @; unknown_types:if cur_type=t-unknown_tag then begin nonlinear_eq(cur_exp,lhs,true); goto done; end else if cur_type=t then begin ring_merge(lhs,cur_exp); goto done; end else if cur_type=pair_type then if t=unknown_path then begin pair_to_path; goto restart; end; transform_type,pair_type:if cur_type=t then @; known,dependent,proto_dependent,independent:if cur_type>=known then begin try_eq(lhs,null); goto done; end; vacuous:do_nothing; @ @= begin if cur_type<=string_type then begin if cur_type=string_type then begin if str_vs_str(v,cur_exp)<>0 then goto not_found; end else if v<>cur_exp then goto not_found; @; goto done; end; print_err("Redundant or inconsistent equation"); @.Redundant or inconsistent equation@> help2("An equation between already-known quantities can't help.")@/ ("But don't worry; continue and I'll just ignore it."); put_get_error; goto done; not_found: print_err("Inconsistent equation"); @.Inconsistent equation@> help2("The equation I just read contradicts what was said before.")@/ ("But don't worry; continue and I'll just ignore it."); put_get_error; goto done; end @ @= begin p:=v+big_node_size[t]; q:=value(cur_exp)+big_node_size[t]; repeat p:=p-2; q:=q-2; try_eq(p,q); until p=v; goto done; end @ The first argument to |try_eq| is the location of a value node in a capsule that will soon be recycled. The second argument is either a location within a pair or transform node pointed to by |cur_exp|, or it is |null| (which means that |cur_exp| itself serves as the second argument). The idea is to leave |cur_exp| unchanged, but to equate the two operands. @= procedure try_eq(@!l,@!r:pointer); label done,done1; var @!p:pointer; {dependency list for right operand minus left operand} @!t:known..independent; {the type of list |p|} @!q:pointer; {the constant term of |p| is here} @!pp:pointer; {dependency list for right operand} @!tt:dependent..independent; {the type of list |pp|} @!copied:boolean; {have we copied a list that ought to be recycled?} begin @; @; if info(p)=null then @ else begin linear_eq(p,t); if r=null then if cur_type<>known then if type(cur_exp)=known then begin pp:=cur_exp; cur_exp:=value(cur_exp); cur_type:=known; free_node(pp,value_node_size); end; end; end; @ @= t:=type(l); if t=known then begin t:=dependent; p:=const_dependency(-value(l)); q:=p; end else if t=independent then begin t:=dependent; p:=single_dependency(l); negate(value(p)); q:=dep_final; end else begin p:=dep_list(l); q:=p; loop@+ begin negate(value(q)); if info(q)=null then goto done; q:=link(q); end; done: link(prev_dep(l)):=link(q); prev_dep(link(q)):=prev_dep(l); type(l):=known; end @ @= begin if abs(value(p))>64 then {off by .001 or more} begin print_err("Inconsistent equation");@/ @.Inconsistent equation@> print(" (off by "); print_scaled(value(p)); print_char(")"); help2("The equation I just read contradicts what was said before.")@/ ("But don't worry; continue and I'll just ignore it."); put_get_error; end else if r=null then @; free_node(p,dep_node_size); end @ @= if r=null then if cur_type=known then begin value(q):=value(q)+cur_exp; goto done1; end else begin tt:=cur_type; if tt=independent then pp:=single_dependency(cur_exp) else pp:=dep_list(cur_exp); end else if type(r)=known then begin value(q):=value(q)+value(r); goto done1; end else begin tt:=type(r); if tt=independent then pp:=single_dependency(r) else pp:=dep_list(r); end; if tt<>independent then copied:=false else begin copied:=true; tt:=dependent; end; @; if copied then flush_node_list(pp); done1: @ @= watch_coefs:=false; if t=tt then p:=p_plus_q(p,pp,t) else if t=proto_dependent then p:=p_plus_fq(p,unity,pp,proto_dependent,dependent) else begin q:=p; while info(q)<>null do begin value(q):=round_fraction(value(q)); q:=link(q); end; t:=proto_dependent; p:=p_plus_q(p,pp,t); end; watch_coefs:=true; @ Our next goal is to process type declarations. For this purpose it's convenient to have a procedure that scans a $\langle\,$declared variable$\,\rangle$ and returns the corresponding token list. After the following procedure has acted, the token after the declared variable will have been scanned, so it will appear in |cur_cmd|, |cur_mod|, and~|cur_sym|. @= function scan_declared_variable:pointer; label done; var @!x:pointer; {hash address of the variable's root} @!h,@!t:pointer; {head and tail of the token list to be returned} @!l:pointer; {hash address of left bracket} begin get_symbol; x:=cur_sym; if cur_cmd<>tag_token then clear_symbol(x,false); h:=get_avail; info(h):=x; t:=h;@/ loop@+ begin get_x_next; if cur_sym=0 then goto done; if cur_cmd<>tag_token then if cur_cmd<>internal_quantity then if cur_cmd=left_bracket then @ else goto done; link(t):=get_avail; t:=link(t); info(t):=cur_sym; end; done: if eq_type(x)<>tag_token then clear_symbol(x,false); if equiv(x)=null then new_root(x); scan_declared_variable:=h; end; @ If the subscript isn't collective, we don't accept it as part of the declared variable. @= begin l:=cur_sym; get_x_next; if cur_cmd<>right_bracket then begin back_input; cur_sym:=l; cur_cmd:=left_bracket; goto done; end else cur_sym:=collective_subscript; end @ Type declarations are introduced by the following primitive operations. @= primitive("numeric",type_name,numeric_type);@/ @!@:numeric_}{\&{numeric} primitive@> primitive("string",type_name,string_type);@/ @!@:string_}{\&{string} primitive@> primitive("boolean",type_name,boolean_type);@/ @!@:boolean_}{\&{boolean} primitive@> primitive("path",type_name,path_type);@/ @!@:path_}{\&{path} primitive@> primitive("pen",type_name,pen_type);@/ @!@:pen_}{\&{pen} primitive@> primitive("picture",type_name,picture_type);@/ @!@:picture_}{\&{picture} primitive@> primitive("transform",type_name,transform_type);@/ @!@:transform_}{\&{transform} primitive@> primitive("pair",type_name,pair_type);@/ @!@:pair_}{\&{pair} primitive@> @ @= type_name: print_type(m); @ Now we are ready to handle type declarations, assuming that a |type_name| has just been scanned. @= procedure do_type_declaration; var @!t:small_number; {the type being declared} @!p:pointer; {token list for a declared variable} @!q:pointer; {value node for the variable} begin if cur_mod>=transform_type then t:=cur_mod@+else t:=cur_mod+unknown_tag; repeat p:=scan_declared_variable; flush_variable(equiv(info(p)),link(p),false);@/ q:=find_variable(p); if q<>null then begin type(q):=t; value(q):=null; end else begin print_err("Declared variable conflicts with previous vardef"); @.Declared variable conflicts...@> help2("You can't use, e.g., `numeric foo[]' after `vardef foo'.")@/ ("Proceed, and I'll ignore the illegal redeclaration."); put_get_error; end; flush_list(p); if cur_cmd; until end_of_statement; end; @ @= begin print_err("Illegal suffix of declared variable will be flushed"); @.Illegal suffix...flushed@> help5("Variables in declarations must consist entirely of")@/ ("names and collective subscripts, e.g., `x[]a'.")@/ ("Are you trying to use a reserved word in a variable name?")@/ ("I'm going to discard the junk I found here,")@/ ("up to the next comma or the end of the declaration."); if cur_cmd=numeric_token then help_line[2]:="Explicit subscripts like `x15a' aren't permitted."; put_get_error; scanner_status:=flushing; repeat get_next; @; until cur_cmd>=comma; {either |end_of_statement| or |cur_cmd=comma|} scanner_status:=normal; end @ \MF's |main_control| procedure just calls |do_statement| repeatedly until coming to the end. Each execution of |do_statement| concludes with |cur_cmd=semicolon|, |end_group|, or |stop|. @p procedure main_control; begin repeat do_statement; if cur_cmd=end_group then begin print_err("Extra `endgroup'"); @.Extra `endgroup'@> help2("I'm not currently working on a `begingroup',")@/ ("so I had better not try to end anything."); flush_error(0); end; until cur_cmd=stop; end; @ @= primitive("end",stop,0);@/ @!@:end_}{\&{end} primitive@> primitive("dump",stop,1);@/ @!@:dump_}{\&{dump} primitive@> @ @= stop:if m=0 then print("end")@+else print("dump"); @* \[44] Commands. Let's turn now to statements that are classified as ``commands'' because of their imperative nature. We'll begin with simple ones, so that it will be clear how to hook command processing into the |do_statement| routine; then we'll tackle the tougher commands. Here's one of the simplest: @= random_seed: do_random_seed; @ @= procedure do_random_seed; begin get_x_next; if cur_cmd<>assignment then begin missing_err(":="); @.Missing `:='@> help1("Always say `randomseed:='."); back_error; end; get_x_next; scan_expression; if cur_type<>known then begin exp_err("Unknown value will be ignored"); @.Unknown value...ignored@> help2("Your expression was too random for me to handle,")@/ ("so I won't change the random seed just now.");@/ put_get_flush_error(0); end else @; end; @ @= begin init_randoms(cur_exp); if selector>=log_only then begin old_setting:=selector; selector:=log_only; print_nl("{randomseed:="); print_scaled(cur_exp); print_char("}"); print_nl(""); selector:=old_setting; end; end @ And here's another simple one (somewhat different in flavor): @= mode_command: begin print_ln; interaction:=cur_mod; @; if log_opened then selector:=selector+2; get_x_next; end; @ @= primitive("batchmode",mode_command,batch_mode); @!@:batch_mode_}{\&{batchmode} primitive@> primitive("nonstopmode",mode_command,nonstop_mode); @!@:nonstop_mode_}{\&{nonstopmode} primitive@> primitive("scrollmode",mode_command,scroll_mode); @!@:scroll_mode_}{\&{scrollmode} primitive@> primitive("errorstopmode",mode_command,error_stop_mode); @!@:error_stop_mode_}{\&{errorstopmode} primitive@> @ @= mode_command: case m of batch_mode: print("batchmode"); nonstop_mode: print("nonstopmode"); scroll_mode: print("scrollmode"); othercases print("errorstopmode") endcases; @ The `\&{inner}' and `\&{outer}' commands are only slightly harder. @= protection_command: do_protection; @ @= primitive("inner",protection_command,0);@/ @!@:inner_}{\&{inner} primitive@> primitive("outer",protection_command,1);@/ @!@:outer_}{\&{outer} primitive@> @ @= protection_command: if m=0 then print("inner")@+else print("outer"); @ @= procedure do_protection; var @!m:0..1; {0 to unprotect, 1 to protect} @!t:halfword; {the |eq_type| before we change it} begin m:=cur_mod; repeat get_symbol; t:=eq_type(cur_sym); if m=0 then begin if t>=outer_tag then eq_type(cur_sym):=t-outer_tag; end else if tcomma; end; @ \MF\ never defines the tokens `\.(' and `\.)' to be primitives, but plain \MF\ begins with the declaration `\&{delimiters} \.{()}'. Such a declaration assigns the command code |left_delimiter| to `\.{(}' and |right_delimiter| to `\.{)}'; the |equiv| of each delimiter is the hash address of its mate. @= delimiters: def_delims; @ @= procedure def_delims; var l_delim,r_delim:pointer; {the new delimiter pair} begin get_clear_symbol; l_delim:=cur_sym;@/ get_clear_symbol; r_delim:=cur_sym;@/ eq_type(l_delim):=left_delimiter; equiv(l_delim):=r_delim;@/ eq_type(r_delim):=right_delimiter; equiv(r_delim):=l_delim;@/ get_x_next; end; @ Here is a procedure that is called when \MF\ has reached a point where some right delimiter is mandatory. @= procedure check_delimiter(@!l_delim,@!r_delim:pointer); label exit; begin if cur_cmd=right_delimiter then if cur_mod=l_delim then return; if cur_sym<>r_delim then begin missing_err(text(r_delim));@/ @.Missing `)'@> help2("I found no right delimiter to match a left one. So I've")@/ ("put one in, behind the scenes; this may fix the problem."); back_error; end else begin print_err("The token `"); print(text(r_delim)); @.The token...delimiter@> print("' is no longer a right delimiter"); help3("Strange: This token has lost its former meaning!")@/ ("I'll read it as a right delimiter this time;")@/ ("but watch out, I'll probably miss it later."); error; end; exit:end; @ The next four commands save or change the values associated with tokens. @= save_command: repeat get_symbol; save_variable(cur_sym); get_x_next; until cur_cmd<>comma; interim_command: do_interim; let_command: do_let; new_internal: do_new_internal; @ @= procedure@?do_statement; forward;@t\2@>@/ procedure do_interim; begin get_x_next; if cur_cmd<>internal_quantity then begin print_err("The token `"); @.The token...quantity@> if cur_sym=0 then print("(%CAPSULE)") else print(text(cur_sym)); print("' isn't an internal quantity"); help1("Something like `tracingonline' should follow `interim'."); back_error; end else begin save_internal(cur_mod); back_input; end; do_statement; end; @ The following procedure is careful not to undefine the left-hand symbol too soon, lest commands like `{\tt let x=x}' have a surprising effect. @= procedure do_let; var @!l:pointer; {hash location of the left-hand symbol} begin get_symbol; l:=cur_sym; get_x_next; if cur_cmd<>equals then if cur_cmd<>assignment then begin missing_err("="); @.Missing `='@> help3("You should have said `let symbol = something'.")@/ ("But don't worry; I'll pretend that an equals sign")@/ ("was present. The next token I read will be `something'."); back_error; end; get_symbol; case cur_cmd of defined_macro,secondary_primary_macro,tertiary_secondary_macro, expression_tertiary_macro: add_mac_ref(cur_mod); othercases do_nothing endcases;@/ clear_symbol(l,false); eq_type(l):=cur_cmd; if cur_cmd=tag_token then equiv(l):=null else equiv(l):=cur_mod; get_x_next; end; @ @= procedure do_new_internal; begin repeat if int_ptr=max_internal then overflow("number of internals",max_internal); @:METAFONT capacity exceeded number of int}{\quad number of internals@> get_clear_symbol; incr(int_ptr); eq_type(cur_sym):=internal_quantity; equiv(cur_sym):=int_ptr; int_name[int_ptr]:=text(cur_sym); internal[int_ptr]:=0; get_x_next; until cur_cmd<>comma; end; @ The various `\&{show}' commands are distinguished by modifier fields in the usual way. @d show_token_code=0 {show the meaning of a single token} @d show_stats_code=1 {show current memory and string usage} @d show_code=2 {show a list of expressions} @d show_var_code=3 {show a variable and its descendents} @d show_dependencies_code=4 {show dependent variables in terms of independents} @= primitive("showtoken",show_command,show_token_code);@/ @!@:show_token_}{\&{showtoken} primitive@> primitive("showstats",show_command,show_stats_code);@/ @!@:show_stats_}{\&{showstats} primitive@> primitive("show",show_command,show_code);@/ @!@:show_}{\&{show} primitive@> primitive("showvariable",show_command,show_var_code);@/ @!@:show_var_}{\&{showvariable} primitive@> primitive("showdependencies",show_command,show_dependencies_code);@/ @!@:show_dependencies_}{\&{showdependencies} primitive@> @ @= show_command: case m of show_token_code:print("showtoken"); show_stats_code:print("showstats"); show_code:print("show"); show_var_code:print("showvariable"); othercases print("showdependencies") endcases; @ @= show_command:do_show_whatever; @ The value of |cur_mod| controls the |verbosity| in the |print_exp| routine: if it's |show_code|, complicated structures are abbreviated, otherwise they aren't. @= procedure do_show; begin repeat get_x_next; scan_expression; print_nl(">> "); @.>>@> print_exp(null,2); flush_cur_exp(0); until cur_cmd<>comma; end; @ @= procedure disp_token; begin print_nl("> "); @.>\relax@> if cur_sym=0 then @ else begin print(text(cur_sym)); print_char("="); if eq_type(cur_sym)>=outer_tag then print("(outer) "); print_cmd_mod(cur_cmd,cur_mod); if cur_cmd=defined_macro then begin print_ln; show_macro(cur_mod,null,100000); end; {this avoids recursion between |show_macro| and |print_cmd_mod|} @^recursion@> end; end; @ @= begin if cur_cmd=numeric_token then print_scaled(cur_mod) else if cur_cmd=capsule_token then begin g_pointer:=cur_mod; print_capsule; end else begin print_char(""""); print(cur_mod); print_char(""""); delete_str_ref(cur_mod); end; end @ The following cases of |print_cmd_mod| might arise in connection with |disp_token|, although they don't correspond to any primitive tokens. @= left_delimiter,right_delimiter: begin if c=left_delimiter then print("lef") else print("righ"); print("t delimiter that matches "); print(text(m)); end; tag_token:if m=null then print("tag")@+else print("variable"); defined_macro: print("macro:"); secondary_primary_macro,tertiary_secondary_macro,expression_tertiary_macro: begin print_cmd_mod(macro_def,c); print("'d macro:"); print_ln; show_token_list(link(link(m)),null,1000,0); end; repeat_loop:print("[repeat the loop]"); internal_quantity:print(int_name[m]); @ @= procedure do_show_token; begin repeat get_next; disp_token; get_x_next; until cur_cmd<>comma; end; @ @= procedure do_show_stats; begin print_nl("Memory usage "); @.Memory usage...@> @!stat print_int(var_used); print_char("&"); print_int(dyn_used); if false then@+tats@t@>@;@/ print("unknown"); print(" ("); print_int(hi_mem_min-lo_mem_max-1); print(" still untouched)"); print_ln; print_nl("String usage "); print_int(str_ptr-init_str_ptr); print_char("&"); print_int(pool_ptr-init_pool_ptr); print(" ("); print_int(max_strings-max_str_ptr); print_char("&"); print_int(pool_size-max_pool_ptr); print(" still untouched)"); print_ln; get_x_next; end; @ Here's a recursive procedure that gives an abbreviated account of a variable, for use by |do_show_var|. @= procedure disp_var(@!p:pointer); var @!q:pointer; {traverses attributes and subscripts} @!n:0..max_print_line; {amount of macro text to show} begin if type(p)=structured then @ else if type(p)>=unsuffixed_macro then @ else if type(p)<>undefined then begin print_nl(""); print_variable_name(p); print_char("="); print_exp(p,0); end; end; @ @= begin q:=attr_head(p); repeat disp_var(q); q:=link(q); until q=end_attr; q:=subscr_head(p); while name_type(q)=subscr do begin disp_var(q); q:=link(q); end; end @ @= begin print_nl(""); print_variable_name(p); if type(p)>unsuffixed_macro then print("@@#"); {|suffixed_macro|} print("=macro:"); if file_offset>=max_print_line-20 then n:=5 else n:=max_print_line-file_offset-15; show_macro(value(p),null,n); end @ @= procedure do_show_var; label done; begin repeat get_next; if cur_sym>0 then if cur_sym<=hash_end then if cur_cmd=tag_token then if cur_mod<>null then begin disp_var(cur_mod); goto done; end; disp_token; done:get_x_next; until cur_cmd<>comma; end; @ @= procedure do_show_dependencies; var @!p:pointer; {link that runs through all dependencies} begin p:=link(dep_head); while p<>dep_head do begin if interesting(p) then begin print_nl(""); print_variable_name(p); if type(p)=dependent then print_char("=") else print(" = "); {extra spaces imply proto-dependency} print_dependency(dep_list(p),type(p)); end; p:=dep_list(p); while info(p)<>null do p:=link(p); p:=link(p); end; get_x_next; end; @ Finally we are ready for the procedure that governs all of the show commands. @= procedure do_show_whatever; begin if interaction=error_stop_mode then wake_up_terminal; case cur_mod of show_token_code:do_show_token; show_stats_code:do_show_stats; show_code:do_show; show_var_code:do_show_var; show_dependencies_code:do_show_dependencies; end; {there are no other cases} if internal[showstopping]>0 then begin print_err("OK"); @.OK@> if interaction= primitive("contour",thing_to_add,contour_code);@/ @!@:contour_}{\&{contour} primitive@> primitive("doublepath",thing_to_add,double_path_code);@/ @!@:double_path_}{\&{doublepath} primitive@> primitive("also",thing_to_add,also_code);@/ @!@:also_}{\&{also} primitive@> primitive("withpen",with_option,pen_type);@/ @!@:with_pen_}{\&{withpen} primitive@> primitive("withweight",with_option,known);@/ @!@:with_weight_}{\&{withweight} primitive@> primitive("dropping",cull_op,drop_code);@/ @!@:dropping_}{\&{dropping} primitive@> primitive("keeping",cull_op,keep_code);@/ @!@:keeping_}{\&{keeping} primitive@> @ @= thing_to_add:if m=contour_code then print("contour") else if m=double_path_code then print("doublepath") else print("also"); with_option:if m=pen_type then print("withpen") else print("withweight"); cull_op:if m=drop_code then print("dropping") else print("keeping"); @ @= function scan_with:boolean; var @!t:small_number; {|known| or |pen_type|} @!result:boolean; {the value to return} begin t:=cur_mod; cur_type:=vacuous; get_x_next; scan_expression; result:=false; if cur_type<>t then @ else if cur_type=pen_type then result:=true else @; scan_with:=result; end; @ @= begin exp_err("Improper type"); @.Improper type@> help2("Next time say `withweight ';")@/ ("I'll ignore the bad `with' clause and look for another."); if t=pen_type then help_line[1]:="Next time say `withpen ';"; put_get_flush_error(0); end @ @= begin cur_exp:=round_unscaled(cur_exp); if (abs(cur_exp)<4)and(cur_exp<>0) then result:=true else begin print_err("Weight must be -3, -2, -1, +1, +2, or +3"); @.Weight must be...@> help1("I'll ignore the bad `with' clause and look for another."); put_get_flush_error(0); end; end @ One of the things we need to do when we've parsed an \&{addto} or similar command is set |cur_edges| to the header of a supposed \&{picture} variable, given a token list for that variable. @= procedure find_edges_var(@!t:pointer); var @!p:pointer; begin p:=find_variable(t); cur_edges:=null; if p=null then begin obliterated(t); put_get_error; end else if type(p)<>picture_type then begin print_err("Variable "); show_token_list(t,null,1000,0); @.Variable x is the wrong type@> print(" is the wrong type ("); print_type(type(p)); print_char(")"); help2("I was looking for a ""known"" picture variable.")@/ ("So I'll not change anything just now."); put_get_error; end else cur_edges:=value(p); flush_node_list(t); end; @ @= add_to_command: do_add_to; @ @= procedure do_add_to; label done, not_found; var @!lhs,@!rhs:pointer; {variable on left, path on right} @!w:integer; {tentative weight} @!p:pointer; {list manipulation register} @!q:pointer; {beginning of second half of doubled path} begin get_x_next; var_flag:=thing_to_add; scan_primary; if cur_type<>token_list then @ else begin lhs:=cur_exp; cur_path_type:=cur_mod;@/ cur_type:=vacuous; get_x_next; scan_expression; if cur_path_type=also_code then @ else @; end; end; @ @= begin exp_err("Not a suitable variable"); @.Not a suitable variable@> help4("At this point I needed to see the name of a picture variable.")@/ ("(Or perhaps you have indeed presented me with one; I might")@/ ("have missed it, if it wasn't followed by the proper token.)")@/ ("So I'll not change anything just now."); put_get_flush_error(0); end @ @= begin find_edges_var(lhs); if cur_edges=null then flush_cur_exp(0) else if cur_type<>picture_type then begin exp_err("Improper `addto'"); @.Improper `addto'@> help2("This expression should have specified a known picture.")@/ ("So I'll not change anything just now."); put_get_flush_error(0); end else begin merge_edges(cur_exp); flush_cur_exp(0); end; end @ @= begin if cur_type=pair_type then pair_to_path; if cur_type<>path_type then begin exp_err("Improper `addto'"); @.Improper `addto'@> help2("This expression should have been a known path.")@/ ("So I'll not change anything just now."); put_get_flush_error(0); flush_token_list(lhs); end else begin rhs:=cur_exp; w:=1; cur_pen:=null_pen; while cur_cmd=with_option do if scan_with then if cur_type=known then w:=cur_exp else @; @; delete_pen_ref(cur_pen); end; end @ We could say `|add_pen_ref(cur_pen)|; |flush_cur_exp(0)|' after changing |cur_pen| here. But that would have no effect, because the current expression will not be flushed. Thus we save a bit of code (at the risk of being too tricky). @= begin delete_pen_ref(cur_pen); cur_pen:=cur_exp; end @ @= find_edges_var(lhs); if cur_edges=null then toss_knot_list(rhs) else begin lhs:=null; if left_type(rhs)=endpoint then if cur_path_type=double_path_code then @ else @ else if cur_path_type=double_path_code then lhs:=htap_ypoc(rhs); cur_wt:=w; rhs:=make_spec(rhs,max_offset(cur_pen),internal[tracing_specs]); @; if max_offset(cur_pen)=0 then fill_spec(rhs) else fill_envelope(rhs); if lhs<>null then begin rev_turns:=true; lhs:=make_spec(lhs,max_offset(cur_pen),internal[tracing_specs]); rev_turns:=false; if max_offset(cur_pen)=0 then fill_spec(lhs) else fill_envelope(lhs); end; not_found: end @ @= if link(rhs)=rhs then @ else begin p:=htap_ypoc(rhs); q:=link(p);@/ right_x(path_tail):=right_x(q); right_y(path_tail):=right_y(q); right_type(path_tail):=right_type(q); link(path_tail):=link(q); free_node(q,knot_node_size);@/ right_x(p):=right_x(rhs); right_y(p):=right_y(rhs); right_type(p):=right_type(rhs); link(p):=link(rhs); free_node(rhs,knot_node_size);@/ rhs:=p; end @ @= begin right_x(rhs):=x_coord(rhs); right_y(rhs):=y_coord(rhs); left_x(rhs):=x_coord(rhs); left_y(rhs):=y_coord(rhs); left_type(rhs):=explicit; right_type(rhs):=explicit; end @ @= begin print_err("Not a cycle"); @.Not a cycle@> help2("That contour should have ended with `..cycle' or `&cycle'.")@/ ("So I'll not change anything just now."); put_get_error; toss_knot_list(rhs); goto not_found; end @ @= if turning_number<=0 then if cur_path_type<>double_path_code then if internal[turning_check]>0 then if (turning_number<0)and(link(cur_pen)=null) then negate(cur_wt) else begin if turning_number=0 then if (internal[turning_check]<=unity)and(link(cur_pen)=null) then goto done else print_strange("Strange path (turning number is zero)") @.Strange path...@> else print_strange("Backwards path (turning number is negative)"); @.Backwards path...@> help3("The path doesn't have a counterclockwise orientation,")@/ ("so I'll probably have trouble drawing it.")@/ ("(See Chapter 27 of The METAFONTbook for more help.)"); @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> put_get_error; end; done: @ @= ship_out_command: do_ship_out; display_command: do_display; open_window: do_open_window; cull_command: do_cull; @ @= @t\4@>@@; procedure do_ship_out; label exit; var @!c:integer; {the character code} begin get_x_next; var_flag:=semicolon; scan_expression; if cur_type<>token_list then if cur_type=picture_type then cur_edges:=cur_exp else begin @; return; end else begin find_edges_var(cur_exp); cur_type:=vacuous; end; if cur_edges<>null then begin c:=round_unscaled(internal[char_code]) mod 256; if c<0 then c:=c+256; @; if internal[proofing]>=0 then ship_out(c); end; flush_cur_exp(0); exit:end; @ @= procedure do_display; label not_found,common_ending,exit; var @!e:pointer; {token list for a picture variable} begin get_x_next; var_flag:=in_window; scan_primary; if cur_type<>token_list then @ else begin e:=cur_exp; cur_type:=vacuous; get_x_next; scan_expression; if cur_type<>known then goto common_ending; cur_exp:=round_unscaled(cur_exp); if cur_exp<0 then goto not_found; if cur_exp>15 then goto not_found; if not window_open[cur_exp] then goto not_found; find_edges_var(e); if cur_edges<>null then disp_edges(cur_exp); return; not_found: cur_exp:=cur_exp*unity; common_ending: exp_err("Bad window number"); @.Bad window number@> help1("It should be the number of an open window."); put_get_flush_error(0); flush_token_list(e); end; exit:end; @ The only thing difficult about `\&{openwindow}' is that the syntax allows the user to go astray in many ways. The following subroutine helps keep the necessary program reasonably short. @= function get_pair(@!c:command_code):boolean; var @!p:pointer; {a pair of values that are known (we hope)} @!b:boolean; {did we find such a pair?} begin if cur_cmd<>c then get_pair:=false else begin get_x_next; scan_expression; if nice_pair(cur_exp,cur_type) then begin p:=value(cur_exp); cur_x:=value(x_part_loc(p)); cur_y:=value(y_part_loc(p)); b:=true; end else b:=false; flush_cur_exp(0); get_pair:=b; end; end; @ @= procedure do_open_window; label not_found,exit; var @!k:integer; {the window number in question} @!r0,@!c0,@!r1,@!c1:scaled; {window coordinates} begin get_x_next; scan_expression; if cur_type<>known then goto not_found; k:=round_unscaled(cur_exp); if k<0 then goto not_found; if k>15 then goto not_found; if not get_pair(from_token) then goto not_found; r0:=cur_x; c0:=cur_y; if not get_pair(to_token) then goto not_found; r1:=cur_x; c1:=cur_y; if not get_pair(at_token) then goto not_found; open_a_window(k,r0,c0,r1,c1,cur_x,cur_y); return; not_found:print_err("Improper `openwindow'"); @.Improper `openwindow'@> help2("Say `openwindow k from (r0,c0) to (r1,c1) at (x,y)',")@/ ("where all quantities are known and k is between 0 and 15."); put_get_error; exit:end; @ @= procedure do_cull; label not_found,exit; var @!e:pointer; {token list for a picture variable} @!keeping:drop_code..keep_code; {modifier of |cull_op|} @!w,@!w_in,@!w_out:integer; {culling weights} begin w:=1; get_x_next; var_flag:=cull_op; scan_primary; if cur_type<>token_list then @ else begin e:=cur_exp; cur_type:=vacuous; keeping:=cur_mod; if not get_pair(cull_op) then goto not_found; while (cur_cmd=with_option)and(cur_mod=known) do if scan_with then w:=cur_exp; @; find_edges_var(e); if cur_edges<>null then cull_edges(floor_unscaled(cur_x+unity-1),floor_unscaled(cur_y),w_out,w_in); return; not_found: print_err("Bad culling amounts"); @.Bad culling amounts@> help1("Always cull by known amounts that exclude 0."); put_get_error; flush_token_list(e); end; exit:end; @ @= if cur_x>cur_y then goto not_found; if keeping=drop_code then begin if (cur_x>0)or(cur_y<0) then goto not_found; w_out:=w; w_in:=0; end else begin if (cur_x<=0)and(cur_y>=0) then goto not_found; w_out:=0; w_in:=w; end @ The \&{everyjob} command simply assigns a nonzero value to the global variable |start_sym|. @= every_job_command: begin get_symbol; start_sym:=cur_sym; get_x_next; end; @ @= @!start_sym:halfword; {a symbolic token to insert at beginning of job} @ @= start_sym:=0; @ Finally, we have only the ``message'' commands remaining. @d message_code=0 @d err_message_code=1 @d err_help_code=2 @= primitive("message",message_command,message_code);@/ @!@:message_}{\&{message} primitive@> primitive("errmessage",message_command,err_message_code);@/ @!@:err_message_}{\&{errmessage} primitive@> primitive("errhelp",message_command,err_help_code);@/ @!@:err_help_}{\&{errhelp} primitive@> @ @= message_command: if m= message_command: do_message; @ @= procedure do_message; var @!m:message_code..err_help_code; {the type of message} begin m:=cur_mod; get_x_next; scan_expression; if cur_type<>string_type then begin exp_err("Not a string"); @.Not a string@> help1("A message should be a known string expression."); put_get_error; end else case m of message_code:begin print_nl(""); slow_print(cur_exp); end; err_message_code:@; err_help_code:@; end; {there are no other cases} flush_cur_exp(0); end; @ The global variable |err_help| is zero when the user has most recently given an empty help string, or if none has ever been given. @= begin if err_help<>0 then delete_str_ref(err_help); if length(cur_exp)=0 then err_help:=0 else begin err_help:=cur_exp; add_str_ref(err_help); end; end @ If \&{errmessage} occurs often in |scroll_mode|, without user-defined \&{errhelp}, we don't want to give a long help message each time. So we give a verbose explanation only once. @= @!long_help_seen:boolean; {has the long \.{\\errmessage} help been used?} @ @=long_help_seen:=false; @ @= begin print_err(""); slow_print(cur_exp); if err_help<>0 then use_err_help:=true else if long_help_seen then help1("(That was another `errmessage'.)") else begin if interaction ("and deduce the truth by inspired guesses."); end; put_get_error; use_err_help:=false; end @* \[45] Font metric data. \TeX\ gets its knowledge about fonts from font metric files, also called \.{TFM} files; the `\.T' in `\.{TFM}' stands for \TeX, but other programs know about them too. One of \MF's duties is to write \.{TFM} files so that the user's fonts can readily be applied to typesetting. @:TFM files}{\.{TFM} files@> @^font metric files@> The information in a \.{TFM} file appears in a sequence of 8-bit bytes. Since the number of bytes is always a multiple of~4, we could also regard the file as a sequence of 32-bit words, but \MF\ uses the byte interpretation. The format of \.{TFM} files was designed by Lyle Ramshaw in 1980. The intent is to convey a lot of different kinds @^Ramshaw, Lyle Harold@> of information in a compact but useful form. @= @!tfm_file:byte_file; {the font metric output goes here} @!metric_file_name: str_number; {full name of the font metric file} @ The first 24 bytes (6 words) of a \.{TFM} file contain twelve 16-bit integers that give the lengths of the various subsequent portions of the file. These twelve integers are, in order: $$\vbox{\halign{\hfil#&$\null=\null$#\hfil\cr |lf|&length of the entire file, in words;\cr |lh|&length of the header data, in words;\cr |bc|&smallest character code in the font;\cr |ec|&largest character code in the font;\cr |nw|&number of words in the width table;\cr |nh|&number of words in the height table;\cr |nd|&number of words in the depth table;\cr |ni|&number of words in the italic correction table;\cr |nl|&number of words in the lig/kern table;\cr |nk|&number of words in the kern table;\cr |ne|&number of words in the extensible character table;\cr |np|&number of font parameter words.\cr}}$$ They are all nonnegative and less than $2^{15}$. We must have |bc-1<=ec<=255|, |ne<=256|, and $$\hbox{|lf=6+lh+(ec-bc+1)+nw+nh+nd+ni+nl+nk+ne+np|.}$$ Note that a font may contain as many as 256 characters (if |bc=0| and |ec=255|), and as few as 0 characters (if |bc=ec+1|). Incidentally, when two or more 8-bit bytes are combined to form an integer of 16 or more bits, the most significant bytes appear first in the file. This is called BigEndian order. @!@^BigEndian order@> @ The rest of the \.{TFM} file may be regarded as a sequence of ten data arrays having the informal specification $$\def\arr$[#1]#2${\&{array} $[#1]$ \&{of} #2} \vbox{\halign{\hfil\\{#}&$\,:\,$\arr#\hfil\cr header&|[0..lh-1]@t\\{stuff}@>|\cr char\_info&|[bc..ec]char_info_word|\cr width&|[0..nw-1]fix_word|\cr height&|[0..nh-1]fix_word|\cr depth&|[0..nd-1]fix_word|\cr italic&|[0..ni-1]fix_word|\cr lig\_kern&|[0..nl-1]lig_kern_command|\cr kern&|[0..nk-1]fix_word|\cr exten&|[0..ne-1]extensible_recipe|\cr param&|[1..np]fix_word|\cr}}$$ The most important data type used here is a |@!fix_word|, which is a 32-bit representation of a binary fraction. A |fix_word| is a signed quantity, with the two's complement of the entire word used to represent negation. Of the 32 bits in a |fix_word|, exactly 12 are to the left of the binary point; thus, the largest |fix_word| value is $2048-2^{-20}$, and the smallest is $-2048$. We will see below, however, that all but two of the |fix_word| values must lie between $-16$ and $+16$. @ The first data array is a block of header information, which contains general facts about the font. The header must contain at least two words, |header[0]| and |header[1]|, whose meaning is explained below. Additional header information of use to other software routines might also be included, and \MF\ will generate it if the \.{headerbyte} command occurs. For example, 16 more words of header information are in use at the Xerox Palo Alto Research Center; the first ten specify the character coding scheme used (e.g., `\.{XEROX TEXT}' or `\.{TEX MATHSY}'), the next five give the font family name (e.g., `\.{HELVETICA}' or `\.{CMSY}'), and the last gives the ``face byte.'' \yskip\hang|header[0]| is a 32-bit check sum that \MF\ will copy into the \.{GF} output file. This helps ensure consistency between files, since \TeX\ records the check sums from the \.{TFM}'s it reads, and these should match the check sums on actual fonts that are used. The actual relation between this check sum and the rest of the \.{TFM} file is not important; the check sum is simply an identification number with the property that incompatible fonts almost always have distinct check sums. @^check sum@> \yskip\hang|header[1]| is a |fix_word| containing the design size of the font, in units of \TeX\ points. This number must be at least 1.0; it is fairly arbitrary, but usually the design size is 10.0 for a ``10 point'' font, i.e., a font that was designed to look best at a 10-point size, whatever that really means. When a \TeX\ user asks for a font `\.{at} $\delta$ \.{pt}', the effect is to override the design size and replace it by $\delta$, and to multiply the $x$ and~$y$ coordinates of the points in the font image by a factor of $\delta$ divided by the design size. {\sl All other dimensions in the\/ \.{TFM} file are |fix_word|\kern-1pt\ numbers in design-size units.} Thus, for example, the value of |param[6]|, which defines the \.{em} unit, is often the |fix_word| value $2^{20}=1.0$, since many fonts have a design size equal to one em. The other dimensions must be less than 16 design-size units in absolute value; thus, |header[1]| and |param[1]| are the only |fix_word| entries in the whole \.{TFM} file whose first byte might be something besides 0 or 255. @ Next comes the |char_info| array, which contains one |@!char_info_word| per character. Each word in this part of the file contains six fields packed into four bytes as follows. \yskip\hang first byte: |@!width_index| (8 bits)\par \hang second byte: |@!height_index| (4 bits) times 16, plus |@!depth_index| (4~bits)\par \hang third byte: |@!italic_index| (6 bits) times 4, plus |@!tag| (2~bits)\par \hang fourth byte: |@!remainder| (8 bits)\par \yskip\noindent The actual width of a character is \\{width}|[width_index]|, in design-size units; this is a device for compressing information, since many characters have the same width. Since it is quite common for many characters to have the same height, depth, or italic correction, the \.{TFM} format imposes a limit of 16 different heights, 16 different depths, and 64 different italic corrections. Incidentally, the relation $\\{width}[0]=\\{height}[0]=\\{depth}[0]= \\{italic}[0]=0$ should always hold, so that an index of zero implies a value of zero. The |width_index| should never be zero unless the character does not exist in the font, since a character is valid if and only if it lies between |bc| and |ec| and has a nonzero |width_index|. @ The |tag| field in a |char_info_word| has four values that explain how to interpret the |remainder| field. \yskip\hang|tag=0| (|no_tag|) means that |remainder| is unused.\par \hang|tag=1| (|lig_tag|) means that this character has a ligature/kerning program starting at |lig_kern[remainder]|.\par \hang|tag=2| (|list_tag|) means that this character is part of a chain of characters of ascending sizes, and not the largest in the chain. The |remainder| field gives the character code of the next larger character.\par \hang|tag=3| (|ext_tag|) means that this character code represents an extensible character, i.e., a character that is built up of smaller pieces so that it can be made arbitrarily large. The pieces are specified in |@!exten[remainder]|.\par \yskip\noindent Characters with |tag=2| and |tag=3| are treated as characters with |tag=0| unless they are used in special circumstances in math formulas. For example, \TeX's \.{\\sum} operation looks for a |list_tag|, and the \.{\\left} operation looks for both |list_tag| and |ext_tag|. @d no_tag=0 {vanilla character} @d lig_tag=1 {character has a ligature/kerning program} @d list_tag=2 {character has a successor in a charlist} @d ext_tag=3 {character is extensible} @ The |lig_kern| array contains instructions in a simple programming language that explains what to do for special letter pairs. Each word in this array is a |@!lig_kern_command| of four bytes. \yskip\hang first byte: |stop_bit|, indicates the final program step if the byte is 128 or more.\par \hang second byte: |next_char|, ``if |next_char| follows the current character, then perform the operation and stop, otherwise continue.''\par \hang third byte: |op_bit|, indicates a ligature step if less than~128, a kern step otherwise.\par \hang fourth byte: |remainder|.\par \yskip\noindent In a ligature step the current character and |next_char| are replaced by the single character whose code is |remainder|. In a kern step, an additional space equal to |@!kern[remainder]| is inserted between the current character and |next_char|. (The value of |kern[remainder]| is often negative, so that the characters are brought closer together by kerning; but it might be positive.) @d stop_flag=128+min_quarterword {value indicating `\.{STOP}' in a lig/kern program} @d kern_flag=128+min_quarterword {op code for a kern step} @d stop_bit(#)==lig_kern[#].b0 @d next_char(#)==lig_kern[#].b1 @d op_bit(#)==lig_kern[#].b2 @d rem_byte(#)==lig_kern[#].b3 @ Extensible characters are specified by an |@!extensible_recipe|, which consists of four bytes called |@!top|, |@!mid|, |@!bot|, and |@!rep| (in this order). These bytes are the character codes of individual pieces used to build up a large symbol. If |top|, |mid|, or |bot| are zero, they are not present in the built-up result. For example, an extensible vertical line is like an extensible bracket, except that the top and bottom pieces are missing. Let $T$, $M$, $B$, and $R$ denote the respective pieces, or an empty box if the piece isn't present. Then the extensible characters have the form $TR^kMR^kB$ from top to bottom, for some |k>=0|, unless $M$ is absent; in the latter case we can have $TR^kB$ for both even and odd values of~|k|. The width of the extensible character is the width of $R$; and the height-plus-depth is the sum of the individual height-plus-depths of the components used, since the pieces are butted together in a vertical list. @d ext_top(#)==exten[#].b0 {|top| piece in a recipe} @d ext_mid(#)==exten[#].b1 {|mid| piece in a recipe} @d ext_bot(#)==exten[#].b2 {|bot| piece in a recipe} @d ext_rep(#)==exten[#].b3 {|rep| piece in a recipe} @ The final portion of a \.{TFM} file is the |param| array, which is another sequence of |fix_word| values. \yskip\hang|param[1]=slant| is the amount of italic slant, which is used to help position accents. For example, |slant=.25| means that when you go up one unit, you also go .25 units to the right. The |slant| is a pure number; it is the only |fix_word| other than the design size itself that is not scaled by the design size. \hang|param[2]=space| is the normal spacing between words in text. Note that character @'40 in the font need not have anything to do with blank spaces. \hang|param[3]=space_stretch| is the amount of glue stretching between words. \hang|param[4]=space_shrink| is the amount of glue shrinking between words. \hang|param[5]=x_height| is the size of one ex in the font; it is also the height of letters for which accents don't have to be raised or lowered. \hang|param[6]=quad| is the size of one em in the font. \hang|param[7]=extra_space| is the amount added to |param[2]| at the ends of sentences. \yskip\noindent If fewer than seven parameters are present, \TeX\ sets the missing parameters to zero. @d slant_code=1 @d space_code=2 @d space_stretch_code=3 @d space_shrink_code=4 @d x_height_code=5 @d quad_code=6 @d extra_space_code=7 @ So that is what \.{TFM} files hold. One of \MF's duties is to output such information, and it does this all at once at the end of a job. In order to prepare for such frenetic activity, it squirrels away the necessary facts in various arrays as they become available. Character dimensions (\&{charwd}, \&{charht}, \&{chardp}, and \&{charic}) are stored respectively in |tfm_width|, |tfm_height|, |tfm_depth|, and |tfm_ital_corr|. Other information about a character (e.g., about its ligatures or successors) is accessible via the |char_tag| and |char_remainder| arrays. Other information about the font as a whole is kept in additional arrays called |header_byte|, |lig_kern|, |kern|, |exten|, and |param|. @= @!bc,@!ec:eight_bits; {smallest and largest character codes shipped out} @!tfm_width:array[eight_bits] of scaled; {\&{charwd} values} @!tfm_height:array[eight_bits] of scaled; {\&{charht} values} @!tfm_depth:array[eight_bits] of scaled; {\&{chardp} values} @!tfm_ital_corr:array[eight_bits] of scaled; {\&{charic} values} @!char_exists:array[eight_bits] of boolean; {has this code been shipped out?} @!char_tag:array[eight_bits] of no_tag..ext_tag; {|remainder| category} @!char_remainder:array[eight_bits] of eight_bits; {the |remainder| byte} @!header_byte:array[1..header_size] of -1..255; {bytes of the \.{TFM} header, or $-1$ if unset} @!lig_kern:array[0..lig_table_size] of four_quarters; {the ligature/kern table} @!nl:0..lig_table_size; {the number of ligature/kern steps so far} @!kern:array[eight_bits] of scaled; {distinct kerning amounts} @!nk:0..256; {the number of distinct kerns so far} @!exten:array[eight_bits] of four_quarters; {extensible character recipes} @!ne:0..256; {the number of extensible characters so far} @!param:array[1..max_font_dimen] of scaled; {\&{fontinfo} parameters} @!np:0..max_font_dimen; {the largest \&{fontinfo} parameter specified so far} @!nw,@!nh,@!nd,@!ni:0..256; {sizes of \.{TFM} subtables} @ @= for k:=0 to 255 do begin tfm_width[k]:=0; tfm_height[k]:=0; tfm_depth[k]:=0; tfm_ital_corr[k]:=0; char_exists[k]:=false; char_tag[k]:=no_tag; char_remainder[k]:=0; end; for k:=1 to header_size do header_byte[k]:=-1; bc:=255; ec:=0; nl:=0; nk:=0; ne:=0; np:=0; @ @= function tfm_check(@!m:small_number):scaled; begin if abs(internal[m])>=fraction_half then begin print_err("Enormous "); print(int_name[m]); @.Enormous charwd...@> @.Enormous chardp...@> @.Enormous charht...@> @.Enormous charic...@> @.Enormous designsize...@> print(" has been reduced"); help1("Font metric dimensions must be less than 2048pt."); put_get_error; if internal[m]>0 then tfm_check:=fraction_half-1 else tfm_check:=1-fraction_half; end else tfm_check:=internal[m]; end; @ @= if cec then ec:=c; char_exists[c]:=true; gf_dx[c]:=internal[char_dx]; gf_dy[c]:=internal[char_dy]; tfm_width[c]:=tfm_check(char_wd); tfm_height[c]:=tfm_check(char_ht); tfm_depth[c]:=tfm_check(char_dp); tfm_ital_corr[c]:=tfm_check(char_ic) @ Now let's consider \MF's special \.{TFM}-oriented commands. @= tfm_command: do_tfm_command; @ @d char_list_code=0 @d lig_table_code=1 @d extensible_code=2 @d header_byte_code=3 @d font_dimen_code=4 @= primitive("charlist",tfm_command,char_list_code);@/ @!@:char_list_}{\&{charlist} primitive@> primitive("ligtable",tfm_command,lig_table_code);@/ @!@:lig_table_}{\&{ligtable} primitive@> primitive("extensible",tfm_command,extensible_code);@/ @!@:extensible_}{\&{extensible} primitive@> primitive("headerbyte",tfm_command,header_byte_code);@/ @!@:header_byte_}{\&{headerbyte} primitive@> primitive("fontdimen",tfm_command,font_dimen_code);@/ @!@:font_dimen_}{\&{fontdimen} primitive@> @ @= tfm_command: case m of char_list_code:print("charlist"); lig_table_code:print("ligtable"); extensible_code:print("extensible"); header_byte_code:print("headerbyte"); othercases print("fontdimen") endcases; @ @= function get_code:eight_bits; {scans a character code value} label found; var @!c:integer; {the code value found} begin get_x_next; scan_expression; if cur_type=known then begin c:=round_unscaled(cur_exp); if c>=0 then if c<256 then goto found; end else if cur_type=string_type then if length(cur_exp)=1 then begin c:=str_pool[str_start[cur_exp]]; goto found; end; exp_err("Invalid code has been replaced by 0"); @.Invalid code...@> help2("I was looking for a number between 0 and 255, or for a")@/ ("string of length 1. Didn't find it; will use 0 instead."); put_get_flush_error(0); c:=0; found: get_code:=c; end; @ @= procedure set_tag(@!c:eight_bits;@!t:small_number;@!r:eight_bits); begin if char_tag[c]=no_tag then begin char_tag[c]:=t; char_remainder[c]:=r; end else @; end; @ @= begin print_err("Character "); if (c>" ")and(c<128) then print(c) else begin print("code "); print_int(c); end; print(" is already "); @.Character c is already...@> case char_tag[c] of lig_tag: print("in a ligtable"); list_tag: print("in a charlist"); ext_tag: print("extensible"); end; {there are no other cases} help2("It's not legal to label a character more than once.")@/ ("So I'll not change anything just now."); put_get_error; end @ @= procedure do_tfm_command; label continue; var @!c,@!cc:eight_bits; {character codes} @!k:0..256; {index into the |kern| array} @!j:integer; {index into |header_byte| or |param|} begin case cur_mod of char_list_code: @; lig_table_code: @; extensible_code: @; header_byte_code, font_dimen_code: begin c:=cur_mod; get_x_next; scan_expression; if (cur_type<>known)or(cur_exp help2("I was looking for a known, positive number.")@/ ("For safety's sake I'll ignore the present command."); put_get_error; end else begin j:=round_unscaled(cur_exp); if cur_cmd<>colon then begin missing_err(":"); @.Missing `:'@> help1("A colon should follow a headerbyte or fontinfo location."); back_error; end; if c=header_byte_code then @ else @; end; end; end; {there are no other cases} end; @ @= begin c:=get_code; while cur_cmd=colon do begin cc:=get_code; set_tag(c,list_tag,cc); c:=cc; end; end @ @= begin continue: c:=get_code; if cur_cmd=colon then @; if cur_cmd=lig_kern_token then begin @; if nl=lig_table_size then overflow("ligtable size",lig_table_size); @:METAFONT capacity exceeded ligtable size}{\quad ligtable size@> incr(nl); if cur_cmd=comma then goto continue; end else begin print_err("Illegal ligtable step"); @.Illegal ligtable step@> help1("I was looking for `=:' or `kern' here."); back_error; end; if nl>0 then stop_bit(nl-1):=stop_flag; end @ @= primitive("=:",lig_kern_token,0); @!@:=:_}{\.{=:} primitive@> primitive("kern",lig_kern_token,kern_flag-min_quarterword); @!@:kern_}{\&{kern} primitive@> @ @= lig_kern_token: if m=0 then print("=:")@+else print("kern"); @ @= begin if nl<256 then set_tag(c,lig_tag,nl) else begin print_err("Ligature program too long"); @.Ligature program too long@> help1("After 256 ligtable steps, no more colons are allowed."); error; end; goto continue; end @ @= next_char(nl):=qi(c); op_bit(nl):=qi(cur_mod); stop_bit(nl):=qi(0); if cur_mod=0 then rem_byte(nl):=qi(get_code) else begin get_x_next; scan_expression; if cur_type<>known then begin exp_err("Improper kern"); @.Improper kern@> help2("The amount of kern should be a known numeric value.")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); put_get_flush_error(0); end; kern[nk]:=cur_exp; k:=0;@+while kern[k]<>cur_exp do incr(k); if k=nk then begin if nk=256 then overflow("kern",256); @:METAFONT capacity exceeded kern}{\quad kern@> incr(nk); end; rem_byte(nl):=qi(k); end @ @d missing_extensible_punctuation(#)== begin missing_err(#); @.Missing `\char`\#'@> help1("I'm processing `extensible c: t,m,b,r'."); back_error; end @= begin if ne=256 then overflow("extensible",256); @:METAFONT capacity exceeded extensible}{\quad extensible@> c:=get_code; set_tag(c,ext_tag,ne); if cur_cmd<>colon then missing_extensible_punctuation(":"); ext_top(ne):=qi(get_code); if cur_cmd<>comma then missing_extensible_punctuation(","); ext_mid(ne):=qi(get_code); if cur_cmd<>comma then missing_extensible_punctuation(","); ext_bot(ne):=qi(get_code); if cur_cmd<>comma then missing_extensible_punctuation(","); ext_rep(ne):=qi(get_code); incr(ne); end @ @= repeat if j>header_size then overflow("headerbyte",header_size); @:METAFONT capacity exceeded headerbyte}{\quad headerbyte@> header_byte[j]:=get_code; incr(j); until cur_cmd<>comma @ @= repeat if j>max_font_dimen then overflow("fontdimen",max_font_dimen); @:METAFONT capacity exceeded fontdimen}{\quad fontdimen@> while j>np do begin incr(np); param[np]:=0; end; get_x_next; scan_expression; if cur_type<>known then begin exp_err("Improper font parameter"); @.Improper font parameter@> help1("I'm zeroing this one. Proceed, with fingers crossed."); put_get_flush_error(0); end; param[j]:=cur_exp; incr(j); until cur_cmd<>comma @ OK: We've stored all the data that is needed for the \.{TFM} file. All that remains is to output it in the correct format. An interesting problem needs to be solved in this connection, because the \.{TFM} format allows at most 256~widths, 16~heights, 16~depths, and 64~italic corrections. If the data has more distinct values than this, we want to meet the necessary restrictions by perturbing the given values as little as possible. \MF\ solves this problem in two steps. First the values of a given kind (widths, heights, depths, or italic corrections) are sorted; then the list of sorted values is perturbed, if necessary. The sorting operation is facilitated by having a special node of essentially infinite |value| at the end of the current list. @= value(inf_val):=fraction_four; @ Straight linear insertion is good enough for sorting, since the lists are usually not terribly long. As we work on the data, the current list will start at |link(temp_head)| and end at |inf_val|; the nodes in this list will be in increasing order of their |value| fields. Given such a list, the |sort_in| function takes a value and returns a pointer to where that value can be found in the list. The value is inserted in the proper place, if necessary. At the time we need to do these operations, most of \MF's work has been completed, so we will have plenty of memory to play with. The value nodes that are allocated for sorting will never be returned to free storage. @d clear_the_list==link(temp_head):=inf_val @p function sort_in(@!v:scaled):pointer; label found; var @!p,@!q,@!r:pointer; {list manipulation registers} begin p:=temp_head; loop@+ begin q:=link(p); if v<=value(q) then goto found; p:=q; end; found: if vd$ such that the covering found by this algorithm would be different. In particular, |min_cover(0)| returns the number of distinct values in the current list and sets |perturbation| to the minimum distance between adjacent values. @p function min_cover(@!d:scaled):integer; var @!p:pointer; {runs through the current list} @!l:scaled; {the least element covered by the current interval} @!m:integer; {lower bound on the size of the minimum cover} begin m:=0; p:=link(temp_head); perturbation:=el_gordo; while p<>inf_val do begin incr(m); l:=value(p); repeat p:=link(p); until value(p)>l+d; if value(p)-l= @!perturbation:scaled; {quantity related to \.{TFM} rounding} @ The smallest |d| such that a given list can be covered with |m| intervals is determined by the |threshold| routine, which is sort of an inverse to |min_cover|. The idea is to increase the interval size rapidly until finding the range, then to go sequentially until the exact borderline has been discovered. @p function threshold(@!m:integer):scaled; var @!d:scaled; {lower bound on the smallest interval size} begin if min_cover(0)<=m then threshold:=0 else begin repeat d:=perturbation; until min_cover(d+d)<=m; while min_cover(d)>m do d:=perturbation; threshold:=d; end; end; @ The |skimp| procedure reduces the current list to at most |m| entries, by changing values if necessary. It also sets |info(p):=k| if |value(p)| is the |k|th distinct value on the resulting list, and it sets |perturbation| to the maximum amount by which a |value| field has been changed. The size of the resulting list is returned as the value of |skimp|. @p function skimp(@!m:integer):integer; var @!d:scaled; {the size of intervals being coalesced} @!p,@!q,@!r:pointer; {list manipulation registers} @!l:scaled; {the least value in the current interval} @!v:scaled; {a compromise value} begin d:=threshold(m); perturbation:=0; q:=temp_head; m:=0; p:=link(temp_head); while p<>inf_val do begin incr(m); l:=value(p); info(p):=m; if value(link(p))<=l+d then @; q:=p; p:=link(p); end; skimp:=m; end; @ @= begin repeat p:=link(p); info(p):=m; until value(link(p))>l+d; v:=half(l+value(p)); if value(p)-v>perturbation then perturbation:=value(p)-v; r:=q; repeat r:=link(r); value(r):=v; until r=p; link(q):=p; {remove duplicate values from the current list} end @ A warning message is issued whenever something is perturbed by more than 1/16\thinspace pt. @p procedure tfm_warning(@!m:small_number); begin print_nl("(some "); print(int_name[m]); @.some charwds...@> @.some chardps...@> @.some charhts...@> @.some charics...@> print(" values had to be adjusted by as much as "); print_scaled(perturbation); print("pt)"); end; @ Here's an example of how we use these routines. The width data needs to be perturbed only if there are 256 distinct widths, but \MF\ must check for this case even though it is highly unusual. An integer variable |k| will be defined when we use this code. The |dimen_head| array will contain pointers to the sorted lists of dimensions. @= clear_the_list; for k:=bc to ec do if char_exists[k] then tfm_width[k]:=sort_in(tfm_width[k]); nw:=skimp(255)+1; dimen_head[1]:=link(temp_head); if perturbation>=@'10000 then tfm_warning(char_wd) @ @= @!dimen_head:array[1..4] of pointer; {lists of \.{TFM} dimensions} @ Heights, depths, and italic corrections are different from widths not only because their list length is more severely restricted, but also because zero values do not need to be put into the lists. @= clear_the_list; for k:=bc to ec do if char_exists[k] then if tfm_height[k]=0 then tfm_height[k]:=zero_val else tfm_height[k]:=sort_in(tfm_height[k]); nh:=skimp(15)+1; dimen_head[2]:=link(temp_head); if perturbation>=@'10000 then tfm_warning(char_ht); clear_the_list; for k:=bc to ec do if char_exists[k] then if tfm_depth[k]=0 then tfm_depth[k]:=zero_val else tfm_depth[k]:=sort_in(tfm_depth[k]); nd:=skimp(15)+1; dimen_head[3]:=link(temp_head); if perturbation>=@'10000 then tfm_warning(char_dp); clear_the_list; for k:=bc to ec do if char_exists[k] then if tfm_ital_corr[k]=0 then tfm_ital_corr[k]:=zero_val else tfm_ital_corr[k]:=sort_in(tfm_ital_corr[k]); ni:=skimp(63)+1; dimen_head[4]:=link(temp_head); if perturbation>=@'10000 then tfm_warning(char_ic) @ @= value(zero_val):=0; info(zero_val):=0; @ Bytes 5--8 of the header are set to the design size, unless the user has some crazy reason for specifying them differently. Error messages are not allowed at the time this procedure is called, so a warning is printed instead. The value of |max_tfm_dimen| is calculated so that $$\hbox{|make_scaled(16*max_tfm_dimen,internal[design_size])|} < \\{three\_bytes}.$$ @d three_bytes==@'100000000 {$2^{24}$} @p procedure fix_design_size; var @!d:scaled; {the design size} begin d:=internal[design_size]; if (d=fraction_half) then begin if d<>0 then print_nl("(illegal design size has been changed to 128pt)"); @.illegal design size...@> d:=@'40000000; internal[design_size]:=d; end; if header_byte[5]<0 then if header_byte[6]<0 then if header_byte[7]<0 then if header_byte[8]<0 then begin header_byte[5]:=d div @'4000000; header_byte[6]:=(d div 4096) mod 256; header_byte[7]:=(d div 16) mod 256; header_byte[8]:=(d mod 16)*16; end; max_tfm_dimen:=16*internal[design_size]-internal[design_size] div @'10000000; if max_tfm_dimen>=fraction_half then max_tfm_dimen:=fraction_half-1; end; @ The |dimen_out| procedure computes a |fix_word| relative to the design size. If the data was out of range, it is corrected and the global variable |tfm_changed| is increased by~one. @p function dimen_out(@!x:scaled):integer; begin if abs(x)>max_tfm_dimen then begin incr(tfm_changed); if x>0 then x:=three_bytes-1@+else x:=1-three_bytes; end else x:=make_scaled(x*16,internal[design_size]); dimen_out:=x; end; @ @= @!max_tfm_dimen:scaled; {bound on widths, heights, kerns, etc.} @!tfm_changed:integer; {the number of data entries that were out of bounds} @ If the user has not specified any of the first four header bytes, the |fix_check_sum| procedure replaces them by a ``check sum'' computed from the |tfm_width| data relative to the design size. @^check sum@> @p procedure fix_check_sum; label exit; var @!k:eight_bits; {runs through character codes} @!b1,@!b2,@!b3,@!b4:eight_bits; {bytes of the check sum} @!x:integer; {hash value used in check sum computation} begin if header_byte[1]<0 then if header_byte[2]<0 then if header_byte[3]<0 then if header_byte[4]<0 then begin @; header_byte[1]:=b1; header_byte[2]:=b2; header_byte[3]:=b3; header_byte[4]:=b4; return; end; for k:=1 to 4 do if header_byte[k]<0 then header_byte[k]:=0; exit:end; @ @= b1:=bc; b2:=ec; b3:=bc; b4:=ec; tfm_changed:=0; for k:=bc to ec do if char_exists[k] then begin x:=dimen_out(value(tfm_width[k]))+(k+4)*@'20000000; {this is positive} b1:=(b1+b1+x) mod 255; b2:=(b2+b2+x) mod 253; b3:=(b3+b3+x) mod 251; b4:=(b4+b4+x) mod 247; end @ Finally we're ready to actually write the \.{TFM} information. Here are some utility routines for this purpose. @d tfm_out(#)==write(tfm_file,#) {output one byte to |tfm_file|} @p procedure tfm_two(@!x:integer); {output two bytes to |tfm_file|} begin tfm_out(x div 256); tfm_out(x mod 256); end; @# procedure tfm_four(@!x:integer); {output four bytes to |tfm_file|} begin if x>=0 then tfm_out(x div three_bytes) else begin x:=x+@'10000000000; {use two's complement for negative values} x:=x+@'10000000000; tfm_out((x div three_bytes) + 128); end; x:=x mod three_bytes; tfm_out(x div unity); x:=x mod unity; tfm_out(x div @'400); tfm_out(x mod @'400); end; @# procedure tfm_qqqq(@!x:four_quarters); {output four quarterwords to |tfm_file|} begin tfm_out(qo(x.b0)); tfm_out(qo(x.b1)); tfm_out(qo(x.b2)); tfm_out(qo(x.b3)); end; @ @= if job_name=0 then open_log_file; pack_job_name(".tfm"); while not b_open_out(tfm_file) do prompt_file_name("file name for font metrics",".tfm"); metric_file_name:=b_make_name_string(tfm_file); @; @; @; @; @; @; @!stat if internal[tracing_stats]>0 then @;@;@+tats@/ print_nl("Font metrics written on "); print(metric_file_name); print_char("."); @.Font metrics written...@> b_close(tfm_file) @ Integer variables |lh| and |k| will be defined when we use this code. @= k:=header_size; while header_byte[k]<0 do decr(k); lh:=(k+3) div 4; {this is the number of header words} if bc>ec then bc:=1; {if there are no characters, |ec=0| and |bc=1|} tfm_two(6+lh+(ec-bc+1)+nw+nh+nd+ni+nl+nk+ne+np); {this is the total number of file words that will be output} tfm_two(lh); tfm_two(bc); tfm_two(ec); tfm_two(nw); tfm_two(nh); tfm_two(nd); tfm_two(ni); tfm_two(nl); tfm_two(nk); tfm_two(ne); tfm_two(np); for k:=1 to 4*lh do begin if header_byte[k]<0 then header_byte[k]:=0; tfm_out(header_byte[k]); end @ @= begin wlog_ln(' '); wlog_ln('(You used ',nw:1,'w,',@| nh:1,'h,',@| nd:1,'d,',@| ni:1,'i,',@| nl:1,'l,',@| nk:1,'k,',@| ne:1,'e,',@| np:1,'p metric file positions'); wlog_ln(' out of ',@| '256w,16h,16d,64i,',@| lig_table_size:1,'l,256k,256e,',@| max_font_dimen:1,'p)'); end @ @= for k:=bc to ec do if not char_exists[k] then tfm_four(0) else begin tfm_out(info(tfm_width[k])); {the width index} tfm_out((info(tfm_height[k]))*16+info(tfm_depth[k])); tfm_out((info(tfm_ital_corr[k]))*4+char_tag[k]); tfm_out(char_remainder[k]); end @ @= tfm_changed:=0; for k:=1 to 4 do begin tfm_four(0); p:=dimen_head[k]; while p<>inf_val do begin tfm_four(dimen_out(value(p))); p:=link(p); end; end @ @= for k:=0 to nl-1 do tfm_qqqq(lig_kern[k]); for k:=0 to nk-1 do tfm_four(dimen_out(kern[k])) @ @= for k:=0 to ne-1 do tfm_qqqq(exten[k]) @ @= for k:=1 to np do if k=1 then if abs(param[1])0 then tfm_four(el_gordo) else tfm_four(-el_gordo); end else tfm_four(dimen_out(param[k])); if tfm_changed>0 then begin if tfm_changed=1 then print_nl("(a font metric dimension") @.a font metric dimension...@> else begin print_nl("("); print_int(tfm_changed); @.font metric dimensions...@> print(" font metric dimensions"); end; print(" had to be decreased)"); end @* \[46] Generic font file format. The most important output produced by a typical run of \MF\ is the ``generic font'' (\.{GF}) file that specifies the bit patterns of the characters that have been drawn. The term {\sl generic\/} indicates that this file format doesn't match the conventions of any name-brand manufacturer; but it is easy to convert \.{GF} files to the special format required by almost all digital phototypesetting equipment. There's a strong analogy between the \.{DVI} files written by \TeX\ and the \.{GF} files written by \MF; and, in fact, the file formats have a lot in common. A \.{GF} file is a stream of 8-bit bytes that may be regarded as a series of commands in a machine-like language. The first byte of each command is the operation code, and this code is followed by zero or more bytes that provide parameters to the command. The parameters themselves may consist of several consecutive bytes; for example, the `|boc|' (beginning of character) command has six parameters, each of which is four bytes long. Parameters are usually regarded as nonnegative integers; but four-byte-long parameters can be either positive or negative, hence they range in value from $-2^{31}$ to $2^{31}-1$. As in \.{TFM} files, numbers that occupy more than one byte position appear in BigEndian order, and negative numbers appear in two's complement notation. A \.{GF} file consists of a ``preamble,'' followed by a sequence of one or more ``characters,'' followed by a ``postamble.'' The preamble is simply a |pre| command, with its parameters that introduce the file; this must come first. Each ``character'' consists of a |boc| command, followed by any number of other commands that specify ``black'' pixels, followed by an |eoc| command. The characters appear in the order that \MF\ generated them. If we ignore no-op commands (which are allowed between any two commands in the file), each |eoc| command is immediately followed by a |boc| command, or by a |post| command; in the latter case, there are no more characters in the file, and the remaining bytes form the postamble. Further details about the postamble will be explained later. Some parameters in \.{GF} commands are ``pointers.'' These are four-byte quantities that give the location number of some other byte in the file; the first file byte is number~0, then comes number~1, and so on. @ The \.{GF} format is intended to be both compact and easily interpreted by a machine. Compactness is achieved by making most of the information relative instead of absolute. When a \.{GF}-reading program reads the commands for a character, it keeps track of two quantities: (a)~the current column number,~|m|; and (b)~the current row number,~|n|. These are 32-bit signed integers, although most actual font formats produced from \.{GF} files will need to curtail this vast range because of practical limitations. (\MF\ output will never allow $\vert m\vert$ or $\vert n\vert$ to get extremely large, but the \.{GF} format tries to be more general.) How do \.{GF}'s row and column numbers correspond to the conventions of \TeX\ and \MF? Well, the ``reference point'' of a character, in \TeX's view, is considered to be at the lower left corner of the pixel in row~0 and column~0. This point is the intersection of the baseline with the left edge of the type; it corresponds to location $(0,0)$ in \MF\ programs. Thus the pixel in \.{GF} row~0 and column~0 is \MF's unit square, comprising the region of the plane whose coordinates both lie between 0 and~1. The pixel in \.{GF} row~|n| and column~|m| consists of the points whose \MF\ coordinates |(x,y)| satisfy |m<=x<=m+1| and |n<=y<=n+1|. Negative values of |m| and~|x| correspond to columns of pixels {\sl left\/} of the reference point; negative values of |n| and~|y| correspond to rows of pixels {\sl below\/} the baseline. Besides |m| and |n|, there's also a third aspect of the current state, namely the @!|paint_switch|, which is always either \\{black} or \\{white}. Each \\{paint} command advances |m| by a specified amount~|d|, and blackens the intervening pixels if |paint_switch=black|; then the |paint_switch| changes to the opposite state. \.{GF}'s commands are designed so that |m| will never decrease within a row, and |n| will never increase within a character; hence there is no way to whiten a pixel that has been blackened. @ Here is a list of all the commands that may appear in a \.{GF} file. Each command is specified by its symbolic name (e.g., |boc|), its opcode byte (e.g., 67), and its parameters (if any). The parameters are followed by a bracketed number telling how many bytes they occupy; for example, `|d[2]|' means that parameter |d| is two bytes long. \yskip\hang|paint_0| 0. This is a \\{paint} command with |d=0|; it does nothing but change the |paint_switch| from \\{black} to \\{white} or vice~versa. \yskip\hang\\{paint\_1} through \\{paint\_63} (opcodes 1 to 63). These are \\{paint} commands with |d=1| to~63, defined as follows: If |paint_switch=black|, blacken |d|~pixels of the current row~|n|, in columns |m| through |m+d-1| inclusive. Then, in any case, complement the |paint_switch| and advance |m| by~|d|. \yskip\hang|paint1| 64 |d[1]|. This is a \\{paint} command with a specified value of~|d|; \MF\ uses it to paint when |64<=d<256|. \yskip\hang|@!paint2| 65 |d[2]|. Same as |paint1|, but |d|~can be as high as~65535. \yskip\hang|@!paint3| 66 |d[3]|. Same as |paint1|, but |d|~can be as high as $2^{24}-1$. \MF\ never needs this command, and it is hard to imagine anybody making practical use of it; surely a more compact encoding will be desirable when characters can be this large. But the command is there, anyway, just in case. \yskip\hang|boc| 67 |c[4]| |p[4]| |min_m[4]| |max_m[4]| |min_n[4]| |max_n[4]|. Beginning of a character: Here |c| is the character code, and |p| points to the previous character beginning (if any) for characters having this code number modulo 256. (The pointer |p| is |-1| if there was no prior character with an equivalent code.) The values of registers |m| and |n| defined by the instructions that follow for this character must satisfy |min_m<=m<=max_m| and |min_n<=n<=max_n|. (The values of |max_m| and |min_n| need not be the tightest bounds possible.) When a \.{GF}-reading program sees a |boc|, it can use |min_m|, |max_m|, |min_n|, and |max_n| to initialize the bounds of an array. Then it sets |m:=min_m|, |n:=max_n|, and |paint_switch:=white|. \yskip\hang|boc1| 68 |c[1]| |@!del_m[1]| |max_m[1]| |@!del_n[1]| |max_n[1]|. Same as |boc|, but |p| is assumed to be~$-1$; also |del_m=max_m-min_m| and |del_n=max_n-min_n| are given instead of |min_m| and |min_n|. The one-byte parameters must be between 0 and 255, inclusive. \ (This abbreviated |boc| saves 19~bytes per character, in common cases.) \yskip\hang|eoc| 69. End of character: All pixels blackened so far constitute the pattern for this character. In particular, a completely blank character might have |eoc| immediately following |boc|. \yskip\hang|skip0| 70. Decrease |n| by 1 and set |m:=min_m|, |paint_switch:=white|. \ (This finishes one row and begins another, ready to whiten the leftmost pixel in the new row.) \yskip\hang|skip1| 71 |d[1]|. Decrease |n| by |d+1|, set |m:=min_m|, and set |paint_switch:=white|. This is a way to produce |d| all-white rows. \yskip\hang|@!skip2| 72 |d[2]|. Same as |skip1|, but |d| can be as large as 65535. \yskip\hang|@!skip3| 73 |d[3]|. Same as |skip1|, but |d| can be as large as $2^{24}-1$. \MF\ obviously never needs this command. \yskip\hang|new_row_0| 74. Decrease |n| by 1 and set |m:=min_m|, |paint_switch:=black|. \ (This finishes one row and begins another, ready to {\sl blacken\/} the leftmost pixel in the new row.) \yskip\hang|@!new_row_1| through |@!new_row_164| (opcodes 75 to 238). Same as |new_row_0|, but with |m:=min_m+1| through |min_m+164|, respectively. \yskip\hang|xxx1| 239 |k[1]| |x[k]|. This command is undefined in general; it functions as a $(k+2)$-byte |no_op| unless special \.{GF}-reading programs are being used. \MF\ generates \\{xxx} commands when encountering a \&{special} string; this occurs in the \.{GF} file only between characters, after the preamble, and before the postamble. However, \\{xxx} commands might appear anywhere in \.{GF} files generated by other processors. It is recommended that |x| be a string having the form of a keyword followed by possible parameters relevant to that keyword. \yskip\hang|@!xxx2| 240 |k[2]| |x[k]|. Like |xxx1|, but |0<=k<65536|. \yskip\hang|xxx3| 241 |k[3]| |x[k]|. Like |xxx1|, but |0<=k<@t$2^{24}$@>|. \MF\ uses this when sending a \&{special} string whose length exceeds~255. \yskip\hang|@!xxx4| 242 |k[4]| |x[k]|. Like |xxx1|, but |k| can be ridiculously large; |k| mustn't be negative. \yskip\hang|yyy| 243 |y[4]|. This command is undefined in general; it functions as a 5-byte |no_op| unless special \.{GF}-reading programs are being used. \MF\ puts |scaled| numbers into |yyy|'s, as a result of \&{numspecial} commands; the intent is to provide numeric parameters to \\{xxx} commands that immediately precede. \yskip\hang|@!no_op| 244. No operation, do nothing. Any number of |no_op|'s may occur between \.{GF} commands, but a |no_op| cannot be inserted between a command and its parameters or between two parameters. \yskip\hang|char_loc| 245 |c[1]| |dx[4]| |dy[4]| |w[4]| |p[4]|. This command will appear only in the postamble, which will be explained shortly. \yskip\hang|@!char_loc0| 246 |c[1]| |@!dm[1]| |w[4]| |p[4]|. Same as |char_loc|, except that |dy| is assumed to be zero, and the value of~|dx| is taken to be |65536*dm|, where |0<=dm<256|. \yskip\hang|pre| 247 |i[1]| |k[1]| |x[k]|. Beginning of the preamble; this must come at the very beginning of the file. Parameter |i| is an identifying number for \.{GF} format, currently 131. The other information is merely commentary; it is not given special interpretation like \\{xxx} commands are. (Note that \\{xxx} commands may immediately follow the preamble, before the first |boc|.) \yskip\hang|post| 248. Beginning of the postamble, see below. \yskip\hang|post_post| 249. Ending of the postamble, see below. \yskip\noindent Commands 250--255 are undefined at the present time. @d gf_id_byte=131 {identifies the kind of \.{GF} files described here} @ Here are the opcodes that \MF\ actually refers to. @d paint_0=0 {beginning of the \\{paint} commands} @d paint1=64 {move right a given number of columns, then black${}\swap{}$white} @d boc=67 {beginning of a character} @d boc1=68 {short form of |boc|} @d eoc=69 {end of a character} @d skip0=70 {skip no blank rows} @d skip1=71 {skip over blank rows} @d new_row_0=74 {move down one row and then right} @d max_new_row=164 {the largest \\{new\_row} command is |new_row_164|} @d xxx1=239 {for \&{special} strings} @d xxx3=241 {for long \&{special} strings} @d yyy=243 {for \&{numspecial} numbers} @d char_loc=245 {character locators in the postamble} @d pre=247 {preamble} @d post=248 {postamble beginning} @d post_post=249 {postamble ending} @ The last character in a \.{GF} file is followed by `|post|'; this command introduces the postamble, which summarizes important facts that \MF\ has accumulated. The postamble has the form $$\vbox{\halign{\hbox{#\hfil}\cr |post| |p[4]| |@!ds[4]| |@!cs[4]| |@!hppp[4]| |@!vppp[4]| |@!min_m[4]| |@!max_m[4]| |@!min_n[4]| |@!max_n[4]|\cr $\langle\,$character locators$\,\rangle$\cr |post_post| |q[4]| |i[1]| 223's$[{\G}4]$\cr}}$$ Here |p| is a pointer to the byte following the final |eoc| in the file (or to the byte following the preamble, if there are no characters); it can be used to locate the beginning of \\{xxx} commands that might have preceded the postamble. The |ds| and |cs| parameters @^design size@> @^check sum@> give the design size and check sum, respectively, which are exactly the values put into the header of the \.{TFM} file that \MF\ produces (or would produce) on this run. Parameters |hppp| and |vppp| are the ratios of pixels per point, horizontally and vertically, expressed as |scaled| integers (i.e., multiplied by $2^{16}$); they can be used to correlate the font with specific device resolutions, magnifications, and ``at sizes.'' Then come |min_m|, |max_m|, |min_n|, and |max_n|, which bound the values that registers |m| and~|n| assume in all characters in this \.{GF} file. (These bounds need not be the best possible; |max_m| and |min_n| may, on the other hand, be tighter than the similar bounds in |boc| commands. For example, some character may have |min_n=-100| in its |boc|, but it might turn out that |n| never gets lower than |-50| in any character; then |min_n| can have any value |<=-50|. If there are no characters in the file, it's possible to have |min_m>max_m| and/or |min_n>max_n|.) @ Character locators are introduced by |char_loc| commands, which specify a character residue~|c|, character escapements (|dx,dy|), a character width~|w|, and a pointer~|p| to the beginning of that character. (If two or more characters have the same code~|c| modulo 256, only the last will be indicated; the others can be located by following backpointers. Characters whose codes differ by a multiple of 256 are assumed to share the same font metric information, hence the \.{TFM} file contains only residues of character codes modulo~256. This convention is intended for oriental languages, when there are many character shapes but few distinct widths.) @^oriental characters@>@^Chinese characters@>@^Japanese characters@> The character escapements (|dx,dy|) are the values of \MF's \&{chardx} and \&{chardy} parameters; they are in units of |scaled| pixels; i.e., |dx| is in horizontal pixel units times $2^{16}$, and |dy| is in vertical pixel units times $2^{16}$. This is the intended amount of displacement after typesetting the character; for \.{DVI} files, |dy| should be zero, but other document file formats allow nonzero vertical escapement. The character width~|w| duplicates the information in the \.{TFM} file; it is a |fix_word| value relative to the design size, and it should be independent of magnification. The backpointer |p| points to the character's |boc|, or to the first of a sequence of consecutive \\{xxx} or |yyy| or |no_op| commands that immediately precede the |boc|, if such commands exist; such ``special'' commands essentially belong to the characters, while the special commands after the final character belong to the postamble (i.e., to the font as a whole). This convention about |p| applies also to the backpointers in |boc| commands, even though it wasn't explained in the description of~|boc|. @^backpointers@> Pointer |p| might be |-1| if the character exists in the \.{TFM} file but not in the \.{GF} file. This unusual situation can arise in \MF\ output if the user had |proofing<0| when the character was being shipped out, but then made |proofing>=0| in order to get a \.{GF} file. @ The last part of the postamble, following the |post_post| byte that signifies the end of the character locators, contains |q|, a pointer to the |post| command that started the postamble. An identification byte, |i|, comes next; this currently equals~131, as in the preamble. The |i| byte is followed by four or more bytes that are all equal to the decimal number 223 (i.e., @'337 in octal). \MF\ puts out four to seven of these trailing bytes, until the total length of the file is a multiple of four bytes, since this works out best on machines that pack four bytes per word; but any number of 223's is allowed, as long as there are at least four of them. In effect, 223 is a sort of signature that is added at the very end. @^Fuchs, David Raymond@> This curious way to finish off a \.{GF} file makes it feasible for \.{GF}-reading programs to find the postamble first, on most computers, even though \MF\ wants to write the postamble last. Most operating systems permit random access to individual words or bytes of a file, so the \.{GF} reader can start at the end and skip backwards over the 223's until finding the identification byte. Then it can back up four bytes, read |q|, and move to byte |q| of the file. This byte should, of course, contain the value 248 (|post|); now the postamble can be read, so the \.{GF} reader can discover all the information needed for individual characters. Unfortunately, however, standard \PASCAL\ does not include the ability to @^system dependencies@> access a random position in a file, or even to determine the length of a file. Almost all systems nowadays provide the necessary capabilities, so \.{GF} format has been designed to work most efficiently with modern operating systems. But if \.{GF} files have to be processed under the restrictions of standard \PASCAL, one can simply read them from front to back. This will be adequate for most applications. However, the postamble-first approach would facilitate a program that merges two \.{GF} files, replacing data from one that is overridden by corresponding data in the other. @* \[47] Shipping characters out. The |ship_out| procedure, to be described below, is given a pointer to an edge structure. Its mission is to describe the the positive pixels in \.{GF} form, outputting a ``character'' to |gf_file|. Several global variables maintain information about the font file as a whole:\ |gf_min_m|, |gf_max_m|, |gf_min_n|, and |gf_max_n| are the minimum and maximum \.{GF} coordinates output so far; |gf_prev_ptr| is the byte number following the preamble or the last |eoc| command in the output; |total_chars| is the total number of characters (i.e., |boc..eoc| segments) shipped out. There's also an array, |char_ptr|, containing the starting positions of each character in the file, as required for the postamble. If character code~|c| has not yet been output, |char_ptr[c]=-1|. @= @!gf_min_m,@!gf_max_m,@!gf_min_n,@!gf_max_n:integer; {bounding rectangle} @!gf_prev_ptr:integer; {where the present/next character started/starts} @!total_chars:integer; {the number of characters output so far} @!char_ptr:array[eight_bits] of integer; {where individual characters started} @!gf_dx,@!gf_dy:array[eight_bits] of integer; {device escapements} @ @= gf_prev_ptr:=0; total_chars:=0; @ The \.{GF} bytes are output to a buffer instead of being sent byte-by-byte to |gf_file|, because this tends to save a lot of subroutine-call overhead. \MF\ uses the same conventions for |gf_file| as \TeX\ uses for its \\{dvi\_file}; hence if system-dependent changes are needed, they should probably be the same for both programs. The output buffer is divided into two parts of equal size; the bytes found in |gf_buf[0..half_buf-1]| constitute the first half, and those in |gf_buf[half_buf..gf_buf_size-1]| constitute the second. The global variable |gf_ptr| points to the position that will receive the next output byte. When |gf_ptr| reaches |gf_limit|, which is always equal to one of the two values |half_buf| or |gf_buf_size|, the half buffer that is about to be invaded next is sent to the output and |gf_limit| is changed to its other value. Thus, there is always at least a half buffer's worth of information present, except at the very beginning of the job. Bytes of the \.{GF} file are numbered sequentially starting with 0; the next byte to be generated will be number |gf_offset+gf_ptr|. @= @!gf_index=0..gf_buf_size; {an index into the output buffer} @ Some systems may find it more efficient to make |gf_buf| a |packed| array, since output of four bytes at once may be facilitated. @^system dependencies@> @= @!gf_buf:array[gf_index] of eight_bits; {buffer for \.{GF} output} @!half_buf:gf_index; {half of |gf_buf_size|} @!gf_limit:gf_index; {end of the current half buffer} @!gf_ptr:gf_index; {the next available buffer address} @!gf_offset:integer; {|gf_buf_size| times the number of times the output buffer has been fully emptied} @ Initially the buffer is all in one piece; we will output half of it only after it first fills up. @= half_buf:=gf_buf_size div 2; gf_limit:=gf_buf_size; gf_ptr:=0; gf_offset:=0; @ The actual output of |gf_buf[a..b]| to |gf_file| is performed by calling |write_gf(a,b)|. It is safe to assume that |a| and |b+1| will both be multiples of 4 when |write_gf(a,b)| is called; therefore it is possible on many machines to use efficient methods to pack four bytes per word and to output an array of words with one system call. @^system dependencies@> @= procedure write_gf(@!a,@!b:gf_index); var k:gf_index; begin for k:=a to b do write(gf_file,gf_buf[k]); end; @ To put a byte in the buffer without paying the cost of invoking a procedure each time, we use the macro |gf_out|. @d gf_out(#)==@+begin gf_buf[gf_ptr]:=#; incr(gf_ptr); if gf_ptr=gf_limit then gf_swap; end @= procedure gf_swap; {outputs half of the buffer} begin if gf_limit=gf_buf_size then begin write_gf(0,half_buf-1); gf_limit:=half_buf; gf_offset:=gf_offset+gf_buf_size; gf_ptr:=0; end else begin write_gf(half_buf,gf_buf_size-1); gf_limit:=gf_buf_size; end; end; @ Here is how we clean out the buffer when \MF\ is all through; |gf_ptr| will be a multiple of~4. @= if gf_limit=half_buf then write_gf(half_buf,gf_buf_size-1); if gf_ptr>0 then write_gf(0,gf_ptr-1) @ The |gf_four| procedure outputs four bytes in two's complement notation, without risking arithmetic overflow. @= procedure gf_four(@!x:integer); begin if x>=0 then gf_out(x div three_bytes) else begin x:=x+@'10000000000; x:=x+@'10000000000; gf_out((x div three_bytes) + 128); end; x:=x mod three_bytes; gf_out(x div unity); x:=x mod unity; gf_out(x div @'400); gf_out(x mod @'400); end; @ Of course, it's even easier to output just two or three bytes. @= procedure gf_two(@!x:integer); begin gf_out(x div @'400); gf_out(x mod @'400); end; @# procedure gf_three(@!x:integer); begin gf_out(x div unity); gf_out((x mod unity) div @'400); gf_out(x mod @'400); end; @ We need a simple routine to generate a \\{paint} command of the appropriate type. @= procedure gf_paint(@!d:integer); {here |0<=d<65536|} begin if d<64 then gf_out(paint_0+d) else if d<256 then begin gf_out(paint1); gf_out(d); end else begin gf_out(paint1+1); gf_two(d); end; end; @ And |gf_string| outputs one or two strings. If the first string number is nonzero, an \\{xxx} command is generated. @= procedure gf_string(@!s,@!t:str_number); var @!k:pool_pointer; @!l:integer; {length of the strings to output} begin if s<>0 then begin l:=length(s); if t<>0 then l:=l+length(t); if l<=255 then begin gf_out(xxx1); gf_out(l); end else begin gf_out(xxx3); gf_three(l); end; for k:=str_start[s] to str_start[s+1]-1 do gf_out(str_pool[k]); end; if t<>0 then for k:=str_start[t] to str_start[t+1]-1 do gf_out(str_pool[k]); end; @ The choice between |boc| commands is handled by |gf_boc|. @d one_byte(#)== #>=0 then if #<256 @= procedure gf_boc(@!min_m,@!max_m,@!min_n,@!max_n:integer); label exit; begin if min_mgf_max_n then gf_max_n:=max_n; if boc_p=-1 then if one_byte(boc_c) then if one_byte(max_m-min_m) then if one_byte(max_m) then if one_byte(max_n-min_n) then if one_byte(max_n) then begin gf_out(boc1); gf_out(boc_c);@/ gf_out(max_m-min_m); gf_out(max_m); gf_out(max_n-min_n); gf_out(max_n); return; end; gf_out(boc); gf_four(boc_c); gf_four(boc_p);@/ gf_four(min_m); gf_four(max_m); gf_four(min_n); gf_four(max_n); exit: end; @ Two of the parameters to |gf_boc| are global. @= @!boc_c,@!boc_p:integer; {parameters of the next |boc| command} @ Here is a routine that gets a \.{GF} file off to a good start. @d check_gf==@t@>@+if output_file_name=0 then init_gf @= procedure init_gf; var @!k:eight_bits; {runs through all possible character codes} @!t:integer; {the time of this run} begin gf_min_m:=4096; gf_max_m:=-4096; gf_min_n:=4096; gf_max_n:=-4096; for k:=0 to 255 do char_ptr[k]:=-1; @; set_output_file_name; gf_out(pre); gf_out(gf_id_byte); {begin to output the preamble} old_setting:=selector; selector:=new_string; print(" METAFONT output "); print_int(round_unscaled(internal[year])); print_char("."); print_dd(round_unscaled(internal[month])); print_char("."); print_dd(round_unscaled(internal[day])); print_char(":");@/ t:=round_unscaled(internal[time]); print_dd(t div 60); print_dd(t mod 60);@/ selector:=old_setting; gf_out(cur_length); str_start[str_ptr+1]:=pool_ptr; gf_string(0,str_ptr); pool_ptr:=str_start[str_ptr]; {flush that string from memory} gf_prev_ptr:=gf_offset+gf_ptr; end; @ @= if internal[hppp]<=0 then gf_ext:=".gf" else begin old_setting:=selector; selector:=new_string; print_char("."); print_int(make_scaled(internal[hppp],59429463)); {$2^{32}/72.27\approx59429463.07$} print("gf"); gf_ext:=make_string; selector:=old_setting; end @ With those preliminaries out of the way, |ship_out| is not especially difficult. @= procedure ship_out(@!c:eight_bits); label done; var @!f:integer; {current character extension} @!prev_m,@!m,@!mm:integer; {previous and current pixel column numbers} @!prev_n,@!n:integer; {previous and current pixel row numbers} @!p,@!q:pointer; {for list traversal} @!prev_w,@!w,@!ww:integer; {old and new weights} @!d:integer; {data from edge-weight node} @!delta:integer; {number of rows to skip} @!cur_min_m:integer; {starting column, relative to the current offset} @!x_off,@!y_off:integer; {offsets, rounded to integers} begin check_gf; f:=round_unscaled(internal[char_ext]);@/ x_off:=round_unscaled(internal[x_offset]); y_off:=round_unscaled(internal[y_offset]); if term_offset>max_print_line-9 then print_ln else if (term_offset>0)or(file_offset>0) then print_char(" "); print_char("["); print_int(c); if f<>0 then begin print_char("."); print_int(f); end; update_terminal; boc_c:=256*f+c; boc_p:=char_ptr[c]; char_ptr[c]:=gf_prev_ptr;@/ if internal[proofing]>0 then @; @; gf_out(eoc); gf_prev_ptr:=gf_offset+gf_ptr; incr(total_chars); print_char("]"); update_terminal; {progress report} if internal[tracing_output]>0 then print_edges(" (just shipped out)",true,x_off,y_off); end; @ @= begin if x_off<>0 then begin gf_string("xoffset",0); gf_out(yyy); gf_four(x_off*unity); end; if y_off<>0 then begin gf_string("yoffset",0); gf_out(yyy); gf_four(y_off*unity); end; end @ @= prev_n:=4096; p:=knil(cur_edges); n:=n_max(cur_edges)-zero_field; while p<>cur_edges do begin @; p:=knil(p); decr(n); end; if prev_n=4096 then @ else if prev_n+y_off= begin gf_boc(0,0,0,0); if gf_max_m<0 then gf_max_m:=0; if gf_min_n>0 then gf_min_n:=0; end @ In this loop, |prev_w| represents the weight at column |prev_m|, which is the most recent column reflected in the output so far; |w| represents the weight at column~|m|, which is the most recent column in the edge data. Several edges might cancel at the same column position, so we need to look ahead to column~|mm| before actually outputting anything. @= if unsorted(p)>void then sort_edges(p); q:=sorted(p); w:=0; prev_m:=-fraction_one; {$|fraction_one|\approx\infty$} ww:=0; prev_w:=0; m:=prev_m; repeat if q=sentinel then mm:=fraction_one else begin d:=ho(info(q)); mm:=d div 8; ww:=ww+(d mod 8)-zero_w; end; if mm<>m then begin if prev_w<=0 then begin if w>0 then @; end else if w<=0 then @; m:=mm; end; w:=ww; q:=link(q); until mm=fraction_one; if w<>0 then {this should be impossible} print_nl("(There's unbounded black in character shipped out!)"); @.There's unbounded black...@> if prev_m-m_offset(cur_edges)+x_off>gf_max_m then gf_max_m:=prev_m-m_offset(cur_edges)+x_off @ @= begin if prev_m=-fraction_one then @ else gf_paint(m-prev_m); prev_m:=m; prev_w:=w; end @ @= begin gf_paint(m-prev_m); prev_m:=m; prev_w:=w; end @ @= begin if prev_n=4096 then begin gf_boc(m_min(cur_edges)+x_off-zero_field, m_max(cur_edges)+x_off-zero_field,@| n_min(cur_edges)+y_off-zero_field,n+y_off); cur_min_m:=m_min(cur_edges)-zero_field+m_offset(cur_edges); end else if prev_n>n+1 then @ else @; gf_paint(m-cur_min_m); {skip to column $m$, painting white} done:prev_n:=n; end @ @= begin delta:=m-cur_min_m; if delta>max_new_row then gf_out(skip0) else begin gf_out(new_row_0+delta); goto done; end; end @ @= begin delta:=prev_n-n-1; if delta<@'400 then begin gf_out(skip1); gf_out(delta); end else begin gf_out(skip1+1); gf_two(delta); end; end @ Now that we've finished |ship_out|, let's look at the other commands by which a user can send things to the \.{GF} file. @= special_command: do_special; @ @= primitive("special",special_command,string_type);@/ @!@:special_}{\&{special} primitive@> primitive("numspecial",special_command,known);@/ @!@:num_special_}{\&{numspecial} primitive@> @ @= procedure do_special; var @!m:small_number; {either |string_type| or |known|} begin m:=cur_mod; get_x_next; scan_expression; if internal[proofing]>=0 then if cur_type<>m then @ else begin check_gf; if m=string_type then gf_string(cur_exp,0) else begin gf_out(yyy); gf_four(cur_exp); end; end; flush_cur_exp(0); end; @ @= begin exp_err("Unsuitable expression"); @.Unsuitable expression@> help1("The expression shown above has the wrong type to be output."); put_get_error; end @ @= begin check_gf; gf_string("title ",cur_exp); end @ @= special_command:if m=known then print("numspecial") else print("special"); @ @= begin cur_exp:=round_unscaled(cur_exp) mod 256; if cur_exp<0 then cur_exp:=cur_exp+256; boolean_reset(char_exists[cur_exp]); cur_type:=boolean_type; end @ At the end of the program we must finish things off by writing the postamble. The \.{TFM} information should have been computed first. An integer variable |k| and a |scaled| variable |x| will be declared for use by this routine. @= begin gf_out(post); {beginning of the postamble} gf_four(gf_prev_ptr); gf_prev_ptr:=gf_offset+gf_ptr-5; {|post| location} gf_four(internal[design_size]*16); for k:=1 to 4 do gf_out(header_byte[k]); {the check sum} gf_four(internal[hppp]); gf_four(internal[vppp]);@/ gf_four(gf_min_m); gf_four(gf_max_m); gf_four(gf_min_n); gf_four(gf_max_n); for k:=0 to 255 do if char_exists[k] then begin x:=gf_dx[k] div unity; if (gf_dy[k]=0)and(x>=0)and(x<256)and(gf_dx[k]=x*unity) then begin gf_out(char_loc+1); gf_out(k); gf_out(x); end else begin gf_out(char_loc); gf_out(k); gf_four(gf_dx[k]); gf_four(gf_dy[k]); end; x:=value(tfm_width[k]); if abs(x)>max_tfm_dimen then if x>0 then x:=three_bytes-1@+else x:=1-three_bytes else x:=make_scaled(x*16,internal[design_size]); gf_four(x); gf_four(char_ptr[k]); end; gf_out(post_post); gf_four(gf_prev_ptr); gf_out(gf_id_byte);@/ k:=4+((gf_buf_size-gf_ptr) mod 4); {the number of 223's} while k>0 do begin gf_out(223); decr(k); end; @; print_nl("Output written on "); print(output_file_name); @.Output written...@> print(" ("); print_int(total_chars); print(" character"); if total_chars<>1 then print_char("s"); print(", "); print_int(gf_offset+gf_ptr); print(" bytes)."); b_close(gf_file); end @* \[48] Dumping and undumping the tables. After \.{INIMF} has seen a collection of macros, it can write all the necessary information on an auxiliary file so that production versions of \MF\ are able to initialize their memory at high speed. The present section of the program takes care of such output and input. We shall consider simultaneously the processes of storing and restoring, so that the inverse relation between them is clear. @.INIMF@> The global variable |base_ident| is a string that is printed right after the |banner| line when \MF\ is ready to start. For \.{INIMF} this string says simply `\.{(INIMF)}'; for other versions of \MF\ it says, for example, `\.{(preloaded base=plain 84.2.29)}', showing the year, month, and day that the base file was created. We have |base_ident=0| before \MF's tables are loaded. @= @!base_ident:str_number; @ @= base_ident:=0; @ @= base_ident:=" (INIMF)"; @ @= @!init procedure store_base_file; var @!k:integer; {all-purpose index} @!p,@!q: pointer; {all-purpose pointers} @!x: integer; {something to dump} @!w: four_quarters; {four ASCII codes} begin @; @; @; @; @; @; @; end; tini @ Corresponding to the procedure that dumps a base file, we have a function that reads one in. The function returns |false| if the dumped base is incompatible with the present \MF\ table sizes, etc. @d off_base=6666 {go here if the base file is unacceptable} @d too_small(#)==begin wake_up_terminal; wterm_ln('---! Must increase the ',#); @.Must increase the x@> goto off_base; end @p @t\4@>@@; function load_base_file:boolean; label off_base,exit; var @!k:integer; {all-purpose index} @!p,@!q: pointer; {all-purpose pointers} @!x: integer; {something undumped} @!w: four_quarters; {four ASCII codes} begin @; @; @; @; @; load_base_file:=true; return; {it worked!} off_base: wake_up_terminal; wterm_ln('(Fatal base file error; I''m stymied)'); @.Fatal base file error@> load_base_file:=false; exit:end; @ Base files consist of |memory_word| items, and we use the following macros to dump words of different types: @d dump_wd(#)==begin base_file^:=#; put(base_file);@+end @d dump_int(#)==begin base_file^.int:=#; put(base_file);@+end @d dump_hh(#)==begin base_file^.hh:=#; put(base_file);@+end @d dump_qqqq(#)==begin base_file^.qqqq:=#; put(base_file);@+end @= @!base_file:word_file; {for input or output of base information} @ The inverse macros are slightly more complicated, since we need to check the range of the values we are reading in. We say `|undump(a)(b)(x)|' to read an integer value |x| that is supposed to be in the range |a<=x<=b|. @d undump_wd(#)==begin get(base_file); #:=base_file^;@+end @d undump_int(#)==begin get(base_file); #:=base_file^.int;@+end @d undump_hh(#)==begin get(base_file); #:=base_file^.hh;@+end @d undump_qqqq(#)==begin get(base_file); #:=base_file^.qqqq;@+end @d undump_end_end(#)==#:=x;@+end @d undump_end(#)==(x>#) then goto off_base@+else undump_end_end @d undump(#)==begin undump_int(x); if (x<#) or undump_end @d undump_size_end_end(#)==too_small(#)@+else undump_end_end @d undump_size_end(#)==if x># then undump_size_end_end @d undump_size(#)==begin undump_int(x); if x<# then goto off_base; undump_size_end @ The next few sections of the program should make it clear how we use the dump/undump macros. @= dump_int(@$);@/ dump_int(mem_min);@/ dump_int(mem_top);@/ dump_int(hash_size);@/ dump_int(hash_prime);@/ dump_int(max_in_open) @ Sections of a \.{WEB} program that are ``commented out'' still contribute strings to the string pool; therefore \.{INIMF} and \MF\ will have the same strings. (And it is, of course, a good thing that they do.) @.WEB@> @^string pool@> @= x:=base_file^.int; if x<>@$ then goto off_base; {check that strings are the same} undump_int(x); if x<>mem_min then goto off_base; undump_int(x); if x<>mem_top then goto off_base; undump_int(x); if x<>hash_size then goto off_base; undump_int(x); if x<>hash_prime then goto off_base; undump_int(x); if x<>max_in_open then goto off_base @ @d dump_four_ASCII== w.b0:=str_pool[k]; w.b1:=str_pool[k+1]; w.b2:=str_pool[k+2]; w.b3:=str_pool[k+3]; dump_qqqq(w) @= dump_int(pool_ptr); dump_int(str_ptr); for k:=0 to str_ptr do dump_int(str_start[k]); k:=0; while k+4= undump_size(0)(pool_size)('string pool size')(pool_ptr); undump_size(0)(max_strings)('max strings')(str_ptr); for k:=0 to str_ptr do begin undump(0)(pool_ptr)(str_start[k]); str_ref[k]:=max_str_ref; end; k:=0; while k+4= sort_avail; var_used:=0; dump_int(lo_mem_max); dump_int(rover); p:=mem_min; q:=rover; x:=0; repeat for k:=p to q+1 do dump_wd(mem[k]); x:=x+q+2-p; var_used:=var_used+q-p; p:=q+node_size(q); q:=rlink(q); until q=rover; var_used:=var_used+lo_mem_max-p; dyn_used:=mem_end+1-hi_mem_min;@/ for k:=p to lo_mem_max do dump_wd(mem[k]); x:=x+lo_mem_max+1-p; dump_int(hi_mem_min); dump_int(avail); for k:=hi_mem_min to mem_end do dump_wd(mem[k]); x:=x+mem_end+1-hi_mem_min; p:=avail; while p<>null do begin decr(dyn_used); p:=link(p); end; dump_int(var_used); dump_int(dyn_used); print_ln; print_int(x); print(" memory locations dumped; current usage is "); print_int(var_used); print_char("&"); print_int(dyn_used) @ @= undump(lo_mem_stat_max+1000)(hi_mem_stat_min-1)(lo_mem_max); undump(lo_mem_stat_max+1)(lo_mem_max)(rover); p:=mem_min; q:=rover; x:=0; repeat for k:=p to q+1 do undump_wd(mem[k]); p:=q+node_size(q); if (p>lo_mem_max)or((q>=rlink(q))and(rlink(q)<>rover)) then goto off_base; q:=rlink(q); until q=rover; for k:=p to lo_mem_max do undump_wd(mem[k]); undump(lo_mem_max+1)(hi_mem_stat_min)(hi_mem_min); undump(null)(mem_top)(avail); mem_end:=mem_top; for k:=hi_mem_min to mem_end do undump_wd(mem[k]); undump_int(var_used); undump_int(dyn_used) @ A different scheme is used to compress the hash table, since its lower region is usually sparse. When |text(p)<>0| for |p<=hash_used|, we output three words: |p|, |hash[p]|, and |eqtb[p]|. The hash table is, of course, densely packed for |p>=hash_used|, so the remaining entries are output in a block. @= dump_int(hash_used); st_count:=frozen_inaccessible-1-hash_used; for p:=1 to hash_used do if text(p)<>0 then begin dump_int(p); dump_hh(hash[p]); dump_hh(eqtb[p]); incr(st_count); end; for p:=hash_used+1 to hash_end do begin dump_hh(hash[p]); dump_hh(eqtb[p]); end; dump_int(st_count);@/ print_ln; print_int(st_count); print(" symbolic tokens") @ @= undump(1)(frozen_inaccessible)(hash_used); p:=0; repeat undump(p+1)(hash_used)(p); undump_hh(hash[p]); undump_hh(eqtb[p]); until p=hash_used; for p:=hash_used+1 to hash_end do begin undump_hh(hash[p]); undump_hh(eqtb[p]); end; undump_int(st_count) @ We have already printed a lot of statistics, so we set |tracing_stats:=0| to prevent them appearing again. @= dump_int(int_ptr); for k:=1 to int_ptr do begin dump_int(internal[k]); dump_int(int_name[k]); end; dump_int(start_sym); dump_int(interaction); dump_int(base_ident); dump_int(bg_loc); dump_int(eg_loc); dump_int(serial_no); dump_int(69069); internal[tracing_stats]:=0 @ @= undump(max_given_internal)(max_internal)(int_ptr); for k:=1 to int_ptr do begin undump_int(internal[k]); undump(0)(str_ptr)(int_name[k]); end; undump(0)(frozen_inaccessible)(start_sym); undump(batch_mode)(error_stop_mode)(interaction); undump(0)(str_ptr)(base_ident); undump(1)(hash_end)(bg_loc); undump(1)(hash_end)(eg_loc); undump_int(serial_no);@/ undump_int(x);@+if (x<>69069)or eof(base_file) then goto off_base @ @= selector:=new_string; print(" (preloaded base="); print(job_name); print_char(" "); print_int(round_unscaled(internal[year]) mod 100); print_char("."); print_int(round_unscaled(internal[month])); print_char("."); print_int(round_unscaled(internal[day])); print_char(")"); if interaction=batch_mode then selector:=log_only else selector:=term_and_log; str_room(1); base_ident:=make_string; str_ref[base_ident]:=max_str_ref;@/ pack_job_name(".base"); while not w_open_out(base_file) do prompt_file_name("base file name",".base"); print_nl("Beginning to dump on file "); @.Beginning to dump...@> print(w_make_name_string(base_file)); flush_string(str_ptr-1); print_nl(base_ident) @ @= w_close(base_file) @* \[49] The main program. This is it: the part of \MF\ that executes all those procedures we have written. Well---almost. We haven't put the parsing subroutines into the program yet; and we'd better leave space for a few more routines that may have been forgotten. @p @@; @@; @ @ We've noted that there are two versions of \MF84. One, called \.{INIMF}, @.INIMF@> has to be run first; it initializes everything from scratch, without reading a base file, and it has the capability of dumping a base file. The other one is called `\.{VIRMF}'; it is a ``virgin'' program that needs @.VIRMF@> to input a base file in order to get started. \.{VIRMF} typically has a bit more memory capacity than \.{INIMF}, because it does not need the space consumed by the dumping/undumping routines and the numerous calls on |primitive|, etc. The \.{VIRMF} program cannot read a base file instantaneously, of course; the best implementations therefore allow for production versions of \MF\ that not only avoid the loading routine for \PASCAL\ object code, they also have a base file pre-loaded. This is impossible to do if we stick to standard \PASCAL; but there is a simple way to fool many systems into avoiding the initialization, as follows:\quad(1)~We declare a global integer variable called |ready_already|. The probability is negligible that this variable holds any particular value like 314159 when \.{VIRMF} is first loaded.\quad(2)~After we have read in a base file and initialized everything, we set |ready_already:=314159|.\quad(3)~Soon \.{VIRMF} will print `\.*', waiting for more input; and at this point we interrupt the program and save its core image in some form that the operating system can reload speedily.\quad(4)~When that core image is activated, the program starts again at the beginning; but now |ready_already=314159| and all the other global variables have their initial values too. The former chastity has vanished! In other words, if we allow ourselves to test the condition |ready_already=314159|, before |ready_already| has been assigned a value, we can avoid the lengthy initialization. Dirty tricks rarely pay off so handsomely. @^dirty \PASCAL@> @^system dependencies@> On systems that allow such preloading, the standard program called \.{MF} should be the one that has \.{plain} base preloaded, since that agrees with {\sl The {\logos METAFONT\/}book}. Other versions, e.g., \.{cmbase}, should also be provided for commonly used bases. @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> @.cmbase@> @.plain@> @= @!ready_already:integer; {a sacrifice of purity for economy} @ Now this is really it: \MF\ starts and ends here. The initial test involving |ready_already| should be deleted if the \PASCAL\ runtime system is smart enough to detect such a ``mistake.'' @^system dependencies@> @p begin @!{|start_here|} history:=fatal_error_stop; {in case we quit during initialization} t_open_out; {open the terminal for output} if ready_already=314159 then goto start_of_MF; @@; if bad>0 then begin wterm_ln('Ouch---my internal constants have been clobbered!', '---case ',bad:1); @.Ouch...clobbered@> goto final_end; end; initialize; {set global variables to their starting values} @!init if not get_strings_started then goto final_end; init_tab; {initialize the tables} init_prim; {call |primitive| for each primitive} tini@/ ready_already:=314159; start_of_MF: @; @; init_str_ptr:=str_ptr; init_pool_ptr:=pool_ptr;@/ max_str_ptr:=str_ptr; max_pool_ptr:=pool_ptr;@/ history:=spotless; {ready to go!} if start_sym>0 then {insert the `\&{everyjob}' symbol} begin cur_sym:=start_sym; back_input; end; main_control; {come to life} final_cleanup; {prepare for death} end_of_MF: close_files_and_terminate; final_end: ready_already:=0; end. @ Here we do whatever is needed to complete \MF's job gracefully on the local operating system. The code here might come into play after a fatal error; it must therefore consist entirely of ``safe'' operations that cannot produce error messages. For example, it would be a mistake to call |str_room| or |make_string| at this time, because a call on |overflow| might lead to an infinite loop. @^system dependencies@> This program doesn't bother to close the input files that may still be open. @= procedure close_files_and_terminate; var @!k:integer; {all-purpose index} @!lh:integer; {the length of the \.{TFM} header, in words} @!p:pointer; {runs through a list of \.{TFM} dimensions} @!x:scaled; {a |tfm_width| value being output to the \.{GF} file} begin @!stat if internal[tracing_stats]>0 then @;@;@+tats@/ wake_up_terminal; @; if log_opened then begin wlog_cr; a_close(log_file); selector:=selector-2; if selector=term_only then begin print_nl("Transcript written on "); @.Transcript written...@> print(log_name); print_char("."); end; end; end; @ We want to finish the \.{GF} file if and only if it has already been started; this will be true if and only if |gf_prev_ptr| is positive. We want to produce a \.{TFM} file if and only if |fontmaking| is positive. The \.{TFM} widths must be computed if there's a \.{GF} file, even if there's going to be no \.{TFM}. We reclaim all of the variable-size memory at this point, so that there is no chance of another memory overflow after the memory capacity has already been exceeded. @= if (gf_prev_ptr>0)or(internal[fontmaking]>0) then begin @; @; fix_design_size; fix_check_sum; if internal[fontmaking]>0 then begin @; internal[fontmaking]:=0; {avoid loop in case of fatal error} @; end; if gf_prev_ptr>0 then @; end @ @= rover:=lo_mem_stat_max+1; link(rover):=empty_flag; lo_mem_max:=hi_mem_min-1; if lo_mem_max-rover>max_halfword then lo_mem_max:=max_halfword+rover; node_size(rover):=lo_mem_max-rover; llink(rover):=rover; rlink(rover):=rover; link(lo_mem_max):=null; info(lo_mem_max):=null @ The present section goes directly to the log file instead of using |print| commands, because there's no need for these strings to take up |str_pool| memory when a non-{\bf stat} version of \MF\ is being used. @= if log_opened then begin wlog_ln(' '); wlog_ln('Here is how much of METAFONT''s memory',' you used:'); @.Here is how much...@> wlog(' ',max_str_ptr-init_str_ptr:1,' string'); if max_str_ptr<>init_str_ptr+1 then wlog('s'); wlog_ln(' out of ', max_strings-init_str_ptr:1);@/ wlog_ln(' ',max_pool_ptr-init_pool_ptr:1,' string characters out of ', pool_size-init_pool_ptr:1);@/ wlog_ln(' ',lo_mem_max-mem_min+mem_end-hi_mem_min+2:1,@| ' words of memory out of ',mem_end+1-mem_min:1);@/ wlog_ln(' ',st_count:1,' symbolic tokens out of ', hash_size:1);@/ wlog_ln(' ',max_in_stack:1,'i,',@| int_ptr:1,'n,',@| max_rounding_ptr:1,'r,',@| max_param_stack:1,'p,',@| max_buf_stack+1:1,'b stack positions out of ',@| stack_size:1,'i,', max_internal:1,'n,', max_wiggle:1,'r,', param_size:1,'p,', buf_size:1,'b'); end @ We get to the |final_cleanup| routine when \&{end} or \&{dump} has been scanned. @= procedure final_cleanup; label exit; var c:small_number; {0 for \&{end}, 1 for \&{dump}} begin c:=cur_mod; if job_name=0 then open_log_file; while cond_ptr<>null do begin print_nl("(end occurred when ");@/ @.end occurred...@> print_cmd_mod(fi_or_else,cur_if); {`\.{if}' or `\.{elseif}' or `\.{else}'} if if_line<>0 then begin print(" on line "); print_int(if_line); end; print(" was incomplete)"); if_line:=if_line_field(cond_ptr); cur_if:=name_type(cond_ptr); cond_ptr:=link(cond_ptr); end; if history<>spotless then if ((history=warning_issued)or(interaction selector:=term_and_log; end; if c=1 then begin @!init store_base_file; return;@+tini@/ print_nl("(dump is performed only by INIMF)"); return; @.dump...only by INIMF@> end; exit:end; @ @= @!init procedure init_prim; {initialize all the primitives} begin @; end; @# procedure init_tab; {initialize other tables} var @!k:integer; {all-purpose index} begin @@; end; tini @ When we begin the following code, \MF's tables may still contain garbage; the strings might not even be present. Thus we must proceed cautiously to get bootstrapped in. But when we finish this part of the program, \MF\ is ready to call on the |main_control| routine to do its work. @= begin @; if (base_ident=0)or(buffer[loc]="&") then begin if base_ident<>0 then initialize; {erase preloaded base} if not open_base_file then goto final_end; if not load_base_file then begin w_close(base_file); goto final_end; end; w_close(base_file); while (loc; if loc"\" then start_input; {\&{input} assumed} end @* \[50] Debugging. Once \MF\ is working, it should be possible to diagnose most errors with the \.{show} commands and other diagnostic features. But for the initial stages of debugging, and for the revelation of really deep mysteries, you can compile \MF\ with a few more aids, including the \PASCAL\ runtime checks and its debugger. An additional routine called |debug_help| will also come into play when you type `\.D' after an error message; |debug_help| also occurs just before a fatal error causes \MF\ to succumb. @^debugging@> @^system dependencies@> The interface to |debug_help| is primitive, but it is good enough when used with a \PASCAL\ debugger that allows you to set breakpoints and to read variables and change their values. After getting the prompt `\.{debug \#}', you type either a negative number (this exits |debug_help|), or zero (this goes to a location where you can set a breakpoint, thereby entering into dialog with the \PASCAL\ debugger), or a positive number |m| followed by an argument |n|. The meaning of |m| and |n| will be clear from the program below. (If |m=13|, there is an additional argument, |l|.) @.debug \#@> @d breakpoint=888 {place where a breakpoint is desirable} @= @!debug procedure debug_help; {routine to display various things} label breakpoint,exit; var @!k,@!l,@!m,@!n:integer; begin loop begin wake_up_terminal; print_nl("debug # (-1 to exit):"); update_terminal; @.debug \#@> read(term_in,m); if m<0 then return else if m=0 then begin goto breakpoint;@\ {go to every label at least once} breakpoint: m:=0; @{'BREAKPOINT'@}@\ end else begin read(term_in,n); case m of @t\4@>@@; othercases print("?") endcases; end; end; exit:end; gubed @ @= 1: print_word(mem[n]); {display |mem[n]| in all forms} 2: print_int(info(n)); 3: print_int(link(n)); 4: begin print_int(eq_type(n)); print_char(":"); print_int(equiv(n)); end; 5: print_variable_name(n); 6: print_int(internal[n]); 7: do_show_dependencies; 9: show_token_list(n,null,100000,0); 10: print(n); 11: check_mem(n>0); {check wellformedness; print new busy locations if |n>0|} 12: search_mem(n); {look for pointers to |n|} 13: begin read(term_in,l); print_cmd_mod(n,l); end; 14: for k:=0 to n do print(buffer[k]); 15: panicking:=not panicking; @* \[51] System-dependent changes. This section should be replaced, if necessary, by changes to the program that are necessary to make \MF\ work at a particular installation. It is usually best to design your change file so that all changes to previous sections preserve the section numbering; then everybody's version will be consistent with the published program. More extensive changes, which introduce new sections, can be inserted here; then only the index itself will get a new section number. @^system dependencies@> @* \[52] Index. Here is where you can find all uses of each identifier in the program, with underlined entries pointing to where the identifier was defined. If the identifier is only one letter long, however, you get to see only the underlined entries. All references are to section numbers instead of page numbers. This index also lists error messages and other aspects of the program that you might want to look up some day. For example, the entry for ``system dependencies'' lists all sections that should receive special attention from people who are installing \MF\ in a new operating environment. A list of various things that can't happen appears under ``this can't happen''. Approximately 25 sections are listed under ``inner loop''; these account for more than 60\% of \MF's running time, exclusive of input and output.