; TeX output 1996.03.11:1325nWn썑0XQ cmr12Allcommercialrighrtsreserved.~MaybSefreelydistributedandusedwiththefollowingexceptions:1.8NocommercialusewithoutexplicitpSermission.2.8ItmarynotbSedistributedwithoutthisnotice.(VANG cmbx12AThezdiagrammacrosb#The lediagramincludesanrumbSerofmacrosfordrarwingcommutativediagramsinspSecialshapSesaswrellasmore exiblemacrostopastetogethermorecomplicateddiagrams.Ingeneral,therearetrwokindsofmacros.#HThe rstkindhasthefollorwingform6߆T cmtt12\shape-name[shapeff}r-parameters]andisinrtendedtobSeusedinthefollowingway:$$\shapename[shape_parameters]$$ThisnwillcreateashapSewiththegivrenparameters. ĉItoperatesbrybuildingitsownpictureenrvironment.TheTsecondkindofproSceduremrustbeusedinsideapictureenrvironmentTandallorwstheuserctoputadiagramofagivrensizeandshapSeatagivenplace.Theusualwayinwhichthisisusedis:$$\bfig\putshape1(x1,y1)[shape_parameters1]...\putshape(xn,yn)[shape_parametersn]\efig$$Hereisanexampleofthe rstkindofmacro.NIwillharvetoexplainafewdetailsbSeforegivingexamplesofthesecond.8ThecoSde$$\square[A`B`C`D;f`g`h`k]$$proSducesthediagramJtȈ~1g cmmi12CאDE$32fd*(pO line10- kbY_AbYB {fd*ά-fHǠ*Ffe*ԟǠ? igHǠ*FfeǠ?  hfandthecoSde$$\Atriangle[A`B`C;f`g`h]$$1*nWnproSducesthediagramB tXB!Cp̞32fdeЍά- hbY7A`9fطLׁ ηL ķL L p̟>p̟> H ,g( ׁ @( @( @( @n>@n>RfThereasonfortheshapSenamewillbeexplainedlater.IftanarrorwlabSelextendsespeciallyhighorlorw,"thespaceautomaticallyexpandstomatcrh.8FVorexample,$$\square[A`B`C`D;\sum_{i=1}^{\infty}`g`h`\Psi^A_k]$$willgetthediagramKoȈ~CאDE$32fd*(pά-  I 22cmmi8ARAkbY_AbYB {fd*ά-f⋟u cmex10P*pK cmsy81̍pi|{Ycmr8=1HǠ*Ffe*ԟǠ? igHǠ*FfeǠ?  hTandsoon.Beforegoingon,&IvwillexplainabSouttheparameters.fAnryshapewillharveoneortrwoparameters~forheighrtand/orlengthaswellasoneparameterforeacharrowinthediagram.These̿areinadditiontotheparametersusedintheactualproScedures.%FVorexample,EtheproScedure$$\square<-1`0`2`-3;1000`700>[A`B`C`D;f`g`h`k]$$givresthesquare:bs􍍍,C 30D鄞32fdd߰ Pάod32fdaYЍ0 kٍgAٍ yBdrfdeWά*0f@Pˍ$g@% ǠAfe%=Ǡ?% @ fe%=@?ʍ)4hThemeaningisasfollorws.qThe rstfourparametersrefertofourarrowsinlinguisticorder(top,left,righrt,bSottom).#Anegativenumbergivesabackwardarrow,whileazerocausesittobSeomitted.AparameterwithanabsolutevXalueof1isordinaryV,*while2givresanarrowwith+atail(monomorphism)and3adoubleheadedarrorw(epimorphism).Thelasttwoparameters3=determinethewidthandheighrt,EbrespSectivelyV,in3=unitsof0.01em(anemisthewidth^ofanM,thewidestletterinafonrt).)rItshouldbSenotedthatitisnotpossibletogivreonlysomeoftheoptionalparameters;eitherallmrustbSegivenornoneis.2:nWn썑HereparetheshapSes.+WVeharvepalreadyseentheshape\square.+Thereareeighrtdi erenttriangles,QallsisoScelesrighrttrianglesindi erentorientations.Thenamesallhavetheform\xtriangle,HwherexistheletterthatmostcloselyresemrblestheactualshapSeofthetriangle.Herearethenames,follorwedbyasampleofeachone:_\btriangleB` BCB32fd*dά- ýhH mbgBׁ @B @B @B @HŸ>@HŸ>RbYbmAߖǠ*FfeǠ?`f_\dtrianglerNBKLCnž32fd*dά- 0=hH`7{fBׁ B B B nŸ>nŸ> bY-ApǠ*FfeǠ? "g_\ptriangleKLʍbYAbY&BŸ{fd*ά-ɾfHڑk> _\qtriangleKLʍbYGAbYFBJ{fd*ά-EFfH Jgzׁ @z @z @z @->@->RHC3(Ǡ*FfefZǠ? h_\AtriangleB` $NB/C ž32fdeЍά- =hbY8-A`{fgBׁ gB gB gB Ÿ> Ÿ> H C"gׁ @  @ @ @">@">R3nWn썑_\VtriangleKLʍbYDAbY/VBƊr{fdeWά-fH,C αrg䢟ׁ @䢟 @䢟 @䢟 @+">@+">RH h!bׁ b  b b >> _\Ctriangle}JH 膘hnŸׁ @nŸ @nŸ @nŸ @B>@B>R-AbYrNB 7{fBBBBnŸnŸ KLCpǠdfeǠ?Ƴk"g_\Dtriangle}JH ^Zh9ׁ 9 9 9 :>:> bYBWˍ^Zg:@:@:@:@9@9RSeAC߇ǠdfeߺڟǠ?HfIn#addition,5qtherearetrwo#spSecialdiagramsthatcomeupoftenenoughtobewrorthhavingaspSecialmacrosfor.4NnWn썑_\AtrianglepairB` 6B4C32fd*dά- $k/FDڞ32fd*(pά- ClbY A`ͼcf:*ׁ :* :* :* >> H?Xl fe?XǠ ferǠ? kBgH  hׁ @ @ @ @!j>@!j>Rf_\VtrianglepairKLʍHhD hׁ @ @ @ @:*>@:*>RbYSAbYvBƙz{fd*ά-QvfbYbY/tC{fd*dά-;gH?Xl fe?XǠ ferǠ? 0kH  l!jׁ j  j j >> FinallyV,thereisonespSecialshapethatisprobablynotusedbryverymanymathematicians.StillIharveitandtheusermighrtaswellshareit.\recurse[A`B`R`C;s`f_0`f`t_0`t]Dˍ`ܺt0d@Dŵ@DŵRHRPHf0dlD׃ D׃ H̝rǠ*FfeФǠ?`у$f.NC(AW2!", cmsy10C32fd`;ά . tY BYAB {fd@wά ሳsHTǠ*FfeǠ?` :dAfvtNoticewhathappSensifthe rstparameterisemptry:\recurse[`B`R`C;s`f_0`f`t_0`t]`t0k@Vŵ@VŵRHRf0klV׃ V׃ H[Ǡ*Ffe؎Ǡ?`AdfCC432fd*}`ά itbYBbYBD{fd*Fչά &sHǠ*FfeF$Ǡ?`fvtThisOFtakrescareofthesimpleproScedures.TheremainingproceduresarethereasbuildingbloScrksformorecomplicateddiagrams.Thesetakretheform\putshape(xpos,ypos)[shapeff}r-parameters]wherex)shapSeisoneofthe11shapes(excluding\recurse)describedaborvex)andtheparmsareasdescribSedthere.Theparametersxposandyposaretheo setsfromthelorwerleftcornerofthepicturemeasuredinumitsof.01emintheusualLDD"A_T UE!Xfashion.TherePisonemoreshapSewhicrhissimplyanarrowwithanattachedlabSel..Thisiscalled\putmorphismanditisusedwiththesynrtax\putmorphism(xpos,ypos)(run,rise)[node1`node2`label]fdistgftypegflocgHerejtheparametersxposandyposareasabSorve.Thejslopeisrise/runwhereriseandrun)aretrwo)numbSersthatgivetheslopSeinaccordancewiththeLDD"A_T UE!Xrules.VdThatisriseanderunmrusthavenocommondivisorandmustnotexceed4inabsolutevXalue. UInaddition,theseeproSceduresarede nedsothatrisemrustnotbepositivreandifitiszero,@thenrunmust5nWnbSepositivre.Inotherwords, allarrowsmustgointhelinguisticdirection, downwardsortotherighrt.8ArrowscanbSemadetogointhereversedirectionasexplainedbSelow.The9nextthreeparametersarethenoSdethatappears rst(inlinguisticorder),MWtheonethatjappSearssecondandthearrorwlabel.1ThenextparamateristhedistanceinthesenseofLDD"A_T UE!XbSetrweenthecenrtersofthenodes.ThismeansitisthehorizonrtalcomponenrtoftheCdistance,junlessthatisnegativre,inwhicrhcaseitistheverticaldistance.bThesecondparameter[(isthecoSdeforthearrorwtype,wan[(integerbSetween-3and3withthesamemeaningasexplainedabSorve.6UsingnegativevXaluesofthisparameterallowsonetodrawanarrowthatgoSesuprwardsorleft.5Note,nhorwever,thatnoSde1isalwraysthenodethatisaborveortothe%leftofnoSde2.Thelastparametershouldharve%thevXaluea,b,l,rorm.ThevXaluesaandbYareusedonlyforhorizonrtalarrowsanddirectthearrowlabSeltobeplacedaborveYorbelorwthearrorw.5ThevXalueslandrareusedforallotherarrowsanddirectthelabSeltobeleftorrighrtofthearrow.F2FinallyV,ŊaverticalarrowonlycanbSegiventheparameterm,ŊinwhichcasethearrorwwillbSegappedandthelabelplacedinthemiddleofthegap.In,general,=^thesimplemacrosaredesignedtobSeusedasindicatedeitherinthedisplaryenrvironmentorinoneoftheothers,usually\begin{equation}\begin{array}{c}...\end{array}\end{equation}The\putshapSemacrosmrustbeusedinsideapictureenrvironmentthatisnormallyplacedinside adisplaryorsimilarenvironment. ThereferencepSointforthepSositioningparametersis"determinedasthelorwer"leftcornerofthesmallestrectanglewithsidesparalleltothecoSordinateaxesthatincludesthecenrterpoinrtsofallnodesonitsborder.5ThisrectanglewillbSedegenerateinthecaseofahorizonrtalorverticalmorphism.ThereferencepSointmayeitherbSeonoroutsidetheactual gure._FVorexample,inthecaseofthebtriangle,itisthecenrterNofthelowerleftnoSde,gwhileforaqtriangleitisoutsidethetriangleitselfbeingthefourthcorneroftheenclosingsquare.The1macrosaremadesothatthey ttogetherwrell. qThatiswhyalldistancesarefromnoSdecenrterstonodecenrters.dbIfavertexispartoftwoshapSes,itisprobablybestnottorepSeatit,hsinceitisconceivXablethatround-o errorswillcauseitstrwoappearancestobeslighrtlyo set.2ItcaneitherbSeomittedor,6sincethewidthisusedtodeterminetheamountto7shortenhorizonrtalarrows,Jreplacedby\phantomversions.BothmethoSdsareillustratedintheexamplegivrenattheend.A?somewhat@baroSque(buttakrenfromanactualtext)exampleillustratesmostofthesepSoinrts.$$\bfig\putmorphism(0,2100)(0,-1)[``T\eta'T]{1400}1l\putmorphism(0,2100)(1,0)[TT`T`\mu]{700}1a\putmorphism(0,2100)(1,-1)[`TTT'`TT\eta']{700}1l\putmorphism(700,2100)(1,-1)[`TT'`T\eta]{700}1r\put(700,1750){\makebox(0,0){1}}\putmorphism(700,1420)(1,0)[\phantom{TTT'}`\phantom{TT'}`\mu YT']{700}1a\putmorphism(700,1380)(1,0)[\phantom{TTT'}`%6-hnWn썑 Y\phantom{TT'}`T\sigma]{700}1b\putsquare<0`1`1`1;700`700>(700,700)[TTT'`TT'`TT'TT'`TT'T';`T\eta'TT'``]\putmorphism(700,700)(1,0)[\phantom{TT'TT'}`% Y\phantom{TT'T'}`TT'\sigma]{700}1a\put(300,1400){\makebox(0,0){2}}\put(950,1050){\makebox(0,0){3}}\putbtriangle<0`1`0;700>(1400,700)[``TT';T\eta'T'`id`]\putmorphism(1400,700)(1,0)[\phantom{TT'T'}`% Y\phantom{TT'}`T\mu']{700}1a\put(1600,1050){\makebox(0,0){6}}\putsquare<1`1`0`1;700`700>(0,0)[TT'T`\phantom{TT'TT'}`T'T`T'TT';% YTT'T\eta'`\sigma,T``T'T\eta']\putmorphism(700,0)(1,0)[\phantom{T'TT'}`% Y\phantom{T'T'}`T'\sigma]{700}1b\putsquare<0`0`1`1;700`700>(1400,0)[``T'T'`T';``\sigma`\mu']\putmorphism(700,700)(0,-1)[``\sigma,TT']{700}1m\putmorphism(1400,700)(0,-1)[``\sigma,T']{700}1m\put(300,350){\makebox(0,0){4}}\put(1050,350){\makebox(0,0){5}}\put(1750,350){\makebox(0,0){7}}\efig$$whicrhproSduces㪍 d@x⟥fey?^!ZQTn920y@ά-;2 d_ˍTTT205ATTn920Nt @Nt% @Nt/ @Nt9 @NtC @NtM @tS_@tS_R d_ˍTT205mTӂ4 @݂4% @4/ @49 @4C @4M @ E4S_@ E4S_R6\}Y1]]4Yrrfd33@ά-TNrTƟ20bX bX 4^$fd33@ά-hb~T2r TT20oTT202 0TT20oT204rfd'@ά-%_ˍTTT20_ˍTT20Y?@[@AfeIԟ?2ATn920 Ǡ*Ffe>@ğǠ? 3gHxҟǠ*FfexǠ? }hSٍ9|BSٍtUCFy$P rfd*dά-ZzZX:h#O@3KbDgFy$'[`@Py$1[`@Zy$;[`@dy$E[`@fG~@fG~R<9OAO@> G*Ffe>@ĠG?5~2{fSٍ;0BSٍv.CHҤP rfd*dά-ZzZZh#O@5~O]fi$'[`_$1[`U$;[`K$E[`HҤG~HҤG~ <vAO@{RG*Ffe{QG?3Kgiٍ9OAiٍt3BFe0rfd*ά-`[0Wfd@9nC> | *Ffe>@Ġ| ?Ë3gd@bDhfw5\5R5H5Fy$zFy$z GApBtIJfd*ά-pEpfHWˍôtg礟@礟@礟@礟@.$@.$RbY.C3R`*Ffef`?ʍhm5 *hn@@䟉n@@ 䟓n@@䟝n@@d^@d^R69$OAqOypBR@gfdD dNdXdbdd 2~${nC(꒟dfe)ğ?n-DgO@q`hdbdldvd䟂W䟂W #SٍgB3Kg'[`@1[`@;[`@E[`@dG~@dG~R< A.CARY dfetY ?R@]f$YBC1V$ٲfdeЍά-F`s?hٍ_mAJ8fQ'G'='3'1V$l1V$l @ᵋxgl d'@v d'@ d'@ d'@Sl@SlRޅY9|BޅYtUCFy$ڞfd*dά- ڍXZkޅYޅYDTڞfd*(pά- ڍ5ltAk MAff\RHFy$1Fy$1 kxҠ fexҠ3` fex3`?zˍugk:ʍh0d@0d@0d@0d@v1@v1Riٍ$OAiٍ0B0e0rfdeWά-`[0`fd@_2CË9"g1V$w5@;V$5@EV$5@OV$5@Qz@QzRd@xhSw5S5xS5nS5l dzl dz ޅY\D:ʍzh礠@礠@礠@礠@.$1@.$1RGApBt粄fd*ά-pEpf${nC粄fd*dά-!gk3R fe3R3` fef3`?:ʍ$kk:ʍ ld ddd11 Added:FVorDhorizonrtalandverticallinesonlyV,Zzarrowparametersof4,Zz5and6willgivredashedlines.HereisadoublecubSe:$$\bfig\putsquare(0,750)[A`B`C`D;```]\putsquare(0,250)[\phantom{C}`\phantom{D}`E`F;```]\putsquare(250,500)[A'`B'`C'`D';```]\putsquare(250,0)[\phantom{C'}`\phantom{D'}`E'`F';```]\putmorphism(0,1250)(1,-1)[```]{250}1a\putmorphism(500,1250)(1,-1)[```]{250}1a\putmorphism(0,750)(1,-1)[```]{250}1a9 LnWn\putmorphism(500,750)(1,-1)[```]{250}1a\putmorphism(0,250)(1,-1)[```]{250}1a\putmorphism(500,250)(1,-1)[```]{250}1a\efig$$?bnCD Rfd*(pά-F`qOyOAqOyBmifd*ά-jܠm5 ђ*Ffeğ?m5 ҟ*Ffe?QfEF=$גfd*dά- Rfd*(p- `ђl*Ffeğl? `ҟl*Ffel?ɯ VCܞ20ɯ DS20t{fd'Vά-ǢˍՑA20ˍsB20TIJfd'0ά-8@H-2`*Ffe`d`?Hr`*Ffe`?fKEE20fK FƟ20t32fd' ά-Y@ɯ ɯ t{fd'V-H-2Ǡ*Ffe`dǠ?HrǠ*FfeǠ?m5=$n@@(W5@(W5Rm5dn@@DW5@DW5R H=$%@(u@(uR Hd%@Du@DuR`=$@(ŵ@(ŵR`d@Dŵ@DŵR10]9;nG ANG cmbx126߆T cmtt122!", cmsy101g cmmi120XQ cmr12K cmsy82cmmi8|{Ycmr8O line10u cmex10cA