Asymptote:TheVectorGraphicsLanguageJohnBowmanandAndyHammerlindlDepartm - PDF document

Asymptote:TheVectorGraphicsLanguageJohnBowmanandAndyHammerlindlDepartmentofMathematicalandStatisticalSciencesUniversityofAlbertaCollaborators:OrestShardt,MichailVidiassovJune30,2010http://asymptote.sf.net/intro.pdf1 Historyˆ1979:TEXandMETAFONT(Knuth)ˆ1986:2DBeziercontrolpointselection(Hobby)ˆ1989:MetaPost(Hobby)ˆ2004:Asymptote{2004:initialpublicrelease(Hammerlindl,Bowman,&Prince){2005:3DBeziercontrolpointselection(Bowman){2008:3DinteractiveTEXwithinPDFles(Shardt&Bowman){2009:3Dbillboardlabelsthatalwaysfacecamera(Bowman){2010:3DPDFenhancements(Vidiassov&Bowman)2 Statistics(asofJune,2010)ˆRunsunderLinux/UNIX,MacOSX,MicrosoftWindows.ˆ4000downloads/monthfromprimaryasymptote.sourceforge.netsitealone.ˆ80000linesoflow-levelC++code.ˆ36000linesofhigh-levelAsymptotecode.3 VectorGraphicsˆRastergraphicsassigncolorstoagridofpixels. ˆVectorgraphicsaregraphicswhichstillmaintaintheirlookwheninspectedatarbitrarilysmallscales. 4 CartesianCoordinatesˆAsymptote'sgraphicalcapabilitiesarebasedonfourprimitivecommands:draw,label,fill,clip[BH08]draw((0,0)--(100,100)); ˆunitsarePostScriptbigpoints(1bp=1/72inch)ˆ--meansjointhepointswithalinearsegmenttocreateapathˆcyclicpath:draw((0,0)--(100,0)--(100,100)--(0,100)--cycle); 5 ScalingtoaGivenSizeˆPostScriptunitsareofteninconvenient.ˆInstead,scaleusercoordinatestoaspeciednalsize:size(3cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); ˆOnecanalsospecifythesizeincm:size(3cm,3cm);draw(unitsquare); 6 LabelsˆAddingandaligningLATEXlabelsiseasy:size(6cm);draw(unitsquare);label("$A$",(0,0),SW);label("$B$",(1,0),SE);label("$C$",(1,1),NE);label("$D$",(0,1),NW); 7 2DBezierSplinesˆUsing..insteadof--speciesaBeziercubicspline:draw(z0..controlsc0andc1..z1,blue); (1�t)3z0+3t(1�t)2c0+3t2(1�t)c1+t3z1;t2[0;1]:8 SmoothPathsˆAsymptotecanchoosecontrolpointsforyou,usingthealgorithmsofHobbyandKnuth[Hob86,Knu86]:pair[]z=f(0,0),(0,1),(2,1),(2,0),(1,0)g;draw(z[0]..z[1]..z[2]..z[3]..z[4]..cycle,grey+linewidth(5));dot(z,linewidth(7)); ˆFirst,linearequationsinvolvingthecurvaturearesolvedtondthedirectionthrougheachknot.Then,controlpointsalongthosedirectionsarechosen: 9 FillingˆThefillprimitivetolltheinsideofapath:pathstar;for(inti=0;i5;++i)star=star--dir(90+144i);star=star--cycle;fill(star,orange+zerowinding);draw(star,linewidth(3));fill(shift(2,0)*star,blue+evenodd);draw(shift(2,0)*star,linewidth(3)); 10 FillingˆUsealistofpathstollaregionwithholes:path[]p=fscale(2)*unitcircle,reverse(unitcircle)g;fill(p,green+zerowinding); 11 ClippingˆPicturescanbeclippedtoapath:fill(star,orange+zerowinding);clip(scale(0.7)*unitcircle);draw(scale(0.7)*unitcircle); 12 AneTransformsˆAnetransformations:shifts,rotations,re ections,andscalingscanbeappliedtopairs,paths,pens,strings,andevenwholepictures:fill(P,blue);fill(shift(2,0)*reflect((0,0),(0,1))*P,red);fill(shift(4,0)*rotate(30)*P,yellow);fill(shift(6,0)*yscale(0.7)*xscale(2)*P,green); 13 C++/Java-likeProgrammingSyntax//Declaration:Declarextobereal:realx;//Assignment:Assignxthevalue1.x=1.0;//Conditional:Testifxequals1ornot.if(x==1.0)fwrite("xequals1.0");gelsefwrite("xisnotequalto1.0");g//Loop:iterate10timesfor(inti=0;i10;++i)fwrite(i);g14 ModulesˆTherearemodulesforFeynmandiagrams, datastructures, 15 algebraicknottheory: (x1;x2;x3;x4;x5)=4b(x1+x4;x2;x3;x5)+4b(x1;x2;x3;x4)+4a(x1;x2+x3;x4;x5)�4b(x1;x2;x3;x4+x5)�4a(x1+x2;x3;x4;x5)�4a(x1;x2;x4;x5):16 TextbookGraphimportgraph;size(150,0);realf(realx)freturnexp(x);gpairF(realx)freturn(x,f(x));gxaxis("$x$");yaxis("$y$",0);draw(graph(f,-4,2,operator..),red);labely(1,E);label("$e^x$",F(1),SE); 17 ScienticGraphimportgraph;size(250,200,IgnoreAspect);realSin(realt)freturnsin(2pi*t);grealCos(realt)freturncos(2pi*t);gdraw(graph(Sin,0,1),red,"$nsin(2npix)$");draw(graph(Cos,0,1),blue,"$ncos(2npix)$");xaxis("$x$",BottomTop,LeftTicks);yaxis("$y$",LeftRight,RightTicks(trailingzero));label("LABEL",point(0),UnFill(1mm));attach(legend(),truepoint(E),20E,UnFill);18 19 DataGraphimportgraph;size(200,150,IgnoreAspect);real[]x=f0,1,2,3g;real[]y=x^2;draw(graph(x,y),red);xaxis("$x$",BottomTop,LeftTicks);yaxis("$y$",LeftRight,RightTicks(Label(fontsize(8pt)),newreal[]f0,4,9g));20 21 ImportedDataGraphimportgraph;size(200,150,IgnoreAspect);filein=input("filegraph.dat").line();real[][]a=in;a=transpose(a);real[]x=a[0];real[]y=a[1];draw(graph(x,y),red);xaxis("$x$",BottomTop,LeftTicks);yaxis("$y$",LeftRight,RightTicks);22 23 LogarithmicGraphimportgraph;size(200,200,IgnoreAspect);realf(realt)freturn1/t;gscale(Log,Log);draw(graph(f,0.1,10));//limits((1,0.1),(10,0.5),Crop);dot(Label("(3,5)",align=S),Scale((3,5)));xaxis("$x$",BottomTop,LeftTicks);yaxis("$y$",LeftRight,RightTicks);24 25 SecondaryAxis 26 ImagesandContours 27 MultipleGraphs 28 Hobby's2DDirectionAlgorithmˆAtridiagonalsystemoflinearequationsissolvedtodetermineanyunspecieddirectionskandkthrougheachknotzk:k�1�2k `k=k+1�2k `k+1: ˆTheresultingshapemaybeadjustedbymodifyingoptionaltensionparametersandcurlboundaryconditions.29 Hobby's2DControlPointAlgorithmˆHavingprescribedoutgoingandincomingpathdirectionseiatnodez0andeiatnodez1relativetothevectorz1�z0,thecontrolpointsaredeterminedas:u=z0+ei(z1�z0)f(;�);v=z1�ei(z1�z0)f(�;);wheretherelativedistancefunctionf(;)isgivenbyHobby[1986]. 30 BezierCurvesin3DˆApplyananetransformationx0i=Aijxj+CitoaBeziercurve:x(t)=3Xk=0Bk(t)Pk;t2[0;1]:ˆTheresultingcurveisalsoaBeziercurve:x0i(t)=3Xk=0Bk(t)Aij(Pk)j+Ci=3Xk=0Bk(t)P0k;whereP0kisthetransformedkthcontrolpoint,noting3Xk=0Bk(t)=1:31 3DGeneralizationofDirectionAlgorithmˆMustreduceto2Dalgorithminplanarcase.ˆDeterminedirectionsbyapplyingHobby'salgorithmintheplanecontainingzk�1,zk,zk+1.ˆTheonlyambiguitythatcanariseistheoverallsignoftheangles,whichrelatestoviewingeach2Dplanefromopposingnormaldirections.ˆAreferencevectorbasedonthemeanunitnormalofsuccessivesegmentscanbeusedtoresolvesuchambiguities[Bow07,BS09]32 3DControlPointAlgorithmˆExpressHobby'salgorithmintermsoftheabsolutedirections!0and!1:u=z0+!0jz1�z0jf(;�);v=z1�!1jz1�z0jf(�;); interpretingandastheanglebetweenthecorrespondingpathdirectionvectorandz1�z0.33 ˆHerethereisanunambiguousreferencevectorfordeterminingtherelativesignoftheanglesand.34 Interactive3DSaddleˆAunitcircleintheX{Yplanemaybeconstructedwith:(1,0,0)..(0,1,0)..(-1,0,0)..(0,-1,0)..cycle: andthendistortedintothesaddle(1,0,0)..(0,1,1)..(-1,0,0)..(0,-1,1)..cycle: 35 LiftingTeXto3DˆGlyphsarerstsplitintosimplyconnectedregionsandthendecomposedintoplanarBeziersurfacepatches[BS09,SB12]: 36 LabelManipulationˆTheycanthenbeextrudedand/orarbitrarilytransformed: 37 BillboardLabels 38 Smooth3Dsurfaces 39 Curved3DArrows 40 SlidePresentationsˆAsymptotehasamoduleforpreparingslides.ˆItevensupportsembeddedhigh-resolutionPDFmovies.title("SlidePresentations");item("Asymptotehasamoduleforpreparingslides.");item("Itevensupportsembeddedhigh-resolutionPDFmovies.");...41 AutomaticSizingˆFigurescanbespeciedinusercoordinates,thenautomaticallyscaledtothedesirednalsize. 42 DeferredDrawingˆWecan'tdrawagraphicalobjectuntilweknowthescalingfactorsfortheusercoordinates.ˆInstead,storeafunctionthat,giventhescalinginformation,drawsthescaledobject.voiddraw(picturepic=currentpicture,pathg,penp=currentpen)fpic.add(newvoid(framef,transformt)fdraw(f,t*g,p);g);pic.addPoint(min(g),min(p));pic.addPoint(max(g),max(p));g43 CoordinatesˆStoreboundingboxinformationasthesumofuserandtrue-sizecoordinates: pic.addPoint(min(g),min(p));pic.addPoint(max(g),max(p));ˆFillingignoresthepenwidth:pic.addPoint(min(g),(0,0));pic.addPoint(max(g),(0,0));ˆCommunicatewithLATEXviaapipetodeterminelabelsizes: 44 SizingˆWhenscalingthenalguretoagivensizeS,werstneedtodetermineascalingfactora�0andashiftbsothatallofthecoordinateswhentransformedwilllieintheinterval[0;S].ˆThatis,ifuandtaretheuserandtruesizecomponents:0au+t+bS:ˆMaximizethevariableasubjecttoanumberofinequalities.ˆUsethesimplexmethodtosolvetheresultinglinearprogrammingproblem.45 SizingˆEveryadditionofacoordinate(t;u)addstworestrictionsau+t+b0;au+t+bS;andeachdrawingcomponentaddstwocoordinates.ˆAgurecouldeasilyproducethousandsofrestrictions,makingthesimplexmethodimpractical.ˆMostoftheserestrictionsareredundant,however.Forinstance,withconcentriccircles,onlythelargestcircleneedstobeaccountedfor. 46 RedundantRestrictionsˆIngeneral,ifuu0andtt0thenau+t+bau0+t0+bforallchoicesofa�0andb,so0au+t+bau0+t0+bS:ˆThisdenesapartialorderingoncoordinates.Whensizingapicture,theprogramrstcomputeswhichcoordinatesaremaximal(orminimal)andonlysendseectiveconstraintstothesimplexalgorithm.ˆInpractice,thelinearprogrammingproblemwillhavelessthanadozenrestraints.ˆAllpicturesizingisimplementedinAsymptotecode.47 InniteLinesˆDeferreddrawingallowsustodrawinnitelines.drawline(P,Q); 48 HelpfulMathNotationˆIntegerdivisionreturnsareal.Usequotientforanintegerresult:3/4==0.75quotient(3,4)==0ˆCaretforrealandintegerexponentiation:2^32.7^32.7^3.2ˆManyexpressionscanbeimplicitlyscaledbyanumericconstant:2pi10cm2x^23sin(x)2(a+b)ˆPairsarecomplexnumbers:(0,1)*(0,1)==(-1,0)49 FunctionCallsˆFunctionscantakedefaultargumentsinanyposition.Argumentsarematchedtotherstpossiblelocation:voiddrawEllipse(realxsize=1,realysize=xsize,penp=blue)fdraw(xscale(xsize)*yscale(ysize)*unitcircle,p);gdrawEllipse(2);drawEllipse(red); ˆArgumentscanbegivenbyname:drawEllipse(xsize=2,ysize=1);drawEllipse(ysize=2,xsize=3,green); 50 RestArgumentsˆRestargumentsallowonetowriteafunctionthattakesanarbitrarynumberofarguments:intsum(...int[]nums)finttotal=0;for(inti=0;inums.length;++i)total+=nums[i];returntotal;gsum(1,2,3,4);//returns10sum();//returns0sum(1,2,3...newint[]f4,5,6g);//returns21intsubtract(intstart...int[]subs)freturnstart-sum(...subs);g51 High-OrderFunctionsˆFunctionsarerst-classvalues.Theycanbepassedtootherfunctions:importgraph;realf(realx)freturnx*sin(10x);gdraw(graph(f,-3,3,300),red); 52 Higher-OrderFunctionsˆFunctionscanreturnfunctions:fn(x)=nsinx n:typedefrealfunc(real);funcf(intn)frealfn(realx)freturnn*sin(x/n);greturnfn;gfuncf1=f(1);realy=f1(pi);for(inti=1;i=5;++i)draw(graph(f(i),-10,10),red); 53 AnonymousFunctionsˆCreatenewfunctionswithnew:pathp=graph(newreal(realx)freturnx*sin(10x);g,-3,3,red);funcf(intn)freturnnewreal(realx)freturnn*sin(x/n);g;gˆFunctiondenitionsarejustsyntacticsugarforassigningfunctionobjectstovariables.realsquare(realx)freturnx^2;gisequivalenttorealsquare(realx);square=newreal(realx)freturnx^2;g;54 StructuresˆAsinotherlanguages,structuresgrouptogetherdata.structPersonfstringfirstname,lastname;intage;gPersonbob=newPerson;bob.firstname="Bob";bob.lastname="Chesterton";bob.age=24;ˆAnycodeinthestructurebodywillbeexecutedeverytimeanewstructureisallocated...structPersonfwrite("Makingaperson.");stringfirstname,lastname;intage=18;gPersoneve=newPerson;//Writes"Makingaperson."write(eve.age);//Writes18.55 ModulesˆFunctionandstructuredenitionscanbegroupedintomodules://powers.asyrealsquare(realx)freturnx^2;grealcube(realx)freturnx^3;gandimported:importpowers;realeight=cube(2.0);draw(graph(powers.square,-1,1));56 Object-OrientedProgrammingˆFunctionsaredenedforeachinstanceofastructure.structQuadraticfreala,b,c;realdiscriminant()freturnb^2-4*a*c;grealeval(realx)freturna*x^2+b*x+c;ggˆThisallowsustoconstruct\methods"whicharejustnormalfunctionsdeclaredintheenvironmentofaparticularobject:Quadraticpoly=newQuadratic;poly.a=-1;poly.b=1;poly.c=2;realf(realx)=poly.eval;realy=f(2);draw(graph(poly.eval,-5,5));57 SpecializationˆCancreatespecializedobjectsjustbyredeningmethods:structShapefvoiddraw();realarea();gShaperectangle(realw,realh)fShapes=newShape;s.draw=newvoid()ffill((0,0)--(w,0)--(w,h)--(0,h)--cycle);g;s.area=newreal()freturnw*h;g;returns;gShapecircle(realradius)fShapes=newShape;s.draw=newvoid()ffill(scale(radius)*unitcircle);g;s.area=newreal()freturnpi*radius^2;greturns;g58 OverloadingˆConsiderthecode:intx1=2;intx2()freturn7;gintx3(inty)freturn2y;gwrite(x1+x2());//Writes9.write(x3(x1)+x2());//Writes11.59 Overloadingˆx1,x2,andx3areneverusedinthesamecontext,sotheycanallberenamedxwithoutambiguity:intx=2;intx()freturn7;gintx(inty)freturn2y;gwrite(x+x());//Writes9.write(x(x)+x());//Writes11.ˆFunctiondenitionsarejustvariabledenitions,butvariablesaredistinguishedbytheirsignaturestoallowoverloading.60 OperatorsˆOperatorsarejustsyntacticsugarforfunctions,andcanbeaddressedordenedasfunctionswiththeoperatorkeyword.intadd(intx,inty)=operator+;write(add(2,3));//Writes5.//Don'ttrythisathome.intoperator+(intx,inty)freturnadd(2x,y);gwrite(2+3);//Writes7.ˆThisallowsoperatorstobedenedfornewtypes.61 OperatorsˆOperatorsforconstructingpathsarealsofunctions:a..controlsbandc..d--eisequivalenttooperator--(operator..(a,operatorcontrols(b,c),d),e)ˆThisallowedustoredeneallofthepathoperatorsfor3Dpaths.62 SummaryˆAsymptote:{usesIEEE oatingpointnumerics;{usesC++/Java-likesyntax;{supportsdeferreddrawingforautomaticpicturesizing;{supportsGrayscale,RGB,CMYK,andHSVcolourspaces;{supportsPostScriptshading,patternlls,andfunctionshading;{canllnonsimplyconnectedregions;{generalizesMetaPostpathconstructionalgorithmsto3D;{liftsTEXto3D;{supports3DbillboardlabelsandPDFgrouping.63 References[BH08]JohnC.BowmanandAndyHammerlindl.Asymptote:Avectorgraphicslanguage.TUGboat:TheCommunicationsoftheTEXUsersGroup,29(2):288{294,2008.[Bow07]JohnC.Bowman.The3DAsymptotegeneralizationofMetaPostBezierinterpolation.ProceedingsinAppliedMathematicsandMechanics,7(1):2010021{2010022,2007.[BS09]JohnC.BowmanandOrestShardt.Asymptote:LiftingTEXtothreedimensions.TUGboat:TheCommunicationsoftheTEXUsersGroup,30(1):58{63,2009.[Hob86]JohnD.Hobby.Smooth,easytocomputeinterpolatingsplines.DiscreteComput.Geom.,1:123{140,1986.[Knu86]DonaldE.Knuth.TheMETAFONTbook.Addison-Wesley,Reading,Massachusetts,1986.[SB12]OrestShardtandJohnC.Bowman.SurfaceparametrizationofnonsimplyconnectedplanarBezierregions.Computer-AidedDesign,44(5):484.e1{10,2012. Asymptote:2D&3DVectorGraphicsLanguage http://asymptote.sf.net(freelyavailableundertheLGPLlicense)64