Mean Value Coordinates for Closed Triangular Meshes Tao Ju Scott Schaefer Joe Warren Rice - PDF document

MeanValueCoordinatesforClosedTriangularMeshesTaoJu,ScottSchaefer,JoeWarrenRiceUniversity(a)(b)(c)(d)Figure1:Originalhorsemodelwithenclosingtrianglecontrolmeshshowninblack(a).Severaldeformationsgeneratedusingour3Dmeanvaluecoordinatesappliedtoamodiedcontrolmesh(b,c,d).AbstractConstructingafunctionthatinterpolatesasetofvaluesdenedatverticesofameshisafundamentaloperationincomputergraphics.Suchaninterpolanthasmanyusesinapplicationssuchasshad-ing,parameterizationanddeformation.Forclosedpolygons,meanvaluecoordinateshavebeenproventobeanexcellentmethodforconstructingsuchaninterpolant.Inthispaper,wegeneralizemeanvaluecoordinatesfromclosed2Dpolygonstoclosedtriangularmeshes.GivensuchameshP,weshowthatthesecoordinatesarecontinuouseverywhereandsmoothontheinteriorofP.ThecoordinatesarelinearonthetrianglesofPandcanreproducelin-earfunctionsontheinteriorofP.Toillustratetheirusefulness,weconcludebyconsideringseveralinterestingapplicationsincludingconstructingvolumetrictexturesandsurfacedeformation.CRCategories:I.3.5[ComputerGraphics]:ComputationalGe-ometryandObjectModeling—Boundaryrepresentations;Curve,surface,solid,andobjectrepresentations;Geometricalgorithms,languages,andsystemsKeywords:barycentriccoordinates,meanvaluecoordinates,vol-umetrictextures,surfacedeformation1IntroductionGivenaclosedmesh,acommonproblemincomputergraphicsistoextendafunctiondenedattheverticesofthemeshtoitsinterior.Forexample,Gouraudshadingcomputesintensitiesattheverticesofatriangleandextendstheseintensitiestotheinteriorusinglinearinterpolation.Givenatrianglewithverticesfp1;p2;p3gandasso-ciatedintensitiesff1;f2;f3g,theintensityatpointvontheinteriorofthetrianglecanbeexpressedintheformˆf[v]=åjwjfjåjwj(1)wherewjistheareaofthetrianglefv;pj1;pj+1g.Inthisformula,notethateachweightwjisnormalizedbythesumoftheweights,åjwjtoformanassociatedcoordinatewjåjwj.Theinterpolantˆf[v]isthensimplythesumofthefjtimestheircorrespondingcoordi-nate.Meshparameterizationmethods[HormannandGreiner2000;Desbrunetal.2002;Khodakovskyetal.2003;Schreineretal.2004;FloaterandHormann2005]andfreeformdeformationmeth-ods[SederbergandParry1986;Coquillart1990;MacCrackenandJoy1996;KobayashiandOotsubo2003]alsomakeheavyuseofinterpolantsofthistype.Bothapplicationsrequirethatapointvberepresentedasanafnecombinationoftheverticesonanenclosingshape.Togeneratethiscombination,wesimplysetthedataval-uesfjtobetheirassociatedvertexpositionspj.Iftheinterpolantreproduceslinearfunctions,i.e.;v=åjwjpjåjwj;thecoordinatefunctionswjåjwjarethedesiredafnecombination.Forconvexpolygonsin2D,asequenceofpapers,[Wachspress1975],[LoopandDeRose1989]and[Meyeretal.2002],havepro-posedandrenedaninterpolantthatislinearonitsboundariesandonlyinvolvesconvexcombinationsofdatavaluesatthever-ticesofthepolygons.Thisinterpolanthasasimple,localdeni-tionasarationalfunctionandreproduceslinearfunctions.[War-ren1996;Warrenetal.2004]alsogeneralizedthisinterpolanttoconvexshapesinhigherdimensions.Unfortunately,Wachspress'sinterpolantdoesnotgeneralizetonon-convexpolygons.Applying (a)(b)(c)(d)Figure2:InterpolatinghuevaluesatpolygonverticesusingWach-spresscoordinates(a,b)versusmeanvaluecoordinates(c,d)onaconvexandaconcavepolygon.theconstructiontosuchapolygonyieldsaninterpolantthathaspoles(divisionsbyzero)ontheinteriorofthepolygon.ThetopportionofFigure2showsWachspress'sinterpolantappliedtotwoclosedpolygons.Notethepolesontheoutsideoftheconvexpoly-gonontheleftaswellasalongtheextensionsofthetwotopedgesofthenon-convexpolygonontheright.Morerecently,severalpapers,[Floater1997;Floater1998;Floater2003],[MalschandDasgupta2003]and[Hormann2004],havefocusedonbuildinginterpolantsfornon-convex2Dpolygons.Inparticular,Floaterproposedanewtypeofinterpolantbasedonthemeanvaluetheorem[Floater2003]thatgeneratessmoothco-ordinatesforstar-shapedpolygons.Givenapolygonwithverticespjandassociatedvaluesfj,Floater'sinterpolantdenesasetofweightfunctionswjoftheformwj=tanhaj12i+tanhaj2ijpjvj:(2)whereajistheangleformedbythevectorpjvandpj+1v.Normalizingeachweightfunctionwjbythesumofallweightfunc-tionsyieldsthemeanvaluecoordinatesofvwithrespecttopj.Inhisoriginalpaper,Floaterprimarilyintendedthisinterpolanttobeusedformeshparameterizationandonlyexploredthebehav-ioroftheinterpolantonpointsinthekernelofastar-shapedpoly-gon.Inthisregion,meanvaluecoordinatesarealwaysnon-negativeandreproducelinearfunctions.Subsequently,Hormann[Hormann2004]showedthat,foranysimplepolygon(ornestedsetofsim-plepolygons),theinterpolantˆf[v]generatedbymeanvaluecoor-dinatesiswell-denedeverywhereintheplane.Bymaintainingaconsistentorientationforthepolygonandtreatingtheajassignedangles,Hormannalsoshowsthatmeanvaluecoordinatesreproducelinearfunctionseverywhere.ThebottomportionofFigure2showsmeanvaluecoordinatesappliedtotwoclosedpolygons.Notethattheinterpolantgeneratedbythesecoordinatespossessesnopolesanywhereevenonnon-convexpolygons.ContributionsHorman'sobservationsuggeststhatFloater'smeanvalueconstructioncouldbeusedtogenerateasimilarin-terpolantforawiderclassofshapes.Inthispaper,weprovidesuchageneralizationforarbitraryclosedsurfacesandshowthattheresultinginterpolantsarewell-behavedandhavelinearpreci-sion.Appliedtoclosedpolygons,ourconstructionreproduces2Dmeanvaluecoordinates.Wethenapplyourmethodtoclosedtri-angularmeshesandconstruct3Dmeanvaluecoordinates.(Inin-dependentcontemporaneouswork,[Floateretal.2005]havepro-posedanextensionofmeanvaluecoordinatesfrom2Dpolygonsto3Dtriangularmeshesidenticaltosection3.2.)Next,wederiveanefcient,stablemethodforevaluatingtheresultingmeanvaluein-terpolantintermsofthepositionsandassociatedvaluesofverticesofthemesh.Finally,weconsiderseveralpracticalapplicationsofsuchcoordinatesincludingasimplemethodforgeneratingclassesofdeformationsusefulincharacteranimation.2MeanvalueinterpolationGivenaclosedsurfacePinR3,letp[x]beaparameterizationofP.(Here,theparameterxistwo-dimensional.)Givenanauxiliaryfunctionf[x]denedoverP,ourproblemistoconstructafunctionˆf[v]wherev2R3thatinterpolatesf[x]onP,i.e.;ˆf[p[x]]=f[x]forallx.OurbasicconstructionextendsanideaofFloaterdevelopedduringtheconstructionof2Dmeanvaluecoordinates.Toconstructˆf[v],weprojectapointp[x]ofPontotheunitsphereSvcenteredatv.Next,weweightthepoint'sassociatedvaluef[x]by1jp[x]vjandintegratethisweightedfunctionoverSv.Toensureafneinvarianceoftheresultinginterpolant,wedividetheresultbytheintegraloftheweightfunction1jp[x]vjtakenoverSv.Puttingthepiecestogether,themeanvalueinterpolanthastheformˆf[v]=Rxw[x;v]f[x]dSvRxw[x;v]dSv(3)wheretheweightfunctionw[x;v]isexactly1jp[x]vj.ObservethatthisformulaisessentiallyanintegralversionofthediscreteformulaofEquation1.Likewise,thecontinuousweightfunctionw[x;v]andthediscreteweightswjofEquation2differonlyintheirnumera-tors.Asweshallsee,thetana2termsinthenumeratorsofthewjaretheresultoftakingtheintegralsinEquation3withrespecttodSv.Theresultingmeanvalueinterpolantsatisesthreeimportantproperties.Interpolation:Asvconvergestothepointp[x]onP,ˆf[v]con-vergestof[x].Smoothness:Thefunctionˆf[v]iswell-denedandsmoothforallvnotonP.Linearprecision:Iff[x]=p[x]forallx,theinterpolantˆf[v]isidenticallyvforallv.Interpolationfollowsfromthefactthattheweightfunctionw[x;v]approachesinnityasp[x]!v.Smoothnessfollowsbecausetheprojectionoff[x]ontoSviscontinuousinthepositionofvandtakingtheintegralofthiscontinuousprocessyieldsasmoothfunc-tion.Theproofoflinearprecisionreliesonthefactthattheintegraloftheunitnormaloverasphereisexactlyzero(duetosymmetry).Specically,Zxp[x]vjp[x]vjdSv=0sincep[x]vjp[x]vjistheunitnormaltoSvatparametervaluex.Rewrit-ingthisequationyieldsthetheorem.v=Zxp[x]jp[x]vjdSv.Zx1jp[x]vjdSv NoticethatiftheprojectionofPontoSvisone-to-one(i.e.;visinthekernelofP),thentheorientationofdSvisnon-negative,whichguaranteesthattheresultingcoordinatefunctionsareposi-tive.Therefore,ifPisaconvexshape,thenthecoordinatefunctionsarepositiveforallvinsideP.However,ifvisnotinthekernelofP,thentheorientationofdSvisnegativeandthecoordinatesfunctionsmaybenegativeaswell.3CoordinatesforpiecewiselinearshapesInpractice,theintegralformofEquation3canbecomplicatedtoevaluatesymbolically1.However,inthissection,wederiveasim-ple,closedformsolutionforpiecewiselinearshapesintermsofthevertexpositionsandtheirassociatedfunctionvalues.Asasimpleexampletoillustrateourapproach,werstre-derivemeanvalueco-ordinatesforclosedpolygonsviameanvalueinterpolation.Next,weapplythesamederivationtoconstructmeanvaluecoordinatesforclosedtriangularmeshes.3.1MeanvaluecoordinatesforclosedpolygonsConsideranedgeEofaclosedpolygonPwithverticesfp1;p2gandassociatedvaluesff1;f2g.Ourrsttaskistoconvertthisdis-cretedataintoacontinuousformsuitableforuseinEquation3.WecanlinearlyparameterizetheedgeEviap[x]=åifi[x]piwheref1[x]=(1x)andf2[x]=x.Wethenusethissamepa-rameterizationtoextendthedatavaluesf1andf2linearlyalongE.Specically,weletf[x]havetheformf[x]=åifi[x]fi:Now,ourtaskistoevaluatetheintegralsinEquation3for0x1.LetEbethecirculararcformedbyprojectingtheedgeEontotheunitcircleSv,wecanrewritetheintegralsofEquation3restrictedtoEasRxw[x;v]f[x]dERxw[x;v]dE=åiwifiåiwi(4)whereweightswi=Rxfi[x]jp[x]vjdE.OurnextgoalistocomputethecorrespondingweightswiforedgeEinEquation4withoutresortingtosymbolicintegration(sincethiswillbedifculttogeneralizeto3D).Observethatthefollowingidentityrelateswitoavector,åiwi(piv)=m:(5)wherem=Rxp[x]vjp[x]vjdEissimplytheintegraloftheoutwardunitnormaloverthecirculararcE.WecallmthemeanvectorofE,asscalingmbythelengthofthearcyieldsthecentroidofthecirculararcE.Basedon2Dtrigonometry,mhasasimpleexpressionintermsofp1andp2.Specically,1ToevaluatetheintegralofEquation3,wecanrelatethedifferentialdSvtodxviadSv=p?[x]:(p[x]v)jp[x]vj2dxwherep?[x]isthecrossproductofthen1tangentvectors¶p[x]¶xitoPatp[x].NotethatthesignofthisexpressioncorrectlycaptureswhetherPhasfoldedbackduringitsprojectionontoSv.m=tan[a=2]((p1v)jp1vj+(p2v)jp2vj)whereadenotestheanglebetweenp1vandp2v.Henceweob-tainwi=tan[a=2]=pivwhichagreeswiththeFloater'sweight-ingfunctiondenedinEquation2for2Dmeanvaluecoordinateswhenrestrictedtoasingleedgeofapolygon.Equation4allowsustoformulateaclosedformexpressionfortheinterpolantˆf[v]inEquation3bysummingtheintegralsforalledgesEkinP(notethatweaddtheindexkforenumerationofedges):ˆf[v]=åkåiwkifkiåkåiwki(6)wherewkiandfkiareweightsandvaluesassociatedwithedgeEk.3.2MeanvaluecoordinatesforclosedmeshesWenowconsiderourprimaryapplicationofmeanvalueinterpo-lationforthispaper;thederivationofmeanvaluecoordinatesfortriangularmeshes.Thesecoordinatesarethenaturalgeneralizationof2Dmeanvaluecoordinates.GiventriangleTwithverticesfp1;p2;p3gandassociatedvaluesff1;f2;f3g,ourrsttaskistodenethefunctionsp[x]andf[x]usedinEquation3overT.Tothisend,wesimplyusethelinearinterpolationformulaofEquation1.Theresultingfunctionf[x]isalinearcombinationofthevaluesfitimesbasisfunctionsfi[x].Asin2D,theintegralofEquation3reducestothesuminEqua-tion6.Inthiscase,theweightswihavetheformwi=Zxfi[x]jp[x]vjdTwhereTistheprojectionoftriangleTontoSv.Toavoidcomputingthisintegraldirectly,weinsteadrelatetheweightswitothemeanvectormforthesphericaltriangleTbyinvertingEquation5.Inmatrixform,fw1;w2;w3g=mfp1v;p2v;p3vg1(7)AllthatremainsistoderiveanexplicitexpressionforthemeanvectormforasphericaltriangleT.Thefollowingtheoremsolvesthisproblem.Theorem3.1GivenasphericaltriangleT,letqibethelengthofitsithedge(acirculararc)andnibetheinwardunitnormaltoitsithedge(seeFigure3(b)).Then,m=åi12qini(8)wherem,themeanvector,istheintegraloftheoutwardunitnor-malsoverT.Proof:ConsiderthesolidtriangularwedgeoftheunitspherewithcapT.Theintegralofoutwardunitnormalsoveraclosedsur-faceisalwaysexactlyzero[Fleming1977,p.342].Thus,wecanpartitiontheintegralintothreetriangularfaceswhoseoutwardnor-malsareniwithassociatedareas12qi.Thetheoremfollowssincemåi12qiniisthenzero.?Notethatasimilarresultholdsin2D,wherethemeanvectormdenedbyEquation3.1foracirculararcEontheunitcirclecanbeinterpretedasthesumofthetwoinwardunitnormalsofthevectorspiv(seeFigure3(a)).In3D,thelengthsqioftheedgesofthesphericaltriangleTaretheanglesbetweenthevectorspi1vandpi+1vwhiletheunitnormalsniareformedbytakingthecross -n-n-n-n-n(a)(b)Figure3:MeanvectormonacirculararcEwithedgenormalsni(a)andonasphericaltriangleTwitharclengthsqiandfacenormalsni.productofpi1vandpi+1v.Giventhemeanvectorm,wenowcomputetheweightswiusingEquation7(butwithoutdoingthematrixinversion)viawi=ni
mni
(piv)(9)Atthispoint,weshouldnotethatprojectingatriangleTontoSvmayreverseitsorientation.Toguaranteelinearprecision,thesefolded-backtrianglesshouldproducenegativeweightswi.IfwemaintainapositiveorientationfortheverticesofeverytriangleT,themeanvectorcomputedusingEquation8pointstowardsthepro-jectedsphericaltriangleTwhenThasapositiveorientationandawayfromTwhenThasanegativeorientation.Thus,theresultingweightshavetheappropriatesign.3.3RobustmeanvalueinterpolationThediscussionintheprevioussectionyieldsasimpleevaluationmethodformeanvalueinterpolationontriangularmeshes.Givenpointvandaclosedmesh,foreachtriangleTinthemeshwithverticesfp1;p2;p3gandassociatedvaluesff1;f2;f3g,1.ComputethemeanvectormviaEquation82.ComputetheweightswiusingEquation93.Updatethedenominatorandnumeratorofˆf[v]denedinEquation6respectivelybyaddingåiwiandåiwifiTocorrectlycomputeˆf[v]usingtheaboveprocedure,however,wemustovercometwoobstacles.First,theweightswicomputedbyEquation9mayhaveazerodenominatorwhenthepointvliesonplanecontainingthefaceT.Ourmethodmusthandlethisdegener-atecasegracefully.Second,wemustbecarefultoavoidnumericalinstabilitywhencomputingwifortriangleTwithasmallprojectedarea.Suchtrianglesarethedominanttypewhenevaluatingmeanvaluecoordinatesonmesheswithlargenumberoftriangles.Nextwediscussoursolutionstothesetwoproblemsandpresentthecom-pleteevaluationalgorithmaspseudo-codeinFigure4.Stability:WhenthetriangleThassmallprojectedareaontheunitspherecenteredatv,computingweightsusingEquation8and9becomesnumericallyunstableduetocancellingofunitnormalsnithatarealmostco-planar.Tothisend,wenextderiveastableformulaforcomputingweightswi.First,wesubstituteEquation8intoEquation9,usingtrigonometryweobtainwi=qicos[yi+1]qi1cos[yi1]qi+12sin[yi+1]sin[qi1]jpkivj;(10)//Robustevaluationonatriangularmeshforeachvertexpjwithvaluesfjdj kpjxkifdjereturnfjuj (pjx)=djtotalF 0totalW 0foreachtrianglewithverticesp1;p2;p3andvaluesf1;f2;f3li kui+1ui1k//fori=1;2;3qi 2arcsin[li=2]h (åqi)=2ifphe//xliesont,use2Dbarycentriccoordinateswi sin[qi]di1di+1return(åwifi)=(åwi)ci (2sin[h]sin[hqi])=(sin[qi+1]sin[qi1])1si sign[det[u1;u2;u3]]p1ci2if9i;jsije//xliesoutsidetonthesameplane,ignoretcontinuewi (qici+1qi1ci1qi+1)=(disin[qi+1]si1)totalF+=åwifitotalW+=åwifx totalF=totalWFigure4:Meanvaluecoordinatesonatriangularmeshwhereyi(i=1;2;3)denotestheanglesinthesphericaltrian-gleT.Notethattheyiarethedihedralanglesbetweenthefaceswithnormalsni1andni+1.WeillustratetheanglesyiandqiinFigure3(b).Tocalculatethecosoftheyiwithoutcomputingunitnormals,weapplythehalf-angleformulaforsphericaltriangles[Beyer1987],cos[yi]=2sin[h]sin[hqi]sin[qi+1]sin[qi1]1;(11)whereh=(q1+q2+q3)=2.SubstitutingEquation11into10,weobtainaformulaforcomputingwithatonlyinvolveslengthspivandanglesqi.Inthepseudo-codefromFig-ure4,anglesqiarecomputedusingarcsin,whichisstableforsmallangles.Co-planarcases:ObservethatEquation9involvesdivisionbyni
(piv),whichbecomeszerowhenthepointvliesonplanecontainingthefaceT.Hereweneedtoconsidertwodifferentcases.IfvliesontheplaneinsideT,thecontinuityofmeanvalueinterpolationimpliesthatˆf[v]convergestothevaluef[x]denedbylinearinterpolationofthefionT.Ontheotherhand,ifvliesontheplaneoutsideT,theweightswibecomezeroastheirintegraldenitionRfi[x]jp[x]vjdTbe-comeszero.WecaneasilytestfortherstcasebecausethesumSiqi=2pforpointsinsideofT.Totestforthesecondcase,weuseEquation11togenerateastablecomputationforsin[yi].Usingthisdenition,vliesontheplaneoutsideTifanyofthedihedralanglesyi(orsin[yi])arezero.4ApplicationsandresultsWhilemeanvaluecoordinatesndtheirmainuseinboundaryvalueinterpolation,thesecoordinatescanbeappliedtoavarietyofappli-cations.Inthissection,webrieydiscussseveraloftheseapplica-tionsincludingconstructingvolumetrictexturesandsurfacedefor-mation.Weconcludewithasectiononourimplementationofthesecoordinatesandprovideevaluationtimesforvariousshapes. Figure5:Originalmodelofacow(top-left)withhuevaluesspec-iedatthevertices.Theplanarcutsillustratetheinteriorofthefunctiongeneratedby3Dmeanvaluecoordinates.4.1BoundaryvalueinterpolationAsmentionedinSection1,thesecoordinatefunctionsmaybeusedtoperformboundaryvalueinterpolationfortriangularmeshes.Inthiscase,functionvaluesareassociatedwiththeverticesofthemesh.Thefunctionconstructedbyourmethodissmooth,interpo-latesthosevertexvaluesandisalinearfunctiononthefacesofthetriangles.Figure5showsanexampleofinterpolatinghuespeciedonthesurfaceofacow.Inthetop-leftistheoriginalmodelthatservesasinputintoouralgorithm.Therestofthegureshowssev-eralslicesofthecowmodel,whichrevealthevolumetricfunctionproducedbyourcoordinates.Noticethatthefunctionissmoothontheinteriorandinterpolatesthecolorsonthesurfaceofthecow.4.2VolumetrictexturesThesecoordinatefunctionsalsohaveapplicationstovolumetrictexturingaswell.Figure6(top-left)illustratesamodelofabunnywitha2Dtextureappliedtothesurface.Usingthetexturecoordi-nates(ui;vi)asthefiforeachvertex,weapplyourcoordinatesandbuildafunctionthatinterpolatesthetexturecoordinatesspeciedattheverticesandalongthepolygonsofthemesh.Ourfunctionextrapolatesthesesurfacevaluestotheinterioroftheshapetocon-structavolumetrictexture.Figure6showsseveralslicesrevealingthevolumetrictexturewithin.4.3SurfaceDeformationSurfacedeformationisoneapplicationofmeanvaluecoordinatesthatdependsonthelinearprecisionpropertyoutlinedinSection2.Inthisapplication,wearegiventwoshapes:amodelandacontrolmesh.Foreachvertexvinthemodel,werstcomputeitsmeanvalueweightfunctionswjwithrespecttoeachvertexpjintheundeformedcontrolmesh.Toperformthedeformation,wemovetheverticesofthecontrolmeshtoinducethedeformationontheoriginalsurface.Letˆpjbethepositionsoftheverticesfromthedeformedcontrolmesh,thenthenewvertexpositionˆvinthede-formedmodeliscomputedasˆv=åjwjˆpjåjwj:Noticethat,duetolinearprecision,ifˆpj=pj,thenˆv=v.Figures1and7showseveralexamplesofdeformationsgeneratedwiththisFigure6:Texturedbunny(top-left).Cutsofthebunnytoexposethevolumetrictextureconstructedfromthesurfacetexture.process.Figure1(a)depictsahorsebeforedeformationandthesurroundingcontrolmeshshowninblack.Movingtheverticesofthecontrolmeshgeneratesthesmoothdeformationsofthehorseshownin(b,c,d).Previousdeformationtechniquessuchasfreeformdeforma-tions[SederbergandParry1986;MacCrackenandJoy1996]re-quirevolumetriccellstobespeciedontheinteriorofthecontrolmesh.Thedeformationsproducedbythesemethodsaredepen-dentonhowthecontrolmeshisdecomposedintovolumetriccells.Furthermore,manyofthesetechniquesrestricttheusertocreatingcontrolmesheswithquadrilateralfaces.Incontrast,ourdeformationtechniqueallowstheartisttospec-ifyanarbitraryclosedtriangularsurfaceasthecontrolmeshanddoesnotrequirevolumetriccellstospantheinterior.Ourtech-niquealsogeneratessmooth,realisticlookingdeformationsevenwithasmallnumberofcontrolpointsandisquitefast.Generatingthemeanvaluecoordinatesforgure1took3:3sand1:9sforg-ure7.However,eachofthedeformationsonlytook0:09sand0:03srespectively,whichisfastenoughtoapplythesedeformationsinreal-time.4.4ImplementationOurimplementationfollowsthepseudo-codefromFigure4veryclosely.However,tospeedupcomputations,itishelpfultopre-computeasmuchinformationaspossible.Figure8containsthenumberofevaluationspersecondforvar-iousmodelssampledona3GHzIntelPentium4computer.Previ-ously,practicalapplicationsinvolvingbarycentriccoordinateshavebeenrestrictedto2Dpolygonscontainingaverysmallnumberoflinesegments.Inthispaper,forthersttime,barycentriccoor-dinateshavebeenappliedtotrulylargeshapes(ontheorderof100;000polygons).Thecoordinatecomputationisaglobalcom-putationandallverticesofthesurfacemustbeusedtoevaluatethefunctionatasinglepoint.However,muchofthetimespentisdeterminingwhetherornotapointliesontheplaneofoneofthetrianglesinthemeshand,ifso,whetherornotthatpointisinsidethattriangle.Thoughwehavenotdoneso,usingvariousspatialpartitioningdatastructurestoreducethenumberoftrianglesthat Figure7:Originalmodelandsurroundingcontrolmeshshowninblack(top-left).Deformingthecontrolmeshgeneratessmoothde-formationsoftheunderlyingmodel.ModelTrisVertsEval/sHorsecontrolmesh(g1)985116281Armadillocontrolmesh(g7)2161117644Cow(g5)58042903328Bunny(g6)696303481720Figure8:Numberofevaluationspersecondforvariousmodels.mustbecheckedforcoplanaritycouldgreatlyenhancethespeedoftheevaluation.5ConclusionsandFutureWorkMeanvaluecoordinatesareasimple,butpowerfulmethodforcre-atingfunctionsthatinterpolatevaluesassignedtotheverticesofaclosedmesh.Perhapsthemostintriguingfeatureofmeanvalueco-ordinatesisthatfactthattheyarewell-denedonboththeinteriorandtheexteriorofthemesh.Inparticular,meanvaluecoordinatesdoareasonablejobofextrapolatingvalueoutsideofthemesh.Weintendtoexploreapplicationsofthisfeatureinfuturework.AnotherinterestingpointistherelationshipbetweenmeanvaluecoordinatesandWachspresscoordinates.In2D,bothcoordinatefunctionsareidenticalforconvexpolygonsinscribedintheunitcir-cle.Asaresult,onemethodforcomputingmeanvaluecoordinatesistoprojecttheverticesoftheclosedpolygonontoacircleandcomputeWachspresscoordinatesfortheinscribedpolygon.How-ever,in3D,thisapproachfails.Inparticular,inscribingtheverticesofatriangularmeshontoaspheredoesnotnecessarilyyieldacon-vexpolyhedron.Eveniftheinscribedpolyhedronhappenstobeconvex,theresultingWachspresscoordinatesarerationalfunctionsofthevertexpositionvwhilethemeanvaluecoordinatesaretran-scendentalfunctionsofv.Finally,weonlyconsidermeshesthathavetriangularfaces.Oneimportantgeneralizationwouldbetoderivemeanvaluecoordinatesforpiecewiselinearmeshwitharbitraryclosedpolygonsasfaces.Onthesefaces,thecoordinateswoulddegeneratetostandard2Dmeanvaluecoordinates.Weplantoaddressthistopicinafuturepaper.AcknowledgementsWe'dliketothankJohnMorrisforhishelpwithdesigningthecon-trolmeshesforthedeformations.ThisworkwassupportedbyNSFgrantITR-0205671.ReferencesBEYER,W.H.1987.CRCStandardMathematicalTables(28thEdition).CRCPress.COQUILLART,S.1990.Extendedfree-formdeformation:asculpturingtoolfor3dge-ometricmodeling.InSIGGRAPH'90:Proceedingsofthe17thannualconferenceonComputergraphicsandinteractivetechniques,ACMPress,187–196.DESBRUN,M.,MEYER,M.,ANDALLIEZ,P.2002.IntrinsicParameterizationsofSurfaceMeshes.ComputerGraphicsForum21,3,209–218.FLEMING,W.,Ed.1977.FunctionsofSeveralVariables.Secondedition.Springer-Verlag.FLOATER,M.S.,ANDHORMANN,K.2005.Surfaceparameterization:atutorialandsurvey.InAdvancesinMultiresolutionforGeometricModelling,N.A.Dodgson,M.S.Floater,andM.A.Sabin,Eds.,MathematicsandVisualization.Springer,Berlin,Heidelberg,157–186.FLOATER,M.S.,KOS,G.,ANDREIMERS,M.2005.Meanvaluecoordinatesin3d.ToappearinCAGD.FLOATER,M.1997.Parametrizationandsmoothapproximationofsurfacetriangula-tions.CAGD14,3,231–250.FLOATER,M.1998.ParametricTilingsandScatteredDataApproximation.Interna-tionalJournalofShapeModeling4,165–182.FLOATER,M.S.2003.Meanvaluecoordinates.Comput.AidedGeom.Des.20,1,19–27.HORMANN,K.,ANDGREINER,G.2000.MIPS-AnEfcientGlobalParametrizationMethod.InCurvesandSurfacesProceedings(SaintMalo,France),152–163.HORMANN,K.2004.Barycentriccoordinatesforarbitrarypolygonsintheplane.Tech.rep.,ClausthalUniversityofTechnology,September.http://www.in.tu-clausthal.de/hormann/papers/barycentric.pdf.KHODAKOVSKY,A.,LITKE,N.,ANDSCHROEDER,P.2003.Globallysmoothpa-rameterizationswithlowdistortion.ACMTrans.Graph.22,3,350–357.KOBAYASHI,K.G.,ANDOOTSUBO,K.2003.t-ffd:free-formdeformationbyusingtriangularmesh.InSM'03:ProceedingsoftheeighthACMsymposiumonSolidmodelingandapplications,ACMPress,226–234.LOOP,C.,ANDDEROSE,T.1989.AmultisidedgeneralizationofB´eziersurfaces.ACMTransactionsonGraphics8,204–234.MACCRACKEN,R.,ANDJOY,K.I.1996.Free-formdeformationswithlatticesofarbitrarytopology.InSIGGRAPH'96:Proceedingsofthe23rdannualconferenceonComputergraphicsandinteractivetechniques,ACMPress,181–188.MALSCH,E.,ANDDASGUPTA,G.2003.Algebraicconstructionofsmoothinter-polantsonpolygonaldomains.InProceedingsofthe5thInternationalMathemat-icaSymposium.MEYER,M.,LEE,H.,BARR,A.,ANDDESBRUN,M.2002.GeneralizedBarycentricCoordinatesforIrregularPolygons.JournalofGraphicsTools7,1,13–22.SCHREINER,J.,ASIRVATHAM,A.,PRAUN,E.,ANDHOPPE,H.2004.Inter-surfacemapping.ACMTrans.Graph.23,3,870–877.SEDERBERG,T.W.,ANDPARRY,S.R.1986.Free-formdeformationofsolidgeo-metricmodels.InSIGGRAPH'86:Proceedingsofthe13thannualconferenceonComputergraphicsandinteractivetechniques,ACMPress,151–160.WACHSPRESS,E.1975.ARationalFiniteElementBasis.AcademicPress,NewYork.WARREN,J.,SCHAEFER,S.,HIRANI,A.,ANDDESBRUN,M.2004.Barycentriccoordinatesforconvexsets.Tech.rep.,RiceUniversity.WARREN,J.1996.BarycentricCoordinatesforConvexPolytopes.AdvancesinComputationalMathematics6,97–108.