Several deformations generated using our 3D mean value coordinates applied to a modi64257ed control mesh bcd Abstract Constructing a function that interpolates a set of values de 64257ned at vertices of a mesh is a fundamental operation in computer ID: 23320
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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-ometryandObjectModelingBoundaryrepresentations;Curve,surface,solid,andobjectrepresentations;Geometricalgorithms,languages,andsystemsKeywords:barycentriccoordinates,meanvaluecoordinates,vol-umetrictextures,surfacedeformation1IntroductionGivenaclosedmesh,acommonproblemincomputergraphicsistoextendafunctiondenedattheverticesofthemeshtoitsinterior.Forexample,Gouraudshadingcomputesintensitiesattheverticesofatriangleandextendstheseintensitiestotheinteriorusinglinearinterpolation.Givenatrianglewithverticesfp1;p2;p3gandasso-ciatedintensitiesff1;f2;f3g,theintensityatpointvontheinteriorofthetrianglecanbeexpressedintheformf[v]=åjwjfjåjwj(1)wherewjistheareaofthetrianglefv;pj 1;pj+1g.Inthisformula,notethateachweightwjisnormalizedbythesumoftheweights,åjwjtoformanassociatedcoordinatewjåjwj.Theinterpolantf[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=tanhaj 12i+tanhaj2ijpj vj:(2)whereajistheangleformedbythevectorpj vandpj+1 v.Normalizingeachweightfunctionwjbythesumofallweightfunc-tionsyieldsthemeanvaluecoordinatesofvwithrespecttopj.Inhisoriginalpaper,Floaterprimarilyintendedthisinterpolanttobeusedformeshparameterizationandonlyexploredthebehav-ioroftheinterpolantonpointsinthekernelofastar-shapedpoly-gon.Inthisregion,meanvaluecoordinatesarealwaysnon-negativeandreproducelinearfunctions.Subsequently,Hormann[Hormann2004]showedthat,foranysimplepolygon(ornestedsetofsim-plepolygons),theinterpolantf[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,ourproblemistoconstructafunctionf[v]wherev2R3thatinterpolatesf[x]onP,i.e.;f[p[x]]=f[x]forallx.OurbasicconstructionextendsanideaofFloaterdevelopedduringtheconstructionof2Dmeanvaluecoordinates.Toconstructf[v],weprojectapointp[x]ofPontotheunitsphereSvcenteredatv.Next,weweightthepoint'sassociatedvaluef[x]by1jp[x] vjandintegratethisweightedfunctionoverSv.Toensureafneinvarianceoftheresultinginterpolant,wedividetheresultbytheintegraloftheweightfunction1jp[x] vjtakenoverSv.Puttingthepiecestogether,themeanvalueinterpolanthastheformf[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,f[v]con-vergestof[x].Smoothness:Thefunctionf[v]iswell-denedandsmoothforallvnotonP.Linearprecision:Iff[x]=p[x]forallx,theinterpolantf[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]=(1 x)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(pi v)=m:(5)wherem=Rxp[x] vjp[x] vjdEissimplytheintegraloftheoutwardunitnormaloverthecirculararcE.WecallmthemeanvectorofE,asscalingmbythelengthofthearcyieldsthecentroidofthecirculararcE.Basedon2Dtrigonometry,mhasasimpleexpressionintermsofp1andp2.Specically,1ToevaluatetheintegralofEquation3,wecanrelatethedifferentialdSvtodxviadSv=p?[x]:(p[x] v)jp[x] vj2dxwherep?[x]isthecrossproductofthen 1tangentvectors¶p[x]¶xitoPatp[x].NotethatthesignofthisexpressioncorrectlycaptureswhetherPhasfoldedbackduringitsprojectionontoSv.m=tan[a=2]((p1 v)jp1 vj+(p2 v)jp2 vj)whereadenotestheanglebetweenp1 vandp2 v.Henceweob-tainwi=tan[a=2]=pi vwhichagreeswiththeFloater'sweight-ingfunctiondenedinEquation2for2Dmeanvaluecoordinateswhenrestrictedtoasingleedgeofapolygon.Equation4allowsustoformulateaclosedformexpressionfortheinterpolantf[v]inEquation3bysummingtheintegralsforalledgesEkinP(notethatweaddtheindexkforenumerationofedges):f[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=mfp1 v;p2 v;p3 vg 1(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-malsare niwithassociatedareas12qi.Thetheoremfollowssincem åi12qiniisthenzero.?Notethatasimilarresultholdsin2D,wherethemeanvectormdenedbyEquation3.1foracirculararcEontheunitcirclecanbeinterpretedasthesumofthetwoinwardunitnormalsofthevectorspi v(seeFigure3(a)).In3D,thelengthsqioftheedgesofthesphericaltriangleTaretheanglesbetweenthevectorspi 1 vandpi+1 vwhiletheunitnormalsniareformedbytakingthecross -n-n-n-n-n(a)(b)Figure3:MeanvectormonacirculararcEwithedgenormalsni(a)andonasphericaltriangleTwitharclengthsqiandfacenormalsni.productofpi 1 vandpi+1 v.Giventhemeanvectorm,wenowcomputetheweightswiusingEquation7(butwithoutdoingthematrixinversion)viawi=nimni(pi v)(9)Atthispoint,weshouldnotethatprojectingatriangleTontoSvmayreverseitsorientation.Toguaranteelinearprecision,thesefolded-backtrianglesshouldproducenegativeweightswi.IfwemaintainapositiveorientationfortheverticesofeverytriangleT,themeanvectorcomputedusingEquation8pointstowardsthepro-jectedsphericaltriangleTwhenThasapositiveorientationandawayfromTwhenThasanegativeorientation.Thus,theresultingweightshavetheappropriatesign.3.3RobustmeanvalueinterpolationThediscussionintheprevioussectionyieldsasimpleevaluationmethodformeanvalueinterpolationontriangularmeshes.Givenpointvandaclosedmesh,foreachtriangleTinthemeshwithverticesfp1;p2;p3gandassociatedvaluesff1;f2;f3g,1.ComputethemeanvectormviaEquation82.ComputetheweightswiusingEquation93.Updatethedenominatorandnumeratoroff[v]denedinEquation6respectivelybyaddingåiwiandåiwifiTocorrectlycomputef[v]usingtheaboveprocedure,however,wemustovercometwoobstacles.First,theweightswicomputedbyEquation9mayhaveazerodenominatorwhenthepointvliesonplanecontainingthefaceT.Ourmethodmusthandlethisdegener-atecasegracefully.Second,wemustbecarefultoavoidnumericalinstabilitywhencomputingwifortriangleTwithasmallprojectedarea.Suchtrianglesarethedominanttypewhenevaluatingmeanvaluecoordinatesonmesheswithlargenumberoftriangles.Nextwediscussoursolutionstothesetwoproblemsandpresentthecom-pleteevaluationalgorithmaspseudo-codeinFigure4.Stability:WhenthetriangleThassmallprojectedareaontheunitspherecenteredatv,computingweightsusingEquation8and9becomesnumericallyunstableduetocancellingofunitnormalsnithatarealmostco-planar.Tothisend,wenextderiveastableformulaforcomputingweightswi.First,wesubstituteEquation8intoEquation9,usingtrigonometryweobtainwi=qi cos[yi+1]qi 1 cos[yi 1]qi+12sin[yi+1]sin[qi 1]jpki vj;(10)//Robustevaluationonatriangularmeshforeachvertexpjwithvaluesfjdj kpj xkifdjereturnfjuj (pj x)=djtotalF 0totalW 0foreachtrianglewithverticesp1;p2;p3andvaluesf1;f2;f3li kui+1 ui 1k//fori=1;2;3qi 2arcsin[li=2]h (åqi)=2ifp he//xliesont,use2Dbarycentriccoordinateswi sin[qi]di 1di+1return(åwifi)=(åwi)ci (2sin[h]sin[h qi])=(sin[qi+1]sin[qi 1]) 1si sign[det[u1;u2;u3]]p1 ci2if9i;jsije//xliesoutsidetonthesameplane,ignoretcontinuewi (qi ci+1qi 1 ci 1qi+1)=(disin[qi+1]si 1)totalF+=åwifitotalW+=åwifx totalF=totalWFigure4:Meanvaluecoordinatesonatriangularmeshwhereyi(i=1;2;3)denotestheanglesinthesphericaltrian-gleT.Notethattheyiarethedihedralanglesbetweenthefaceswithnormalsni 1andni+1.WeillustratetheanglesyiandqiinFigure3(b).Tocalculatethecosoftheyiwithoutcomputingunitnormals,weapplythehalf-angleformulaforsphericaltriangles[Beyer1987],cos[yi]=2sin[h]sin[h qi]sin[qi+1]sin[qi 1] 1;(11)whereh=(q1+q2+q3)=2.SubstitutingEquation11into10,weobtainaformulaforcomputingwithatonlyinvolveslengthspi vandanglesqi.Inthepseudo-codefromFig-ure4,anglesqiarecomputedusingarcsin,whichisstableforsmallangles.Co-planarcases:ObservethatEquation9involvesdivisionbyni(pi v),whichbecomeszerowhenthepointvliesonplanecontainingthefaceT.Hereweneedtoconsidertwodifferentcases.IfvliesontheplaneinsideT,thecontinuityofmeanvalueinterpolationimpliesthatf[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.Letpjbethepositionsoftheverticesfromthedeformedcontrolmesh,thenthenewvertexpositionvinthede-formedmodeliscomputedasv=åjwjpjåjwj:Noticethat,duetolinearprecision,ifpj=pj,thenv=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,187196.DESBRUN,M.,MEYER,M.,ANDALLIEZ,P.2002.IntrinsicParameterizationsofSurfaceMeshes.ComputerGraphicsForum21,3,209218.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,157186.FLOATER,M.S.,KOS,G.,ANDREIMERS,M.2005.Meanvaluecoordinatesin3d.ToappearinCAGD.FLOATER,M.1997.Parametrizationandsmoothapproximationofsurfacetriangula-tions.CAGD14,3,231250.FLOATER,M.1998.ParametricTilingsandScatteredDataApproximation.Interna-tionalJournalofShapeModeling4,165182.FLOATER,M.S.2003.Meanvaluecoordinates.Comput.AidedGeom.Des.20,1,1927.HORMANN,K.,ANDGREINER,G.2000.MIPS-AnEfcientGlobalParametrizationMethod.InCurvesandSurfacesProceedings(SaintMalo,France),152163.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.22,3,350357.KOBAYASHI,K.G.,ANDOOTSUBO,K.2003.t-ffd:free-formdeformationbyusingtriangularmesh.InSM'03:ProceedingsoftheeighthACMsymposiumonSolidmodelingandapplications,ACMPress,226234.LOOP,C.,ANDDEROSE,T.1989.AmultisidedgeneralizationofB´eziersurfaces.ACMTransactionsonGraphics8,204234.MACCRACKEN,R.,ANDJOY,K.I.1996.Free-formdeformationswithlatticesofarbitrarytopology.InSIGGRAPH'96:Proceedingsofthe23rdannualconferenceonComputergraphicsandinteractivetechniques,ACMPress,181188.MALSCH,E.,ANDDASGUPTA,G.2003.Algebraicconstructionofsmoothinter-polantsonpolygonaldomains.InProceedingsofthe5thInternationalMathemat-icaSymposium.MEYER,M.,LEE,H.,BARR,A.,ANDDESBRUN,M.2002.GeneralizedBarycentricCoordinatesforIrregularPolygons.JournalofGraphicsTools7,1,1322.SCHREINER,J.,ASIRVATHAM,A.,PRAUN,E.,ANDHOPPE,H.2004.Inter-surfacemapping.ACMTrans.Graph.23,3,870877.SEDERBERG,T.W.,ANDPARRY,S.R.1986.Free-formdeformationofsolidgeo-metricmodels.InSIGGRAPH'86:Proceedingsofthe13thannualconferenceonComputergraphicsandinteractivetechniques,ACMPress,151160.WACHSPRESS,E.1975.ARationalFiniteElementBasis.AcademicPress,NewYork.WARREN,J.,SCHAEFER,S.,HIRANI,A.,ANDDESBRUN,M.2004.Barycentriccoordinatesforconvexsets.Tech.rep.,RiceUniversity.WARREN,J.1996.BarycentricCoordinatesforConvexPolytopes.AdvancesinComputationalMathematics6,97108.