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1、GraphicalModels68(2006)237–254www.elsevier.com/locate/gmodManifoldsplinesXianfengGu,YingHe*,HongQinCenterforVisualComputing(CVC),DepartmentofComputerScience,StonyBrookUniversity,StonyBrook,NY11794-4400,USAReceived21February2005;receivedinrevisedform1December2005;a
2、ccepted6March2006AbstractConstructingsplineswhoseparametricdomainisanarbitrarymanifoldandeffectivelycomputingsuchsplinesinreal-worldapplicationsareoffundamentalimportanceinsolidandshapemodeling,geometricdesign,graphics,etc.Thispaperpresentsageneraltheoreticalandcom
3、putationalframework,inwhichsplinesurfacesdefinedoverplanardomainscanbesystematicallyextendedtomanifolddomainswitharbitrarytopologywithorwithoutboundaries.Westudytheaffinestructureofdomainmanifoldsindepthandprovethattheexistenceofmanifoldsplinesisequivalenttotheexiste
4、nceofamanifold’saffineatlas.Basedonourtheoreticalbreakthrough,wealsodevelopasetofpracticalalgorithmstogen-eralizetriangularB-splinesurfacesfromplanardomainstomanifolddomains.WechoosetriangularB-splinesmainlybecauseofitsgeneralityandmanyofitsattractiveproperties.Asar
5、esult,ournewsplinesurfacedefinedoveranymanifoldisapiecewisepolynomialsurfacewithhighparametriccontinuitywithouttheneedforanypatchingand/ortrimmingoper-ations.Throughourexperiments,wehopetodemonstratethatournovelmanifoldsplinesarebothpowerfulandefficientinmodelingarbi
6、trarilycomplicatedgeometryandrepresentingcontinuouslyvaryingphysicalquantitiesdefinedovershapesofarbitrarytopology.Ó2006ElsevierInc.Allrightsreserved.Keywords:Geometricmodeling;Manifoldspline;Riemannsurface;Conformalstructure;Affineatlas1.Introductionandmotivationpas
7、t.Despitetheirmodelingadvantagesforarbi-trarilycomplicatedgeometryandtopology,subdivi-Real-worldvolumetricobjectsareoftentimesofsionsurfaceshavetwodrawbacks:(1)accuratecomplexgeometryandarbitrarytopology.Onefun-surfaceevaluationisfrequentlyconductedviadamentalgoal
8、ofsolidandphysicalmodelingistoexplicit,recursivesubdivisionsincemostsubdivi-seekaccurateandeffectivetechniquesforthecom-sionschemes(especiallythoseinterp
