ComputerAidedGeometricDesign16(1999)691–700 - PDF document

ComputerAidedGeometricDesign16(1999)691–700 DiscreteCoonspatchesGeraldFarin,DianneHansford Abstract G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–700.u;v/;v/;v//1�uu]x.0;0/x.0;1/x.1;0/x.1;1/1�vv:(1)Fromnowon,wewillrefertothismethodastheCoonspatchWhiletheboundarycurves;v/;;v/maybetotallyarbitrary,intheearlydaystheboundarypolygonswerediscretizedcurveswithmanypointsonthem.AmoremodernuseforCAD/CAMwouldbetotreattheboundarypolygonsasBéziercontrolpolygonsofanarrayofpointsi;j;:::;m;j;:::;n.Aconguration3lookslikethis:i;jisassociatedwithaparameterpair.u;v/.i=m;j=n/,theGrevilleabscissae(Farin,1996).Theinteriori;jaredenedbythediscreteversionof(1),whichwewillcallthediscreteCoonspatchi;ji=m/i=mm;jj=n/j=ni;nn1�i=mi=mm;nj=nj=nfor0and0.Fig.1illustrates.InterpretingtheboundarypolygonsasBézierpolygons,theresultingCoonspatchwouldthenbethecontrolpolygonofaBéziersurfacethatadherestothegivenboundaryinformation.In(Farin,1992)itisshownthat,infact,theBéziercontrolpointsdenethesamepatchasifCoonswasappliedtothetransniteboundarycurves. Fig.1.DiscreteCoonspatches:anexample.Thegivenboundaryverticesaremarkeddark;thecomputedinterioronesareshowninalightercolor. G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–700 Fig.2.Permanencepatches:an“optimal”controlnetfor257.2.TheminimumprincipleCoonspatchesminimizethetwistinthesensethatisminimalexactlyfortheCoonspatch,withtheintegralbeingtakenovertheunitsquareSeeNielsonetal.(1978).Asurface.u;v/minimizingthisvariationalprinciplesatisestheEuler–LagrangePDEuuvvTheCoonspatchisknowntoproducelessthandesirableshapesinmanycases.ItappearsthattheafnityoftheCoonspatchwithzerotwistsaccountsfortheseshapedefects.ConsideringFig.1,itisclearthatthediscreteCoonspatchistooat—a“good”surfacewouldlookliketheoneinFig.2.Ifweapplytheminimumprinciple(3)tothediscreteCoonspatch,wehavethati;jisminimalifthei;jformadiscreteCoonspatch,wherei;ji;ji;j “Good”herereferstothetraditionaldesigner'sparadigmthattheinteriorofasurfaceshouldnothavedifferentshapecharacteristicsthanthoseimpliedbytheboundarycurves.ClearlythehorizontalstraightlineontopoftheCoonspatchfromFig.1isnotimpliedbythecircle-shapedboundarypolygons. G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–7003.ThepermanenceprincipleTheCoonspatchsatisesapermanenceprinciple:lettwopointsdenearectangleinthedomainoftheCoonspatch.ThefourboundariesofthissubpatchwillmaptofourcurvesontheCoonspatch.OnemayaskwhattheCoonspatchtothosefourboundarycurvesis.Theanswer:theoriginalCoonspatch,restrictedtothe.Infact,allschemeswhoseconstructionssatisfyavariationalprinciplesharethispermanenceprincipleproperty.Onecanapplythisprincipletoa33gridfromthediscreteCoonspatch,suchasi;ji;ji;jIfthecontrolpointsoftheboundaryofthis33gridareknown,thenasaconsequenceofthepermanenceprinciple,theinteriorpointcouldbedeterminedbyi;j 4.bi�11CbiC11Cbi�11CbiC11/C1 i;ji;jAneaterwayofwritingthisisusingamaski;j ThismaskisindeedthediscreteformoftheEuler–LagrangePDE(4).ThediscreteCoonspatchhasvertices;ofthese,areunknown.Eq.(5)givesoneequationforeachunknown.ThuswemayndthediscreteCoonspatchasthesolutionofalinearsystemwithequationsinasmanyunknowns.Intheinteriorofthepatch,theequationsjustrelatetheunknownstoeachother;neartheboundaries,theyrelatethemtotheknownsandunknowns.Ofcourse,thisisaveryexpensivewaytocomputethediscreteCoonspatch;yetitofferssomenewinsights,and,moreimportantly,someimprovements.ThelinearsystemforthediscreteCoonspatchemploysamaskoftheformi;j25and5.ThissuggeststhepossibilityofdifferentchoicesforNotethatwealwaysneed41inorderfor(6)toutilizebarycentric(orafne)combinations.Byallowingothervaluesof,weobtainanewclassofcontrolnetgenerationschemes—wecallthesepermanencepatchesIf,forthedatausedinFig.1,weuse257,andsolvetheresultinglinearsystem,weobtainFig.2.Theresultingshapeismuchclosertoany“designer'sintent”:thegiven G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–700 Fig.3.Permanencepatches:a“minimal”controlnetforpolygonscamefromatorus-likeshape,andnowwerecapturethatshape.NotethattheoriginaldiscreteCoonspatchfromFig.1failedmiserablyinthisshapesense.Ifweselect0,weobtainFig.3.ThismaskisthediscreteformoftheLaplacePDEandhencetheresultingnetverymuchresemblesaminimalsurfacetbetweenthegivenboundarypolygons.Themaskin(6)is,infact,ablendoftheEuler–LagrangeandLaplaceequationsi;j 010010Anasymmetricmaskmaybedesiredif.ThisgeneralizationcorrespondstoanasymmetricLaplacemaskaboutthedirections.Formoreliteratureonvariationalprinciplesforfairsurfaceconstruction,see(Greiner,1994;Kobbelt,1997).4.MoreonpermanencepatchesApointonaCoonspatchdependsoneightpointsonly—Coonspatchesarelocalthatsense.Ontheotherhand,apointonapermanencepatch(for6D�25)dependsonallboundarypointsandisthereforeglobal.Webelievethisaccountsforthepotentiallyimprovedshapes.Sincethediscretepartialderivativesaredependentupon,soistheaffectofHowever,aninterestingobservationisthatforagiven,asinglechoiceofnotalwaysproduce“good”shape.Theappropriatevaluedependsonthegeometryoftheboundarycurves.Fig.4illustrates.Inotherwords,nogeometry-independentcombinationofEuler–LagrangeandLaplacemaskswillbesufcientforallgeometries.Itisnotclearwhetheranasymmetricmaskisnecessarytoachieve“good”shape. Thevalueforthisexamplewasfoundempirically. G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–700 Fig.4.Permanencepatchesfortwo33nets.Toachievethedesiredshape,thetorus-likedatasetontheleftrequires75,whereasthe“tent”ontherightrequires25(Coons).Permanencepatcheshavebilinearprecisioninthefollowingsense.Letthefourcornerpointsdetermineabilinearpatch.PlacetheremainingedgeBézierpointsequallyspacedalongtheedges.Theresultingpermanencepatchpointsreproducethebilinearpatch.Toseethis,substitutethelinearexpressionsfortheedgepointsrelativetothecornerpointsintothemaskin(6),andthemiddleBézierpointtakestheformi;j/:Themaskwasconstructedtopreservebarycentriccombinations,so4.Sincethisisapermanencepatch,thisbilinearpropertywillholdforarbitrary5.TriangularpermanencepatchesThecontrolnetofatriangularBézierpatchisapiecewiselinearsurface(see(Farin,1986)).Onemaythenask:giventhreeboundarycontrolpolygons,whatisa“good”controlnettotinbetweenthem?Varioustriangularmethods(BarnhillandGregory,1975;Nielsonetal.,1978,1979;Nielson,1980;Perronnet,1997)maybeemployedhere,butapermanenceprinciplecanalsobeestablishedbyutilizingamaskoftheformwith31.WewillcallthepatchesformedwiththismasktriangularpermanencepatchesGeneralizingrectangularpatches,oneinterpretationofthismaskisasfollows.ForcubicBézierpatchestherearenineinterioredges,andassociatedwitheachedgeisaquadrilateral.Formnineequationsfor,requiringthequadrilateralstobeascloseaspossibletoparallelograms.Theleastsquaressolutionforresultsinthemask(8)with9.AnexampleisillustratedinFig.5.Aswedidforrectangularpatches,letusconsiderthisproblemintermsofdiscretizedPDEs.AsnotedinNielsonetal.(1978),fourthorderpartialsarenotappropriate.Additionally,therstorderderivativesyieldanasymmetricmask.Theonlyderivativesweneedtoconsiderareofsecondorder.TheseareillustratedinFig.7.Asnotedabove, G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–700 Fig.5.Triangularpermanencepatch:Coonsgeneralizationwith Fig.6.Triangularpermanencepatch:Coonsgeneralizationwith Fig.7.Aschematicdescriptionofthesecondorderpartialsofatriangularpatch.Left,secondorderderivativesfortheLaplacemask.Right,thequadrilateralsforthecornertwistminimizingmask.therearenine“twistequations”;wedifferentiatethemasthethreecornertwistsandthesixinteriortwists.Duetothecyclicnatureoftheinteriortwists,themaskthattheseequationsgenerateisidenticaltothatoftheLaplacemask.Therefore,themask(8)canbedescribedasablendofacornertwistminimizingmaskandaLaplacemask: Ourtwist-minimizingmask(9)whichwederivedasthesolutiontoaleastsquaresproblem,isactuallyablendofthePDEmaskswith3.Thepatchcorrespondingto3,or1,whichissimplythecornertwistminimizer.ItisillustratedinFig.6. G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–700 Fig.8.Triangularpermanencepatch:a“minimal”surfacewithWethusseethatthetriangularcasediffersfromtherectangularoneinthatnowthesolutiontotheleastsquaressystemisnotthesameasthattothecornertwistminimizer.5.1.QuadraticprecisionFor6,wecreateaquadraticprecisionconguration.Thetriangularperma-nencepatchesresultingfrom6enjoyaquadraticprecisionpropertyinthefollow-ingsense:Letbeaquadratic(discrete)patch.Degreeelevateittoanarbitrarydegreeresultinginapatch.Thenapplythepermanenceconstructiontotheboundarycurves,resultinginapatch.WeclaimthatForaproof,observethatisthecontrolnetofaquadraticBézierpatchwhichwasdegreeelevatedtodegree.Thusallthirdderivativesofthispatcharezero.Thecoefcientsofathirdderivativeareobtainedbyaveragingall“cubic”subnetsof.Ifallthoseaveragesaretovanish,theneachofthesubnetshadtobe“quadratic”itself,i.e.,ithadtosatisfyarelationshipoftheform(8)with6.Thisispreciselywhatthepermanenceconstructionyieldswhenappliedtotheboundariesof,thusprovingourclaim.5.2.MoreontriangularpermanencepatchesFor0,weobtainthediscreteLaplacemask,resultinginsurfaceswhichareclosetominimalinthesenseofdifferentialgeometry.Fig.8illustrates.Forallvalues,triangularpermanencepatcheshavelinearprecisioninthesensethatifthegivenedgecontrolpointsareequallyspacedalongtheedgesthenlinearfunctionsarereproduced.Theproofiscompletelyanalogoustotheoneforrectangularpatches.Justaswiththerectangularpermanencepatch,therearevalueswhichproducesingularities.Additionally,thebehaviorofthepatchcorrespondingtoaparticularvalueisdependentonthedegreeofthepatch.Anautomaticmethodfordeterminingthe“optimal”isunderdevelopment.InFig.9,anvaluehasbeenselectedthatcertainlyproducesanicershapethanthatofFig.5. AnothermaskwhichhasquadraticprecisionwasgivenbyBarron(1988),howeverthismaskisnotsymmetric. G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–700 Fig.9.Triangularpermanencepatch:an“optimal”6.ConclusionsWereformulatedthediscreteCoonspatchandgeneralizedittopermanencepatches,bothfortherectangularandthetriangularcases.ThegeneralizationallowedustoproduceshapeswhicharemoredesirablethanthestandardCoonsshapes.ThesemethodswerealsodescribedintermsofdiscretePDEs.InFig.4,weillustrated(forrectangularpatches)thatnosingleblendofEuler–LagrangeandLaplacevariationalprinciplescanproduce“good”shapeforallboundarycurvegeometries.Wethusconjecturethatonehastoemployshapedescriptorsthatadapttothegivenboundaryinformationinsteadofusingarigidblendofvariationalprinciples.Onealsoneedsconditionsontheshapeparameterstoensuresolvabilityofthelinearsystem.AcknowledgementsWeappreciatethecommentsbytherefereesregardingtherelevanceofthediscretizedPDEapproach.ReferencesBarnhill,R.(1982),Coons'patches,ComputersinIndustry3,37–43.Barnhill,R.andGregory,J.(1975),Polynomialinterpolationtoboundarydataontriangles,Math.Computation29(131),726–735.Barron,P.(1988),Thegeneralformulaofthequadraticallyprecisenine-parameterinterpolant,TechnicalReport,UniversityofUtah,manuscript.Coons,S.(1964),Surfacesforcomputeraideddesign,TechnicalReport,MIT.AvailableasAD663504fromtheNationalTechnicalInformationservice,Springeld,VA22161.Farin,G.(1986),TriangularBernstein–Bézierpatches,ComputerAidedGeometricDesign3(2),Farin,G.(1992),CommutativityofCoonsandtensorproductoperators,RockyMtn.J.Math.22(2), G.Farin,D.Hansford/ComputerAidedGeometricDesign16(1999)691–700Farin,G.(1996),CurvesandSurfacesforComputerAidedGeometricDesign,4thed.,AcademicPress.Gordon,W.(1969),Free-formsurfaceinterpolationthroughcurvenetworks,TechnicalReportGMR-921,GeneralMotorsResearchLaboratories.Greiner,G.(1994),Variationaldesignandfairingofsplinesurfaces,ComputerGraphicsForum13,Kobbelt,L.(1997),Discretefairing,in:Goodman,T.andMartin,R.,eds.,TheMathematicsofSurfacesVII,Winchester,InformationGeometers,101–131.Nielson,G.(1980),Minimumnorminterpolationintriangles,SIAMJ.Numer.Analysis17(1),46–Nielson,G.,Thomas,D.andWixom,J.(1978),Boundarydatainterpolationontriangulardomains,TechnicalReportGMR-2834,GeneralMotorsResearchLaboratories.Nielson,G.,Thomas,D.andWixom,J.(1979),Interpolationintriangles,Bull.Austral.Math.Society20,115–130.Perronnet,A.(1997),Triangle,tetrahedron,pentahedrontransniteinterpolations.Applicationstothegenerationof-continuousalgebraicmeshes,in:Proc.6thInternationalConferenceonNumericalGridGenerationinComputationalFieldSimulation,AveryHillCampus,Greenwich,England,467–476.