DynamicNURBSSwungSurfacesforPh ysicsBased ShapeDesign HongQin and DemetriT erzopoulos - PDF document

DynamicNURBSSwungSurfacesforPhShapeDesignHongQinDemetriTerzopoulostofComputerScience,UnivyofT10King'sCollegeRoad,Tto,Ontario,M5S1A4PublishedindDesign(2):111{127,1995.edevelopadynamic,free-formsurfacemodelwhichisusefulforrepresentingabroadclassofobjectswithsymmetriesandtopologicalv.ThenewmodelisbaseduponswungNURBSsurfaces,anditinheritstheirdesirablecross-sectionaldesignproperties.Itmeldsthesegeometricfeatureswiththedemonstratedconeniencesofsurfacedesignwithinaphysics-basedframework.WedemonstratesevapplicationsofdynamicNURBSswungsurfaces,includinginesculptingthroughtheimpositionofforcesandtheadjustmentofphysicalparameterssuchasmass,damping,andelasticit.Additionalapplicationsincludesurfacedesignwithgeometricandphysicalconstraints,byroundingsolids,andthroughthettingofunstructureddata.WederivetheequationsofmotionforthedynamicNURBSswungsurfacemodelusingLagrangianmechanicsofanelasticsurfaceandtheniteelementmethod.ealsoshowthatthesesurfacesareaspecialcaseofD-NURBSsurfaces,arecentlyproposedphbasedgeneralizationofstandardgeometricNURBS.Ourfree-form,rationalmodelnotonlyprovidesasystematicanduniedapproachforavyofCAGDproblemssuchasconstraint-basedoptimization,ariationaldesign,automaticwtselection,shapeapproximation,etc.,butitalsosupportsinsculptingusingphysics-basedmanipulationtools.GD,NURBS,NURBSSwinging,DeformableModels,Dynamics,Constraints,Finitets,SolidRounding,SurfaceBlending,ScatteredDataFitting,IneSculpting. w,CanadianInstituteforAdvancedResearcE-mailaddresses:qin@cs.toronto.edu;dt@cs.toron PublishedindDesign(2):111{127,1995. Figure1:ConstructionofacubicalNURBSswungsurface.(a)NURBSprolecurveonx-zplane,NURBStrajectorycurveonx-yplane.(b)Cubesurfacewireframe.troductionAmongthesurfacerepresentationschemesinCAGD,non-uniformrationalB-splines(NURBS)haebe-comeanindustrystandard[1].Oneoftheirmostsignicantadvtagesisthattheyareauniedrep-tationofbothcomplexfree-formshapesandstandardanalyticshapes[1,2,3].NURBSobjectsaredesignedbyadjustingcontrolpointsandwtsthatareassociatedwithNURBSsurfacepatcyobjectsofinterest,especiallymanufacturedobjects,exhibitsymmetries.Oftenitiscontomodelsymmetricobjectsthroughcross-sectionaldesignbyspecifyingprolecurves[4].oodww]introducedtheswingingoperatorbyextendingthesphericalcross-productwithascalingfactor,andappliedittogeneratesurfaceswithB-splineprolecurves(seealso[6]).Piegl[1]carriedtheswingingideaertoNURBScurves.HeproposedNURBSswungsurfaces,aspecialtypeofNURBSsurfacesformedyswingingoneplanarNURBSprolecurvealongasecondNURBStrajectorycurve.Thetogeneratoresmaybesmooth,ortheymayhaedisconuities.Forexample,Fig.1illustratesthedesignofacubicalNURBSswungsurfacefromtoNURBSprolecurvTheNURBSswungsurfaceretainsaconsiderablebreadthofgeometriccoItcanrepresencommongeometricprimitivessuchasspheres,tori,cubes,quadrics,surfacesofrevolution,etc.Fig.4illustratesfourNURBSswungsurfaceswithdistincttopologicalstructures.TheNURBSswungsurfaceistcomparedtoageneralNURBSsurface,inasmhasitcanrepresentabroadclassofshapeswithtiallyasfewdegreesoffreedomasittakestospecifythetogeneratorcurves.Severalgeometricshapedesignsystems,includingtherecentonein[7],includesomeformofswinging(orsweeping)amongtheirrepertoireoftecInthispaper,wedevelopaphysics-basedgeneralizationofthegeometricNURBSswungsurface.WrefertoournewmodelsasdynamicNURBSswungsurfacAlthoughplanarcurvedesignismheasierthangeneralsurfacedesign,inmanyreal-worldcircumstancesitishardtoacesatisfactoryresultsquic.Normallythedesignerobtains(quasi-global)controlo PublishedindDesign(2):111{127,1995.thefree-formNURBSswungsurfacebyadjustingthecontrolpointsandwtsofthetoNURBScurvAlthoughthereareavyofalgorithmsandinteractiondevices,thisindirectdesignprocess,whichisharacteristicofgeometricdesignwithNURBSandotherfree-formsurfacerepresentationsingeneral,canbeclumsyandtimeconsuming.Moreoer,relevtdesignrequirementsareusuallyshapeorientedandnotcontrolpointandwtoriented.Becauseofthegeometric\redundancy"ofNURBS,traditionalgeometricshaperenementremainsadhoandambiguous.Inparticular,makingmeaningfuluseofwisoneofthemostimportantareasofcurrentNURBSresearch.Furthermore,typicaldesignrequiremenybeposedinbothquaneandqualitativeterms.Itcanbeveryfrustratingwithindirectdesignto,forexample,shapea\fair"surfacethatapproximatesunstructured3Ddata.Unstructuredshapetsareespeciallyproblematicforcross-sectionaldesign.ysics-basedmodelingtechniquesprovideameansofoercomingthesediculties.Itispossibletoconstructfree-formsurfaceswithnaturaldynamicbehaviorgoernedbyphysicallaws[8,9,10].TzopoulosandFleischer[11]demonstratedsimpleinesculptingusingviscoelasticandplasticmodels.erandGossard[12]developedaninterestingprototypesystemforinefree-formdesignbasedonthisideaandthenite-elementoptimizationofenergyfunctionals.BloorandWilson[13]usedsimilarenergiesoptimizedthroughnumericalmethodsandtheyemploedB-splinesforthispurpose.SubsequenhandWitkin[14]extendedtheapproachtotrimmedhierarchicalB-splines.ThingvoldandCohen[15proposedahybriddeformableB-splinewhosecontrolpointsaremasspointsconnectedbyelasticspringsandhinges.MoretonandSequin[16]interpolatedaminimumenergycurvenetorkwithquinticBezierhesbyminimizingthevariationofcurvature.Halsteadetal.[17]implementedsmoothinterpolationwithCatmull-Clarksurfacesusingathin-plateenergyfunctional[18ThedynamicNURBSswungsurfacesproposedinthepresentpaperwereinspiredbyD-NURBS,ourysics-basedgeneralizationofstandardgeometricNURBS[19].LikeD-NURBS,dynamicNURBSswungsurfaceshaeconuousmassanddampingdistributions,aswellasadeformationenergy.Withproperhoiceofphysicalparameters,theybehaelikephysicalsurfaces.Thisallowsadesignertoinsculptanddirectlymanipulateshapesinanaturalandpredictablewyusingavyofforce-basedtools.ThesurfacesinFig.4wereinelysculptedinthisfashionfromtheprototypeshapesindicatedinthecaption.AnimportantadvtageofadynamicmodelbuiltuponthestandardNURBSgeometricfoundationisthatshapedesignmayproceedinelyorautomaticallyatthephysicallevel,whileexistinggeometrictoolkitsareconcurrentlyapplicableatthegeometriclevInthephysics-basedapproach,functionaldesignrequirementscanbereadilyimplementedassurfacedeformation(fairness)energiesandasgeometricconstraintsonthesurface.Asadynamicsurfacereacequilibrium,itactsasanonlinearshapeoptimizersinceitminimizesitsenergysubjecttotheimposeder,shapedesignisgenerallyatime-varyingprocess|adesignerisofteninnotonlyinthenalequilibriumshapebutalsointheintermediateshapevariationduetoparameterhanges.Timeisfundamentaltothephysics-basedformulation.Physics-basedmodelsaregoernedbdynamicdierentialequationswhich,whenintegratednumericallythroughtime,canconuouslyevtrolpointsandwtsinresponsetoappliedforcestoproducephysicallymeaningfulandinpredictableshapevariation.Akeyadvtageofourphysics-basedframeworkisthatitpermitsappropriatealuestobedeterminedautomaticallyinaccordancewithvariousphysicalparametersandgeometriceuseLagrangianmechanicstoformulatetheequationsofmotionofdynamicNURBSswungsur-faces,andniteelementanalysistoreducetheseequationstoecientalgorithmsthatcanbesimumericallyusingstandardtechniques.OneofthechallengesinthiseortiscopingwiththenonlineardynamicformulationstemmingfromtheunderlyingswungNURBSgeometry.TheNURBSswungsurfaceisinherentlynonlinearwithrespecttoitsdegreesoffreedom,evenifbothNURBSgeneratorcurvesarereducedtosimpleB-splinesbyxingtheirwtstounitAsaconsequenceofthenonlinearit,the AparticularshapecanoftenberepresentednonuniquelyusingNURBS,withdierenaluesofknots,controlpoints,and PublishedindDesign(2):111{127,1995.mass,damping,andstinessmatricesinthedynamicformulationmustberecomputedateachsimtimestep.Section2deneskinematicversionsofthebasicNURBScurvegeneratorsintheswungsurfaceandgivthekinematicequations.InSection3,weformulatethedynamicNURBSswungsurfaceandderivetheirequationsofmotion.ediscussthenumericalsimulationoftheseequationsinSection4.Section5discussestheuseofforcesandconstraintsforphysics-baseddesign.Section6presentsapplicationsofdynamicNURBSswungsurfacestoinesculpting,scattereddatatting,androunding/blendinganddiscussestheresults.InSection7weshowthatdynamicNURBSswungsurfacesareaspecialcaseofD-NURBSsurfaces[19]thathaebeensubjectedtoadimensionality-reducingnonlinearconstrainSection8concludesthepaper.KinematicNURBSCurvAkinematicNURBScurveextendsthegeometricNURBSdenitionbyexplicitlyincorporatingtime.Thekinematiccurveisafunctionofboththeparametricvandtimeu;t wherethe)aretheusualrecursivelydenedpiecewiserationalbasisfunctions[20,21)arethe+1controlpoints,and)areassociatednon-negativts.Assumingbasisfunctionsofdegree1,thecurvehas+1knotsinnon-decreasingsequence:Inmanapplications,theendknotsarerepeatedwithminordertointerpolatetheinitialandnaltrolpoinosimplifynotation,wedenethevectorofgeneralizedcoordinates)andw)asdenotestransposition.Wethenexpressthecurve(1)as)inordertoemphasizeitsdepen-denceonwhosecomponentsarefunctionsoftime.Thevelocityofthekinematicsplineiswheretheokdotdenotesatimederiveand)istheJacobianmatrix.Becauseisa3-componenaluedfunctionandisan4(+1)dimensionalvisa3+1)matrix,hisexpressedas @p00@cy @p00@cz @p775@c 775(3)where@cx @pi;x=@cy @pi;y=@cz @pi;z=wiBi;k Pnj=0wjBj;k;@c =Pnjpipj)wjBi;kBj;k Thesubscript,anddenotethecomponentofa3-vector.Furthermore,wecanexpressthecurveas PublishedindDesign(2):111{127,1995.theproductoftheJacobianmatrixandthegeneralizedcoordinatevTheproofof(4)canbefoundin[19DynamicNURBSSwungSurfaceInthissection,weformulatetheunderlyinggeometryofthedynamicswungsurfacesandderivetheJacobianandbasisfunctionmatricesthatleadtosuccinctexpressionsanalogousto(2)and(4)fortheelocityandpositionfunctionsofthesurface,respectiv.Thisallowsustoderiveequationsofmotionforthedynamicswungsurfaceincludingmass,damping,anddeformationenergydistributions.,adynamicswungsurfaceisgeneratedfromtoplanarkinematicNURBSprolecurvthroughtheswingingoperation[1](Fig1).Letthetogeneratorcurv)and)beoftheform(1).Theswungsurfaceisthendenedasu;v;tisanarbitraryscalar.Thesecondsubscriptdenotesthecomponentofa3-vAssumethathasbasisfunctionsofdegree1andthatithas+1controlpoin)and).Similarlyhasbasisfunctionsofdegree1andthatithas+1controlpoinandw).Therefore,Therefore,a�0;wa0;:::;;b�0;wb0;:::;arethegeneralizedcoordinatevectorsoftheprolecurves.WecollecttheseintothegeneralizedcoordinateThisvectorhasdimensionalit=1+4(+1)+4(+1).Thusthemodelhas)degreesoffreedom,comparedto)forgeneralNURBSsurfaces.JacobianMatrixDenotingtheJacobianmatricesofthetoprolecurvesas)and),thecurvepositionandelocityfunctionstaketheformof(2)and(4):isa3+1)matrix,andisa3+1)matrix.Bothareoftheform(3).IfweexpresseachroectoroftheJacobianmatricesexplicitlyasecanwritetheblockforms: PublishedindDesign(2):111{127,1995.Theswungsurfaceisthereforewrittenasu;v;Thevelocityoftheswungsurfaceisu;v;u;v;)istheJacobianmatrixwithrespecttothegeneralizedcoordinatevcomprisesthev=@,and,whicharegivenasfollo 264(X1a)(X2b)(X1a)(Y2b)0375=C)c1whereA(64X1a000X1a0000375;B(64X2b00Y2b00001375;C=264000000001375;@s ;v;+(1;and ;u;.Hence,weexpresstheJacobianmatrixasNotethat,andare33matrices.Therefore,isa3visa3+1)matrix,andisa3+1)matrix.Thisa3BasisFunctionMatrixin(4),cannotalsoserveasthebasisfunctionmatrixoftheswungsurface.LetItisstraighardtoverifythatu;v;uswehau;v; PublishedindDesign(2):111{127,1995.isthe3basisfunctionmatrix.EquationsofMotionTheequationsofmotionofourdynamicNURBSswungsurfacearederivedfromthework-energyversionofLagrangiandynamics[22].ToproceedwiththeLagrangianformulation,weexpressthekineticenergyduetoaprescribedmassdistributionfunctionu;vertheparametricdomainofthesurfaceandaRaleighdissipationenergyduetoadampingdensityfunctionu;v).Todeneanelasticpotentialenergyadoptthethin-plateundertensionenergymodelwhicasproposedin[18]andalsousedin[12,14,17,19 21;1@s� s 2;2@s� s 1;1@2s� @2s +1;2@2s� @u@v @u@v @2s dudv:u;v)andu;v)areelasticityfunctionswhichcontroltensionandrigidit,respectiv,intheoparametriccoordinatedirections.Otherenergiesareapplicable,includingthenonquadratic,curvbasedenergiesin[8,16ApplyingtheLagrangianformulation,weobtainthesecond-ordernonlinearequationsofmotionwherethemassmatrixisdudv;thedampingmatrixisdudv;andthestinessmatrixisdudv(thesubscriptsondenoteparametricpartialderivmatrices.Thegeneralizedforce,obtainedthroughtheprincipleofvirtualwork[22]donebytheappliedforcedistributionu;v;t)isu;v;tdudv:Becauseofthegeometricnonlinearit,generalizedinertialforcesdudvarealsoassociatedwiththemodels.Thederivationoftheequationsofmotion(12)proceedsinthesamemannerasforD-NURBS(see[19]forthedetails).NumericalSimTheevolutionof,determinedby(12)withtime-varyingmatrices,cannotbesolvedanalyticallyingeneral.Instead,wepursueanecienumericalimplementationusingnite-elementtechniques[23 PublishedindDesign(2):111{127,1995.Standardniteelementcodesexplicitlyassembletheglobalmatricesthatappearinthediscreteequa-tionsofmotion[23].Weuseaniterativematrixsolvertoaoidthecostofassemblingtheglobal,and.Inthisworkwiththeindividualelementmatricesandconstructniteelementdatastructuresthatpermittheparallelcomputationofelementmatrices.MatrixStructureandComputationeexaminethemassanddampingmatrices.Bothmatricesinetheintegrationofinthepara-metricdomainwhereisgivenin(8).Basedon(8),isdecomposedintothefollowingblockmatrices:,andSeeAppendixAforthedetailsaboutthestinessmatrix.tDataStructuresedeneanelementdatastructurewhichcontainsthegeometricspecicationofthesurfacepatchel-talongwithitsphysicalproperties.Acompletedynamicswungsurfaceisthenimplementedasadatastructurewhichconsistsofanorderedarrayofelementswithadditionalinformation.Theelemenstructureincludespointerstoappropriatecomponentsoftheglobalvtrolpointsandwboringelementswillsharesomegeneralizedcoordinates.Thesharedvariableswillhapointersimpingingonthem.Wealsoallocateineachelementanelementalmass,damping,andstinessmatrix,andincludeintheelementdatastructurethequantitiesneededtocomputethesematrices.Thesetitiesincludethemassu;v),dampingu;v),andelasticitu;vu;v)densityfunctions,hmayberepresentedasanalyticfunctionsorasparametricarraysofsamplevCalculationofElementMatricesTheintegralexpressionsforthemass,damping,andstinessmatricesassociatedwitheachelementarealuatednumericallyusingGaussianquadrature[24].Weshallexplainthecomputationoftheelemenmassmatrix;thecomputationofthedampingandstinessmatricesfollowsuit.Assumingtheparametricdomainoftheelementis[s[v0;v1],theexpressionforenofthemassmatrixtakestheinu;vu;vdudareentriesofthematrixin(13).GiveninecanndGausswandabscissasinthetoparametricdirectionssuchthatcanbeapproximatedby[24 PublishedindDesign(2):111{127,1995.eapplythedeBooralgorithm[25]toev).Ingeneral,Gaussianquadratureevaluatesthetegralexactlywithtsandabscissasforpolynomialsofdegree21orless.Inoursystemwhoosetobeintegersbeteen4and7.Ourexperimentsindicatethatmatricescomputedinthiswyleadtostable,contsolutions.Notethatinthecasewherethemass,damping,andstinesspropertiesareuniformoerthesurfaceand,therefore,reducetoscalarquantities,thedoublesumintheGaussianintegrationformuladecomposestotheproductoftoindependentsumsoereachoftheunivariatedomainsofthegeneratorcurvesanditbecomesmhmoreecienDiscreteDynamicsEquationsointegrate(12)inaninemodelingent,itisimportanttoprovidethemodelerwithvisualfeedbackabouttheevolvingstateofthedynamicmodel.Ratherthanusingcostlytimeinmethodsthattakethelargestpossibletimesteps,itismorecrucialtoprovideasmoothanimationbtainingtheconyofthedynamicsfromonesteptothenext.Hence,lesscostlyyetstabletimetegrationmethodsthattakemodesttimestepsaredesirable.ThestateofthedynamicNURBSswungsurfaceattimeisintegratedusingpriorstatesattimeomaintainthestabilityoftheintegrationscheme,weuseanimplicittimeinmethod,whichemploysdiscretederivesofusingbacarddierenceseobtainthetimeintegrationform+2
=2
)+4wherethesuperscriptsdenoteevaluationofthequantitiesattheindicatedtimes.Thematricesandforcesareevaluatedattimeeemploytheconjugategradientmethodtoobtainaniterativesolution[24].Toaceinulationrates,welimitthenberofconjugategradientiterationspertimestepto10.Wehaeobservthat2iterationstypicallysucetoconergetoaresidualoflessthan10.Morethan2iterationstendtobenecessarywhenthephysicalparameters(mass,damping,tension,stiness,appliedforces)arectlyduringdynamicsimulation.Hence,ourimplementationpermitsthereal-timesimulationofdynamicswungsurfacesoncommongraphicsworkstations.Quadraticandcubicsurfaceswithmorethan200constrainedcontrolpointscanbesimulatedatinerates.Theequationsofmotionallowrealisticdynamicssuchaswouldbedesirableforphysics-basedcomputergraphicsanimation.Itispossible,hoer,tomakesimplicationsthatfurtherreducethecomputationalcostof(14)toinelysculptlargersurfaces.Forexample,inCAGDapplicationssuchasdatattingwherethemodelerisinterestedonlyinthenalequilibriumcongurationofthemodel,itmakessensetosimplify(12)bysettingthemassdensityfunctionu;v)tozero,sothattheinertialtermsvanish.Thiseconomizesonstorageandmakesthealgorithmmoreecient.Withzeromassdensit,(12)reducestotherst-ordersystemDiscretizingthederivesofin(15)withbacarddierences,weobtaintheintegrationform PublishedindDesign(2):111{127,1995.ysics-BasedShapeDesignInthephysics-basedshapedesignapproach,designrequirementsmaybesatisedthroughtheuseofenergies,forces,andconstraints.Thedesignermayapplytime-varyingforcestosculptshapesinortooptimallyapproximatedata.Certainaestheticconstraintssuchas\fairness"areexpressibleintermsofelasticenergiesthatgiverisetospecicstinessmatricesOtherconstraintsincludepositionornormalspecicationatsurfacepoints,andconyrequirementsbeteenadjacentsurfacepatches.BybuildingthedynamicswungsurfaceuponthestandardgeometryoftheNURBSswungsurface,weallothemodelertoconuetousethewholespectrumofadvancedgeometricdesigntoolsthathaebecomet,amongthem,theimpositionofgeometricconstraintsthatthenalshapemustsatisfyAppliedFSculptingtoolsmaybeimplementedasappliedforces.Theforceu;v;t)representstheneteectofallappliedforces.ypicalforcefunctionsarespringforces,repulsionforces,gravitationalforces,in ationforces,etc.[8].orexample,considerconnectingamaterialpoin)ofadynamicswungsurfacetoapoinspacewithanidealHookeanspringofstiness.Thenetappliedspringforceisu;v;tu;v;tdudv;wheretheistheunitdeltafunction.Equation(17)impliesthat))andanisheselsewhereonthesurface,butwecangeneralizeitbyreplacingthefunctionwithasmoothk(e.g.,aunitGaussian)tospreadtheappliedforceoeragreaterportionofthesurface.Furthermore,thepoints()andneednotbeconstant,ingeneral.Wecancontroleitherorbothusingamousetoobtainaninespringforce.Inpracticalapplications,designrequirementsmaybeposedasasetofphysicalparametersorasgeometricts.NonlinearconstraintscanbeenforcedthroughLagrangemultipliertechniques[26,27,28].Thishincreasesthenberofdegreesoffreedom,hencethecomputationalcost,byaddingunknownasLagrangemultipliers,whichdeterminethemagnitudesoftheconstraintforces.TheaugmenLagrangianmethod[26]combinestheLagrangemultiplierswiththesimplerpenaltymethod[9].Baumgartestabilizationmethod[29]solvesconstrainedequationsofmotionthroughlinearfeedbackcon(seealso[10,19]).ThesetechniquesareappropriateforthedynamicswungsurfaceswithconstrainLineargeometricconstraintssuchaspoint,curve,andsurfacenormalconstraintscanbeeasilyincorpo-ratedintodynamicswungsurfacebyreducingthematricesandvectorsin(12)toaminimalunconstrainedsetofgeneralizedcoordinates.Forexample,thetogeneratorcurvesmustbeembeddedinplanes,respectiv.Ifthemodelisconnedasasurfaceofrevolution,thedegreesoffreedomassociatedwiththesecondprolesmustbeconstrainedgeometricallytoadmitacircle.Linearconstraintscanbetedbyapplyingthesamenumericalsolveronanunconstrainedsubsetof.See[19]foradetaileddiscussiononconstraintsinthecontextofD-NURBS.DynamicsurfacesconstructedfromNURBSgeometryhaeaninterestingidiosyncrasyduetothets.Whilethecontrolpointcomponentsofytakearbitrarynitevaluesin,negativycausethedenominatortovanishatsomeevaluationpoints,causingthematricestodiverge.Althoughnotforbidden,negativtsarenotuseful.Weenforcepositivityofwtsateachsimulationtimestepbysimplyprojectinganaluethathasdriftedbelowasmallpositivethresholdbacktothiserbound(nominally0.1).Anotherpotentialdicultyisthatloerwaluestendto attenthesurfaceinthevicinityofthecontrolpoints,loeringthedeformationenergy;thusthewtsmaytendtodecrease.Onesolutionistouseamorecomplexdeformationenergythatdoesnotfaor atsurfacesas PublishedindDesign(2):111{127,1995.in[16].Alternativecancounteractthetendencyandalsogivethedesignertheoptionofconstrainingthewtsnearcertaindesiredtargetvyincludinginthesurfaceenergythepenaltyterm),wheretrolsthetightnessoftheconstrainApplicationsandResultsehaedevelopedaprototypemodelingsystembasedondynamicNURBSswungsurfaces.Curren,thesystemimplementssurfaceswithbasisfunctionsoforder2,3,or4(i.e.,fromlineartocubic)andgeometricts.ThesystemiswritteninCandispacagedasanineIrisExplorermoduleonSiliconGraphicsworkstations.ItmaybecombinedwithexistingExplorermodulesfordatainputandsurfacevisualization.OurparallelizediterativumericalalgorithmtakesadvtageofanSGIIris4D/380Vultiprocessor.Userscansculptsurfaceshapesincontionalgeometricwys,suchasbyskhingcontrolpolygonsofarbitraryprolecurves,repositioningcontrolpoints,andadjustingassociatedwts,oraccordingtothephysics-basedparadigmthroughtheuseofforces.Theycansatisfydesignrequirementsbyadjustingtheinternalphysicalparameterssuchasthemass,damping,andstinessdensities,alongwithforcegainfactors,inelythroughExplorercontrolpanels.ThefollowingsectionsdemonstrateapplicationsofdynamicNURBSswungsurfacestoroundingandblending,scattereddatatting,andinesculpting.RoundingandBlendingTheroundingandblendingofsurfacesisusuallyattemptedgeometricallyyenforcingconyrequire-tsonthelletwhichinterpolatesbeteentoormoresurfaces.Bycontrast,thedynamicNURBSswungsurfacecanproduceasmoothlletbyminimizingitsinternaldeformationenergysubjecttoposi-tionandnormalconstraints.Thedynamicsimulationautomaticallyproducesthedesirednalshapeasitesequilibrium.Fig.5demonstratestheroundingofapolyhedraltoroid.TheprolecurveontheplaneisaquadraticNURBScurvewith17controlpoints.ThetrajectorycurveontheplaneisalsoaquadraticNURBSwith17controlpoints.Notethat,thecornersofthecurvescanberepresentedexactlywithmultipleconpointsorapproximatelybysettingaverylargewt.IfthismodelwereageneralNURBSsurface,itouldhae289controlpointsandwts.Asaswungsurfaceithasonly34controlpointsandwhareconsideredthegeneralizedcoordinatesofthedynamicmodel.ThewireframeandshadedshapesisshowninFig.5(a)andFig.5(b).Afterinitiatingthephysicalsimulation,thecornersandsharpedgesareroundedasthenalshapeequilibratesintotheminimalenergystateshowninFig.5(c-d).Fig.6illustratestheroundingofacubicalsolid.TheproleisaquadraticNURBSwith15conpoints,andthetrajectoryisalinearNURBSwith5controlpoints.Theroundingoperationisappliedinthevicinityofthemiddleedge.InFig.6(a-b)showsthewireframeandshadedobjects,respectiv.TheroundedshapeisshowninFig.6(c-d).Fig.7showsablendingexampleinolvingacylindricalpipe.Thecircularproleisaquadraticcurvwith7controlpoints.Thepiecewiselineartrajectoryhas5controlpoints.Theinitialright-anglepipeandthenalroundedpipeareshowninFig.7(a-d).odemonstrateautomaticwariationinthedynamicmodelinaccordancewithphysicalparame-ters,wehaeconductedanexperimentwiththepolyhedraltoroidshowninFig.5(a).Wexall137degreesoffreedomofthetoroid,whichoccupiesthecubex;y;z;1,exceptforonewtassociatedwiththecontrolpoint(10)whichisinitiallysetto500toformthecorner.Thiswtispermittedtovarysubjecttovaryingphysicalparametersandexternalforce(Fig.2).Initially=1000andtheremainingparametersarezero(
248).Startingthedynamicsimulation,werstgraduallyincreasefromzeroto1500.AsshowninFig.2,thevalueofthefreewtdecreasesquicklytoardsitserlimit03inunder100iterations(dashedplot).Thisreducesthedeformationenergybyroundingthecorner.Next,weattachaspringforcefromthepoint(10)tothenearestsurfacepoint.As PublishedindDesign(2):111{127,1995. Tension parameter increasing Spring force increasing Rigidity parameter increasing Spring force increasing |0|200|400|600|800|1000|0.0|5.0|10.0|15.0|20.0|25.0|30.0|35.0|40.0|45.0|50.0 Figure2:Freewariationsubjecttovaryingphysicalparametersandexternalforce(seetext).thestinessconstantofthisspringisslowlyincreasedfromzeroto10,thewtincreases(solidplotinFig.2)thusrecreatingacorner.Next,wfromzeroto1000(dottedplotinFig.2).Theincreasedrigidityofthesurfacecausesthewttodecrease,againroundingthecorner.Finallyincreasethespringstinessto100(stippledplotinFig.2),thuscounteractingtheincreasedstinessandcausingthewttoincreasestoagainrecreatethecorner.Thisexperimentdemonstratesthatwhentrolpointsarexed,afreewtwillautomaticallydecreasetodecreasethedeformationenergyb atteningthesurfacelocally,unlessexternalforcescounteractthistendency.Whencontrolpointsarefree,theywillalsovaryaswelltoproduceananalogousphysicaleect.ScatteredDataFittingAusefulmodelingtechniqueisbasedonttingsurfacestounstructuredconstraints,generallyknownasscattereddatatting.Interestingsituationsarisewhentherearefewerormoredatapointsthantherearedegreesoffreedominthemodel,leadingtounderconstrainedoroerconstrainedttingproblems.Theinclusionofanelasticenergyinourdynamicsurfacesmakesthemapplicabletosuchproblems.Thedatainterpolationproblemisamenabletocommonconstrainttechniques[26canbeacedbyphysicallycouplingthedynamicNURBSswungsurfacetothedatathroughHookspringforces(17).Weinin(17)asthedatapoint(generallyin)and()astheparametriccoordinatesassociatedwiththedatapoint(whichmaybethenearestmaterialpointofthesurface).Thespringconstandeterminestheclosenessofttothedatapoinondtheclosestpointonthemodelforarbitrarilysampleddata(),weexploitthespecialsymmetricstructureofNURBSswungsurfacethroughthefollowingto-stepsearchscheme.Werstnd alidationprovidesaprincipledapproachtochoosingtherelevtphysicalparameters|typicallytheratioofdataforcespringconstantstosurfacestinesses|forgivendatasets[30].Forthespecialcaseofzero-meanGaussiandataerrors,optimalapproximationintheleastsquaresresidualsenseresultswhenisproportionaltotheinersevarianceofdataerrors. PublishedindDesign(2):111{127,1995. Weights fixed Weights free (from iteration 131) Weights free (from iteration 0) |0|50|100|150|200|250|300|350|400|0.060|0.065|0.070|0.075|0.080|0.085|0.090|0.095|0.100|0.105|0.110 Figure3:LeastSquaredFittingErrorsForNRCCPot(seetext).hthat)isnearestto().Thenwesearchtheisoparametriccurvu;v)andndthehthat)istheclosestto().Experimentsshowthatthisapproximationapproacleadstosatisfactoryresultsbecausethemappingisrecomputedateachsimulationstep.Moreimportanereducethecomplexityofoptimalmatchingfrom)forageneralD-NURBSsurface[19]to).Forlarge,thedynamicsimulationisspeededupsignican.OthertechniquessucasnonlinearoptimizationareapplicabletondingtheclosestpoinAnimportantadvtageofourmodels,despitethefactthattheyareprolesurfaces,isthattheycanbettedtoarbitrarilydistributedempiricaldatathatarenotalignedalonganyparticularisoparametricepattern.Werstuseadynamicswungsurfacegeneratedboquadraticproleswith10and7trolpointstoreconstructaclaypotwhichhasbeendenselysampledbyacylindricallaserscannertoproduceabout1datapoints.The7controlpointsandwtsofthecross-sectionproleareconstrainedsoastopermitonlysurfacesofcircularcrosssection.erandomlyselectedonly20datapointsfromwhichtoreconstructasurfaceofrevolution.Fig.8showsthesamplepoints,thecylindricalinitialcondition,andthenalttedshaperenderedwiththetexturemapoftheobjectacquiredbytheTheelasticenergyofthesurfaceallowsittointerpolatebeteendatapoinThephparametersusedinthisexperimeneremass0,damping=600,bendingstinessparameter=140whilealltheothersarezero,dataspringconstan=9000.Thesurfacettingstabilizesinafewsecondswithatimestep
ereporttomoreexperimentswiththeaboesurfacettingscenarioinordertoinestigatetheleast-squaresttingerrorsunderdierentcircumstances.Intherstexperiment,weinitiallyxall10tsoftheprolecurve.Anoptimaltisacedafter81iterations.ThedecreasingerrorisplottedythedashedplotinFig.3.Thenwefreethewtsatiteration131andthemodelisabletofurtherreducethettingerrorbecauseitnowhasmoredegreesoffreedomatitsdisposal.Thedottedplotinthegureindicatestheimproedt.Inthesecondexperiment,thewtsaresetfreefromthestartofthedynamicsimulation.After295iterations,theoptimalttingisrecordedbythesolidplotinFig.3.The PublishedindDesign(2):111{127,1995. ationNo. Weights w0 w1 w2 w3 w4 w5 w6 w7 w8 w9 initial 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 80 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.011 1.011 1.096 0.884 0.978 0.998 1.019 0.957 1.023 1.030 1.041 1.260 0.513 0.941 1.031 1.029 0.893 1.065 1.030 1.041 1.260 0.513 0.941 1.031 1.029 0.893 1.065 nal 1.042 1.030 1.041 1.260 0.513 0.941 1.031 1.029 0.893 1.065 able1:Variationofwtsinexperiment1. ationNo. Weights w0 w1 w2 w3 w4 w5 w6 w7 w8 w9 initial 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 80 1.004 1.008 1.004 1.060 0.925 0.997 1.001 1.010 0.977 1.008 1.045 1.059 1.350 0.392 0.785 1.066 1.026 0.864 1.087 1.058 1.076 1.377 0.304 0.623 1.126 1.020 0.845 1.106 1.079 1.098 1.357 0.300 0.587 1.144 1.013 0.826 1.130 nal 1.094 1.079 1.098 1.357 0.300 0.587 1.144 1.013 0.826 1.130 able2:Variationofwtsinexperiment2.aluesinthetoexperimentsaregiveninTable1and2,respectiv.Theexperimentsindicatethat,atleastinthissituation,aslightlybettersurfacetisobtainedwhenthemodelispermittedtousealloftheaailabledegreesoffreedomfromthestartofthettingprocess.Ob,thenalsurfaceintherstexperimenthadattainedalocaloptimNext,weusethesamesurfacemodeltoapproximate10datapointssampledfromavase.ThewireframeandtexturedimagesofthedynamicswungsurfaceareillustratedinFig.9.Fig.10(a-d)showsthenalreconstructedshapesfromfourotherttingexperimentsusingsyntheticdatatorecoeranotherpot,aase,abottle,andawineglass.Thenberofrandomlysampleddataare10,13,14,and17,respectiveSculptingInthephysics-basedmodelingapproach,notonlycanthedesignermanipulatetheindividualdegreesoffreedomwithcontionalgeometricmethods,buthecanalsomoetheobjectorreneitsshapewithesculptingforces.Thephysics-basedmodelingapproachisidealforinesculptingofsurfaces.Itprovidesdirectmanipulationofthedynamicsurfacetorenetheshapeofthesurfacethroughtheapplicationofinsculptingtoolsintheformofforces.Fig.4(a)illustratestheresultsoffourinesculptingsessionsusingspringforces.Aspherewasgeneratedusingtoquadraticcurveswith4and7controlpointsandassculptedintotheooidshowninFig.4(a).Atoruswhosetoprolecurvesarequadraticwith7and7controlpoints,respectiv,hasbeendeformedintotheshapeinFig.4(b).Ahatshapewascreatedfromocurveswith9and6controlpointsandwasthendeformedbyspringforcesintotheshapeinFig4(d).egeneratedawineglassshapeusingtocurveswith7and5controlpointsandsculpteditinthemorepleasingshapeshowninFig4(c).ConstrainedD-NURBSFItisknownthatageometricNURBSswungsurfaceisaNURBSsurface[1].Inthissection,weshowthatdynamicNURBSswungsurfacesare,analogously,D-NURBSsurfaces[19]thathaebeensubjectedtoa PublishedindDesign(2):111{127,1995.y-reducingnonlinearconstrainD-NURBSSurfaceAD-NURBSsurfacegeneralizesthegeometricNURBSsurface:u;v;t The(+1)(+1)controlpoin)andw),whicharefunctionsoftime,comprisetheD-NURBSgeneralizedcoordinates.Weconcatenatethese=4(+1)(+1)coordinatesintothevSimilarto(2)and(4),wehau;v;u;v;u;v;)isthe3JacobianmatrixoftheD-NURBSsurfacewithrespectto.ThemotionequationsofD-NURBSsurfacesarewherethemassmatrix,thedampingmatrix,andthestinessmatrixareallisthegeneralizedforcevectoractingon,andisthegeneralizedinertialforce.See[19]forthedetailsoftheD-NURBSformoreducetheD-NURBSsurfacetoadynamicswungsurface,weapplythenonlinearconstrain,and,for;:::;m;:::;n,aredenedasinSection3.Dieren(21),weobtainUsingthenotationsinSection3,wecanrewrite(21)and(22)inthematrixformmatriceswith=(4+9).Substituting(23)into(20),wearriveattheequationsofmotionforthedynamicNURBSswungsurface(12),wherethemass,damping,andstinessmatricesaregivenbThegeneralizedforceswithrespectto PublishedindDesign(2):111{127,1995.Theconstraintreducesthe4(+1)generalizedcoordinatesoftheD-NURBSsurfacetothe4generalizedcoordinatesofthedynamicNURBSswungsurface.ehaeproposeddynamicNURBSswungsurfaces.LiketheD-NURBSpredecessor[19],thenewmodelisaphysics-basedgeneralizationofitsgeometriccounterpart.WederiveditsystematicallythroughLa-grangianmechanicsandimplementedinesoftareusingconceptsfromniteelementanalysisandumericalmethods.WehaeshownthatthedynamicNURBSswungsurfacemodelcanbefor-ulatedintodierenys:(i)constructivelyfromtoNURBSprolecurvesand(ii)sinceswungsurfacesareNURBSsurfaces,byapplyinganonlinearconstrainttoaD-NURBSsurface.Swungsurfaces)degreesoffreedom.Intherstformulationthesurfaceinherits,byconstruction,thedegreesoffreedomofeachgeneratorcurv)).Inthesecondformulation,thenonlinearconstraineliminatesmostofthe)degreesoffreedomofageneralD-NURBSsurfacetoproducetheswungform,butthesurfacecanreverttoitsgeneralformiftheconstraintisremoThephysics-basedmodelrespondstoappliedsimulatedforceswithnaturalandpredictabledynamicswiththegeometricparameters,includingthecontrolpointsandwts,varyingautomatically.Designerscanemployforce-based\tools"toperformdirectmanipulationandinesculpting.Additionalconertheshapeisaailablethroughthemodicationofphysicalparameters.Elasticenergyfunctionalsallothequalitativeimpositionoffairnesscriteriathroughquanemeans.Linearornonlinearconstrainybeimposedeitherashardconstraintsthatmustnotbeviolated,orassoftconstraintstobesatisedximatelyintheformofsimpleforces.Inparticular,dynamicNURBSswungsurfacesmaybettoverysparse,unstructureddata.Energy-basedshapeoptimizationforsurfacedesignandfairingisanautomaticconsequenceofthedynamicmodelachievingstaticequilibriumsubjecttodataconstrainOurprototypeinemodelingsystemprovidesapromisingapproachtoadvancedsurfacedesignproblemssuchasblendingandcross-sectiondesign.Itdemonstratesthe exibilityofdynamicNURBSswungsurfacesinavyofapplications,includingconstraint-basedoptimizationforsurfacedesignandfairing,automaticsettingsofwtsinsurfacetting,andinesculptingthroughappliedforces.Ourresultsindicatethatthesenon-xed-topologydynamicsurfacesoerbroadgeometriccoerageduringdesign.Sinceourmodelsarebuiltontheindustry-standardNURBSgeometricsubstrate,designerswwiththemcanconuetomakeuseoftheexistingarrayofgeometricdesigntoolkits. PublishedindDesign(2):111{127,1995.StructureofStinessMatrixInaddition,sinceisnotafunctionofehaedecomposetotomatrices.Let dudv dudvSo,inviewof(10),itiseasytovoexaminethestructureofeconsiderwithoutlossofgeneralityonlythesecondcrossetermfor.Theentryistheintegralof(,andNext,wediscuss.Because(,(25)canbesimpliedas dudveconsidertherstderiveenUsingtheforegoingnotations,itiseasytoverifythat,andissymmetric.Also,isobviouslysymmetric.Therefore,issymmetric. PublishedindDesign(2):111{127,1995.egratefullyacwledgethecooperationofDrs.MarcRioux,LucCournoer,andPierreGariepyoftheNationalResearchCouncilofCanadawhoprovidedthegeometryandtexturedataoftheclaypot.ThisassupportedbygrantsfromtheNaturalSciencesandEngineeringResearchCouncilofCanadaandtheInformationThnologyResearchCenterofOn PublishedindDesign(2):111{127,1995.[1]L.Piegl.OnNURBS:AsurvIEEEComputerGraphicsandApplic,11(1):55{71,Jan.1991.[2]L.PieglandW.Tiller.CurveandsurfaceconstructionsusingrationalB-splines.,19(9):485{498,1987.[3]W.Tiller.RationalB-splinesforcurveandsurfacerepresenIEEEComputerGraphicsand,3(6):61{69,Sept.1983.[4]I.D.FauxandM.J.Pratt.ComputationalGeometryforDesignandManufacturEllisHorwood,hester,UK,1979.[5]C.Woodward.Cross-sectionaldesignofB-splinesurfaces.ComputersandGr,11(2):193{201[6]E.Cobb.DesignofSculpturdSurfacesUsingtheB-splineR.PhDthesis,UnivyofUtah,1984.[7]J.SnyderandJ.Kajiya.Generativemodeling:Asymbolicsystemforgeometricmodeling.,26(2):369{378,1992.[8]D.Terzopoulos,J.Platt,A.Barr,andK.Fleischer.Elasticallydeformablemodels.ComputerGr21(4):205{214,1987.[9]J.PlattandA.Barr.Constraintsmethodsfor exiblemodels.ComputerGr,22(4):279{288[10]D.MetaxasandD.Terzopoulos.Dynamicdeformationofsolidprimitiveswithconstrain,26(2):309{312,1992.[11]D.TerzopoulosandK.Fleischer.Deformablemodels.TheVisualComputer,4(6):306{331,1988.[12]G.CelnikerandD.Gossard.Deformablecurveandsurfacenite-elementforfree-formshapedesign.ComputerGr,25(4):257{266,1991.[13]M.I.G.BloorandM.J.Wilson.RepresentingPDEsurfacesintermsofB-splines.,22(6):324{331,1990.[14]W.WhandA.Witkin.Variationalsurfacemodeling.ComputerGr,26(2):157{166,1992.[15]J.A.ThingvoldandE.Cohen.PhysicalmodelingwithB-splinesurfacesforinedesignandComputerGr,24(2):129{137,1990.Proceedings,1990SymposiumonIn3DGraphics.[16]H.P.MoretonandC.H.Sequin.Functionaloptimizationforfairsurfacedesign.ComputerGr26(2):167{176,1992.[17]M.Halstead,M.Kass,andT.DeRose.Ecient,fairinterpolationusingCatmull-Clarksurfaces.InComputerGrProceedings,AnnualConferenceSeries,Proc.ACMSiggraph'93(Anaheim,CA,Aug.,1993),pages35{44,1993.[18]D.Terzopoulos.RegularizationofinersevisualproblemsinolvingdisconIEEETonPatternAnalysisandMachineIntelligenc,8(4):413{424,1986.[19]D.TerzopoulosandH.Qin.DynamicNURBSwithgeometricconstraintsforinesculpting.CMTansactionsonGr,13(2):103{136,1994. PublishedindDesign(2):111{127,1995.[20]G.FCurvesandSurfacesforComputeraidedGeometricDesign:APralGuide.AcademicPress,secondedition,1990.[21]M.E.Mortenson.ometricMo.JohnWileyandSons,1985.[22]B.R.Gossic'sPrincipleandPhysicalSystems.AcademicPress,NewYorkandLondon,[23]H.Kardestuncer.FiniteElementHandb.McGraw{Hill,NewYork,1987.[24]W.Press,B.Flanney,S.T,andW.ValRes:TheArtofScientic.CambridgeUnivyPress,Cambridge,1986.[25]C.deBoor.OncalculatingwithB-Splines.JournalofApproximationThe,6(1):50{62,1972.[26]M.Minoux.alPr.Wiley,NewYork,1986.[27]G.Strang.ductiontoAppliedMathematicsbridgePress,MA,1986.[28]J.Platt.AgeneralizationofdynamicconstrainGIP:GralModelsandImagePr54(6):516{525,1992.[29]J.Baumgarte.Stabilizationofconstraintsandintegralsofmotionindynamicalsystems.Comp.Meth.inAppl.Mech.andEng.,1:1{16,1972.[30]G.WSplineModelsforObservationalData.SIAM,Philadelphia,PA,1990. PublishedindDesign(2):111{127,1995. (a)(b) Figure4:AssortedDynamicNURBSSwungSurfaces.Openandclosedsurfacesshownweresculptedelyfromprototypeshapesnotedinparentheses(a)Eggshape(sphere).(b)Deformedtoroid(torus).(c)Hat(opensurface).(d)Wineglass(cylinder). PublishedindDesign(2):111{127,1995. (a)(b) Figure5:Roundingofpolyhedraltoroid.(a)Wireframe.(b)Shadedobject.(c-d)Finalroundedshape. PublishedindDesign(2):111{127,1995. (a)(b) Figure6:Roundingofcubicalsolid.(a)Wireframe.(b)Shadedobject.(c-d)Finalroundedshape. PublishedindDesign(2):111{127,1995. (a)(b) Figure7:Surfaceblendingofpipe.(a)Wireframe.(b)Shadedobject.(c-d)Finalsmoothblend. PublishedindDesign(2):111{127,1995. (a)(b) Figure8:Fittingof3Dlaserscannerdata.(a)Originalcylinderwireframe.(b)Reconstructedpotwireframe.(c-d)Texturedpot. PublishedindDesign(2):111{127,1995. (a)(b) Figure9:Fittingofsyntheticdata.(a)cylinderwireframe.(b)Reconstructedvasewireframe.exturedv PublishedindDesign(2):111{127,1995. (a)(b) Figure10:Fourttedshapes.(a)Pot.(b)Vase.(c)Glass.(d)Bottle.