DynamicNURBSSwungSurfacesforPh ysicsBased ShapeDesign HongQin and DemetriT erzopoulos Departmen t of Computer Science Univ ersit yofT oron to KingsCollegeRoadT oron toOn tarioMSA Publishedin ComputerA - PDF document

DynamicNURBSSwungSurfacesforPh ysicsBased ShapeDesign HongQin and DemetriT erzopoulos Departmen t of Computer Science Univ ersit yofT oron to KingsCollegeRoadT oron toOn tarioMSA Publishedin ComputerA
DynamicNURBSSwungSurfacesforPh ysicsBased ShapeDesign HongQin and DemetriT erzopoulos Departmen t of Computer Science Univ ersit yofT oron to KingsCollegeRoadT oron toOn tarioMSA Publishedin ComputerA

Embed / Share - DynamicNURBSSwungSurfacesforPh ysicsBased ShapeDesign HongQin and DemetriT erzopoulos Departmen t of Computer Science Univ ersit yofT oron to KingsCollegeRoadT oron toOn tarioMSA Publishedin ComputerA


Presentation on theme: "DynamicNURBSSwungSurfacesforPh ysicsBased ShapeDesign HongQin and DemetriT erzopoulos Departmen t of Computer Science Univ ersit yofT oron to KingsCollegeRoadT oron toOn tarioMSA Publishedin ComputerA"— Presentation transcript


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-basedmodelsaregoernedbdynamicdi erentialequationswhich,whenintegratednumericallythroughtime,canconuouslyevtrolpointsandwtsinresponsetoappliedforcestoproducephysicallymeaningfulandinpredictableshapevariation.Akeyadvtageofourphysics-basedframeworkisthatitpermitsappropriatealuestobedeterminedautomaticallyinaccordancewithvariousphysicalparametersandgeometriceuseLagrangianmechanicstoformulatetheequationsofmotionofdynamicNURBSswungsur-faces,and niteelementanalysistoreducetheseequationstoecientalgorithmsthatcanbesimumericallyusingstandardtechniques.Oneofthechallengesinthise ortiscopingwiththenonlineardynamicformulationstemmingfromtheunderlyingswungNURBSgeometry.TheNURBSswungsurfaceisinherentlynonlinearwithrespecttoitsdegreesoffreedom,evenifbothNURBSgeneratorcurvesarereducedtosimpleB-splinesby xingtheirwtstounitAsaconsequenceofthenonlinearit,the AparticularshapecanoftenberepresentednonuniquelyusingNURBS,withdi erenaluesofknots,controlpoints,and PublishedindDesign(2):111{127,1995.mass,damping,andsti nessmatricesinthedynamicformulationmustberecomputedateachsimtimestep.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 2 1;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;andthesti nessmatrixisdudv(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:,andSeeAppendixAforthedetailsaboutthesti nessmatrix.tDataStructuresede neanelementdatastructurewhichcontainsthegeometricspeci cationofthesurfacepatchel-talongwithitsphysicalproperties.Acompletedynamicswungsurfaceisthenimplementedasadatastructurewhichconsistsofanorderedarrayofelementswithadditionalinformation.Theelemenstructureincludespointerstoappropriatecomponentsoftheglobalvtrolpointsandwboringelementswillsharesomegeneralizedcoordinates.Thesharedvariableswillhapointersimpingingonthem.Wealsoallocateineachelementanelementalmass,damping,andsti nessmatrix,andincludeintheelementdatastructurethequantitiesneededtocomputethesematrices.Thesetitiesincludethemassu;v),dampingu;v),andelasticitu;vu;v)densityfunctions,hmayberepresentedasanalyticfunctionsorasparametricarraysofsamplevCalculationofElementMatricesTheintegralexpressionsforthemass,damping,andsti nessmatricesassociatedwitheachelementarealuatednumericallyusingGaussianquadrature[24].Weshallexplainthecomputationoftheelemenmassmatrix;thecomputationofthedampingandsti nessmatricesfollowsuit.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,andsti nesspropertiesareuniformoerthesurfaceand,therefore,reducetoscalarquantities,thedoublesumintheGaussianintegrationformuladecomposestotheproductoftoindependentsumsoereachoftheunivariatedomainsofthegeneratorcurvesanditbecomesmhmoreecienDiscreteDynamicsEquationsointegrate(12)inaninemodelingent,itisimportanttoprovidethemodelerwithvisualfeedbackabouttheevolvingstateofthedynamicmodel.Ratherthanusingcostlytimeinmethodsthattakethelargestpossibletimesteps,itismorecrucialtoprovideasmoothanimationbtainingtheconyofthedynamicsfromonesteptothenext.Hence,lesscostlyyetstabletimetegrationmethodsthattakemodesttimestepsaredesirable.ThestateofthedynamicNURBSswungsurfaceattimeisintegratedusingpriorstatesattimeomaintainthestabilityoftheintegrationscheme,weuseanimplicittimeinmethod,whichemploysdiscretederivesofusingbacarddi erenceseobtainthetimeintegrationform+2=2)+4wherethesuperscriptsdenoteevaluationofthequantitiesattheindicatedtimes.Thematricesandforcesareevaluatedattimeeemploytheconjugategradientmethodtoobtainaniterativesolution[24].Toaceinulationrates,welimitthenberofconjugategradientiterationspertimestepto10.Wehaeobservthat2iterationstypicallysucetoconergetoaresidualoflessthan10.Morethan2iterationstendtobenecessarywhenthephysicalparameters(mass,damping,tension,sti ness,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)withbacarddi erences,weobtaintheintegrationform PublishedindDesign(2):111{127,1995.ysics-BasedShapeDesignInthephysics-basedshapedesignapproach,designrequirementsmaybesatis edthroughtheuseofenergies,forces,andconstraints.Thedesignermayapplytime-varyingforcestosculptshapesinortooptimallyapproximatedata.Certainaestheticconstraintssuchas\fairness"areexpressibleintermsofelasticenergiesthatgiverisetospeci csti nessmatricesOtherconstraintsincludepositionornormalspeci cationatsurfacepoints,andconyrequirementsbeteenadjacentsurfacepatches.BybuildingthedynamicswungsurfaceuponthestandardgeometryoftheNURBSswungsurface,weallothemodelertoconuetousethewholespectrumofadvancedgeometricdesigntoolsthathaebecomet,amongthem,theimpositionofgeometricconstraintsthatthe nalshapemustsatisfyAppliedFSculptingtoolsmaybeimplementedasappliedforces.Theforceu;v;t)representsthenete ectofallappliedforces.ypicalforcefunctionsarespringforces,repulsionforces,gravitationalforces,in ationforces,etc.[8].orexample,considerconnectingamaterialpoin)ofadynamicswungsurfacetoapoinspacewithanidealHookeanspringofsti ness.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,andsti nessdensities,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).thesti nessconstantofthisspringisslowlyincreasedfromzeroto10,thewtincreases(solidplotinFig.2)thusrecreatingacorner.Next,wfromzeroto1000(dottedplotinFig.2).Theincreasedrigidityofthesurfacecausesthewttodecrease,againroundingthecorner.Finallyincreasethespringsti nessto100(stippledplotinFig.2),thuscounteractingtheincreasedsti nessandcausingthewttoincreasestoagainrecreatethecorner.Thisexperimentdemonstratesthatwhentrolpointsare xed,afreewtwillautomaticallydecreasetodecreasethedeformationenergyb atteningthesurfacelocally,unlessexternalforcescounteractthistendency.Whencontrolpointsarefree,theywillalsovaryaswelltoproduceananalogousphysicale ect.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|typicallytheratioofdataforcespringconstantstosurfacesti nesses|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,bendingsti nessparameter=140whilealltheothersarezero,dataspringconstan=9000.Thesurface ttingstabilizesinafewsecondswithatimestepereporttomoreexperimentswiththeaboesurface ttingscenarioinordertoinestigatetheleast-squares ttingerrorsunderdi erentcircumstances.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,andthesti nessmatrixareallisthegeneralizedforcevectoractingon,andisthegeneralizedinertialforce.See[19]forthedetailsoftheD-NURBSformoreducetheD-NURBSsurfacetoadynamicswungsurface,weapplythenonlinearconstrain,and,for;:::;m;:::;n,arede nedasinSection3.Di eren(21),weobtainUsingthenotationsinSection3,wecanrewrite(21)and(22)inthematrixformmatriceswith=(4+9).Substituting(23)into(20),wearriveattheequationsofmotionforthedynamicNURBSswungsurface(12),wherethemass,damping,andsti nessmatricesaregivenbThegeneralizedforceswithrespectto PublishedindDesign(2):111{127,1995.Theconstraintreducesthe4(+1)generalizedcoordinatesoftheD-NURBSsurfacetothe4generalizedcoordinatesofthedynamicNURBSswungsurface.ehaeproposeddynamicNURBSswungsurfaces.LiketheD-NURBSpredecessor[19],thenewmodelisaphysics-basedgeneralizationofitsgeometriccounterpart.WederiveditsystematicallythroughLa-grangianmechanicsandimplementedinesoftareusingconceptsfrom niteelementanalysisandumericalmethods.WehaeshownthatthedynamicNURBSswungsurfacemodelcanbefor-ulatedintodi erenys:(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-topologydynamicsurfaceso erbroadgeometriccoerageduringdesign.Sinceourmodelsarebuiltontheindustry-standardNURBSgeometricsubstrate,designerswwiththemcanconuetomakeuseoftheexistingarrayofgeometricdesigntoolkits. PublishedindDesign(2):111{127,1995.StructureofSti nessMatrixInaddition,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.

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DynamicNURBSSwungSurfacesforPh ysicsBased ShapeDesign HongQin and DemetriT erzopoulos Departmen t of Computer Science Univ ersit yofT oron to KingsCollegeRoadT oron toOn tarioMSA Publishedin ComputerA - Description


Abstract e dev elop a dynamic freeform surface mo del whic h is useful for represen ting a broad class of ob jects with symmetries and top ological v ariabilit The new mo del is based up on swung NURBS surfaces and it inherits their desirable cross ID: 4570 Download Pdf

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