MIT-SCIENCE-Lectures-09_curve_surf

MITEECS6.837,DurandandCutlerCurves&SurfacesMITEECS6.837,DurandandCutlerSchedule•SundayOctober5th,*3-5PM*ReviewSessionforQuiz1•ExtraOfficeHoursonMonday•TuesdayOctober7th:Quiz1:Inclass1hand-written8.5x11sheetofnotesallowed•WednesdayOctober15th:Assignment4(GridAcceleration)dueMITEECS6.837,DurandandCutlerLastTime:•AccelerationDataStructuresMITEECS6.837,DurandandCutlerQuestions?MITEECS6.837,DurandandCutlerToday•Review•Motivation–LimitationsofPolygonalModels–PhongNormalInterpolation–SomeModelingTools&Definitions•Curves•Surfaces/Patches•SubdivisionSurfaces•ProceduralTexturingMITEECS6.837,DurandandCutlerLimitationsofPolygonalMeshes•planarfacets•fixedresolution•deformationisdifficult•nonaturalparameterizationMITEECS6.837,DurandandCutlerCanWeDisguisetheFacets?MITEECS6.837,DurandandCutlerPhongNormalInterpolation•NotPhongShadingfromAssignment3•Insteadofusingthenormalofthetriangle,interpolateanaveragednormalateachvertexacrosstheface•MustberenormalizedCourtesyofLeonardMcMillan.Usedwithpermission.MITEECS6.837,DurandandCutler10Kfacets1Kfacets1Ksmooth10KsmoothMITEECS6.837,DurandandCutlerBetter,butnotalwaysgoodenough•Stilllowresolution(missingfinedetails)•Stillhavepolygonalsilhouettes•Intersectiondepthisplanar•Collisionsinasimulation•SolidTexturing•...SomeNon-PolygonalModelingToolsMITEECS6.837,DurandandCutlerExtrusionSplineSurfaces/PatchesSurfaceofRevolutionQuadricsandotherimplicitpolynomialsMITEECS6.837,DurandandCutlerContinuitydefinitions:•C0continuous–curve/surfacehasnobreaks/gaps/holes–watertight•C1continuous–curve/surfacederivativeiscontinuous–lookssmooth,nofacets•C2continuous–curve/surface2ndderivativeiscontinuous–ActuallyimportantforshadingMITEECS6.837,DurandandCutlerQuestions?MITEECS6.837,DurandandCutlerToday•Review•Motivation•Curves–What'saSpline?–LinearInterpolation–InterpolationCurvesvs.ApproximationCurves–Bézier–BSpline(NURBS)•Surfaces/Patches•SubdivisionSurfaces•ProceduralTexturingMITEECS6.837,DurandandCutlerDefinition:What'saSpline?•Smoothcurvedefinedbysomecontrolpoints•MovingthecontrolpointschangesthecurveInterpolationBézier(approximation)BSpline(approximation)CourtesyofSethTeller.Usedwithpermission.MITEECS6.837,DurandandCutlerInterpolationCurves/Splines•SimplestcurvebetweentwopointsMITEECS6.837,DurandandCutlerInterpolationCurves•Curveisconstrainedtopassthroughallcontrolpoints•GivenpointsP0,P1,...Pn,findlowestdegreepolynomialwhichpassesthroughthepointsx(t)=an-1tn-1+....+a2t2+a1t+a0y(t)=bn-1tn-1+....+b2t2+b1t+b0Interpolationvs.ApproximationCurvesMITEECS6.837,DurandandCutlerInterpolationcurvemustpassthroughcontrolpointsApproximationcurveisinfluencedbycontrolpointsInterpolationvs.ApproximationCurvesMITEECS6.837,DurandandCutler•InterpolationCurve–overconstrained→lotsof(undesirable?)oscillations•ApproximationCurve–morereasonable?MITEECS6.837,DurandandCutlerCubicBézierCurve•4controlpoints•Curvepassesthroughfirst&lastcontrolpoint•CurveistangentatP0to(P0-P1)andatP4to(P4-P3)MITEECS6.837,DurandandCutlerCubicBézierCurve•deCasteljau'salgorithmforconstructingBéziercurvesttttttMITEECS6.837,DurandandCutlerCubicBézierCurveMITEECS6.837,DurandandCutlerConnectingCubicBézierCurves•HowcanweguaranteeC0continuity(nogaps)?•HowcanweguaranteeC1continuity(tangentvectorsmatch)?•Asymmetric:CurvegoesthroughsomecontrolpointsbutmissesothersMITEECS6.837,DurandandCutlerHigher-OrderBézierCurves•4controlpoints•BernsteinPolynomialsasthebasisfunctions•Everycontrolpointaffectstheentirecurve–Notsimplyalocaleffect–MoredifficulttocontrolformodelingCourtesyofSethTeller.Usedwithpermission.MITEECS6.837,DurandandCutlerCubicBSplines•≥4controlpoints•Locallycubic•CurveisnotconstrainedtopassthroughanycontrolpointsCourtesyofSethTeller.Usedwithpermission.MITEECS6.837,DurandandCutlerCubicBSplines•IterativemethodforconstructingBSplinesShirley,FundamentalsofComputerGraphicsCourtesyofSethTeller.Usedwithpermission.MITEECS6.837,DurandandCutlerCubicBSplinesCourtesyofSethTeller.Usedwithpermission.MITEECS6.837,DurandandCutlerCubicBSplines•canbechainedtogether•bettercontrollocally(windowing)CourtesyofSethTeller.Usedwithpermission.MITEECS6.837,DurandandCutlerBézierisnotthesameasBSpline•RelationshiptothecontrolpointsisdifferentBézierBSplineCourtesyofSethTeller.Usedwithpermission.MITEECS6.837,DurandandCutlerBezierisnotthesameasBspline•Butwecanconvertbetweenthecurvesusingthebasisfunctions:MITEECS6.837,DurandandCutlerNURBS(generalizedBSplines)•BSpline:uniformcubicBSpline•NURBS:Non-UniformRationalBSpline–non-uniform=differentspacingbetweentheblendingfunctions,a.k.a.knots–rational=ratioofpolynomials(insteadofcubic)MITEECS6.837,DurandandCutlerQuestions?MITEECS6.837,DurandandCutlerToday•Review•Motivation•SplineCurves•SplineSurfaces/Patches–TensorProduct–BilinearPatches–BezierPatches•SubdivisionSurfaces•ProceduralTexturingMITEECS6.837,DurandandCutlerTensorProduct•Oftwovectors:•Similarly,wecandefineasurfaceasthetensorproductoftwocurves....Farin,CurvesandSurfacesforComputerAi