CurvesandSurfaces(I).ppt

CurvesandSurfaces(I),BasedonEA,Chapter10.,姜明北京大学数学科学学院,更新时间2019年11月19日星期二1时6分35秒,Introduction,Incomputergraphics,flatobjectsarepopularinthisvirtualworldGraphicsystemscanrenderthemathighrates.Weneedmethodstomodelcurvedobjects.,Outline,RepresentationofCurvesandSurfacesDesignCriteriaParametricCubicPolynomialCurvesCubicInterpolatingPolynomial,RepresentationofCurvesandSurfaces,ThreemajorwaysofobjectrepresentationExplicitRepresentationImplicitRepresentationsParametricFormPolynomialRepresentationsParametricPolynomialCurvesParametricPolynomialSurfaces,ExplicitRepresentation,Noguaranteethatthisrepresentationexistsforagivencurve/surfacez=f(x,y)cannotrepresenta(full)sphere.Somecurvesandsurfacesmaynothaveanexplicitrepresentation.Coordinate-system-dependenteffect.Easytoobtainpointsonthem.,ImplicitRepresentations,f(x,y,z)=0in3Dorf(x,y)=0in2D.Lesscoordinate-system-dependent:itdoesrepresentalllines/circles.Allowtodeterminewhetherpointslieonthecurve/surface.Difficulttofindpointsonthecurve/surface.Curvesin3Darenoteasilyrepresentedinimplicitformbecauseittakestwoequationstorepresentacurvein3Df(x,y,z)=0andg(x,y,z)=0“Ingeneral,mostofthecurvesandsurfacesthatariseinrealapplicationshaveimplicitrepresentations.Theiruseislimitedbythedifficultinobtainingpointsonthem.”EA,p.600Algebraicsurfaces:f(x,y,z)isapolynomial.Ofparticularimportancearethequadricsurfaces.,ParametricForm,Sameformin2Dand3D.Mostflexibleandrobustforcomputergraphicsthantheothers.Stillcoordinate-system-dependent.Coordinate-system-independentrepresentationsarepossibleUsingFrenetframeforcurves.Difficulttodetermineifapointisonthecurve/surface.,ParametricPolynomialCurves/Surfaces,Parametricrepresentationsarenotunique.Parametricpolynomialformsareofmostuseincomputergraphics.,SurfacePatch,CurveSegment,DesignCriteria,Localcontrolofshape:Asingleglobaldescriptionisgenerallyoutofthequestionandtoocomplex.Wewouldlike/havetoworkinteractivelywiththeshape,carefullymoldingitmeetspecificationsSmoothnessandcontinuityatjointpointsAcurvewithdiscontinuityisoflittleinteresttous.Notonlywilleachsegmenthavetobesmooth,butalsowewantadegreeofsmoothnesswherethesegmentsmeetatjointpoint.Smoothnessismeasuredwithderivativesalongthecurves/surfaces.Abilitytoevaluatederivativesisneededtoevaluatesmoothnessandnormals.StabilityItisnecessarytobendcurves/surfacestothedesiredshapethroughlocalcontrolpoints.Whenwemakeachange,thischangewillonlyaffecttheshapeinonlytheareawhereweareworking.Weareusuallysatisfiedifthecurve/surfacepassesclosetothecontrolpoints,EaseofRenderingGoodmathrepresentationsmaybeoflimitedvalueiftheycannotbeefficientlyrendered.,Splines,Splinesaretypesofcurves,originallydevelopedforship-buildinginthedaysbeforecomputermodeling.Navalarchitectsneededawaytodrawasmoothcurvethroughasetofpoints.Thesolutionwastoplacemetalweights(calledknots)atthecontrolpoints,andbendathinmetalorwoodenbeam(calledaspline)throughtheweights.Thephysicsofthebendingsplinemeantthattheinfluenceofeachweightwasgreatestatthepointofcontact,anddecreasedsmoothlyfurtheralongthespline.Togetmorecontroloveracertainregionofthespline,thedraftsmansimplyaddedmoreweights.Thisschemehadobviousproblemswithdataexchange!Peopleneededamathematicalwaytodescribetheshapeofthecurve.CubicPolynomialsSplinesarethemathematicalequivalentofthedraftsmanswoodenbeam.PolynomialswereextendedtoB-splines(forBasissplines),whicharesumsoflower-levelpolynomialsplines.ThenB-splineswereextendedtocreateamathematicalrepresentationcalledNURBS.ReadHistoryofSplines,Editedfrom,ParametricCubicPolynomialCurves,Wemustchoosethedegreeofpolynomials.IfwechoosehigherdegreeWewillhavemoreparametersthatwecansettofromthedesiredshapeCubicisenough.Theevaluationofpointswillbecostly.Thereismoredangerthatthecurvewillbecomerougheratsomepoints.Ifwepicktoolowadegree,wemaynothaveenoughparameterstoworkandtomakethemtojoinsmoothly.Ifwedesigneachcurvesegmentoverashortinterval,wecanachievemanyofourpurposeswithlow-degreecurves.Cubicpolynomialsarechosen,atleastinitially.,cistobedeterminedfromthecontrol-pointdata.Typesofcubiccurvesdifferinhowtheyusethecontrol-pointdata.ThedatamaybeInterpolating:thepolynomialmustpasssomepoints.Higherorderinterpolatingatcertainparametervalues.Smoothnessconditionsatjoinpoints.Approximative:thecurvepassclosetosomedatapoints.,CubicInterpolatingPolynomial,Supposethatwehavefourcontrolpointsin3D.Weseekthecoefficientscsuchthatthepolynomialp(u)=uTcpassesthroughthefourpoints.Weassumetheintervalis0,1,thecurvepassesfourpointsatu=0,1/3,2/3,and1.,Joiningofinterpolatingsegments.,Becauseofthelackofderivativecontinuityatjointpoints,theinterpolatingpolynomialisoflimiteduseincomputergraphics.,BlendingFunctions,CubicInterp