CurvesandSurfaces(I).ppt

CurvesandSurfaces I Basedon EA Chapter10 姜明北京大学数学科学学院 更新时间2020年2月22日星期六11时43分21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 600 Algebraicsurfaces 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 CubicPolynomialsSplinesarethemathematicalequivalentofthedraftsman swoodenbeam 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 Weassumetheintervalis 0 1 thecurvepassesfourpointsatu 0 1 3 2 3 and1 Joiningofinterpolatingsegments Becauseofthelackofderivativecontinuityatjointpoints theinterpolatingpolynomialisoflimiteduseincomputergraphics BlendingFunctions CubicInterpolatingPatch AbicubicsurfacepatchcanbewrittenintheformSupposethatwehave163Dcontrolpointspij Wecanusethosepointstodefineaninterpolatingbicubicsurfacepatchatinterpolatingpoints0 1 3 2 3 1 Itiseasytofindtheinterpolatingpolynomialbytensorproduct