AngelCG38山东大学软件学院图形学教材.ppt

RenderingCurvesandSurfaces EdAngelProfessorofComputerScience ElectricalandComputerEngineering andMediaArtsUniversityofNewMexico 2 Angel InteractiveComputerGraphics4E Addison Wesley2005 Objectives IntroducemethodstodrawcurvesApproximatewithlinesFiniteDifferencesDerivetherecursivemethodforevaluationofBeziercurvesandsurfacesLearnhowtoconvertallpolynomialdatatodataforBezierpolynomials 3 Angel InteractiveComputerGraphics4E Addison Wesley2005 EvaluatingPolynomials SimplestmethodtorenderapolynomialcurveistoevaluatethepolynomialatmanypointsandformanapproximatingpolylineForsurfaceswecanformanapproximatingmeshoftrianglesorquadrilateralsUseHorner smethodtoevaluatepolynomialsp u c0 u c1 u c2 uc3 3multiplications evaluationforcubic 4 Angel InteractiveComputerGraphics4E Addison Wesley2005 FiniteDifferences Forequallyspaced uk wedefinefinitedifferences Forapolynomialofdegreen thenthfinitedifferenceisconstant 5 Angel InteractiveComputerGraphics4E Addison Wesley2005 BuildingaFiniteDifferenceTable p u 1 3u 2u2 u3 6 Angel InteractiveComputerGraphics4E Addison Wesley2005 FindingtheNextValues Startingatthebottom wecanworkupgeneratingnewvaluesforthepolynomial 7 Angel InteractiveComputerGraphics4E Addison Wesley2005 deCasteljauRecursion WecanusetheconvexhullpropertyofBeziercurvestoobtainanefficientrecursivemethodthatdoesnotrequireanyfunctionevaluationsUsesonlythevaluesatthecontrolpointsBasedontheideathat anypolynomialandanypartofapolynomialisaBezierpolynomialforproperlychosencontroldata 8 Angel InteractiveComputerGraphics4E Addison Wesley2005 SplittingaCubicBezier p0 p1 p2 p3determineacubicBezierpolynomialanditsconvexhull Considerlefthalfl u andrighthalfr u 9 Angel InteractiveComputerGraphics4E Addison Wesley2005 l u andr u Sincel u andr u areBeziercurves weshouldbeabletofindtwosetsofcontrolpoints l0 l1 l2 l3 and r0 r1 r2 r3 thatdeterminethem 10 Angel InteractiveComputerGraphics4E Addison Wesley2005 ConvexHulls l0 l1 l2 l3 and r0 r1 r2 r3 eachhaveaconvexhullthatthatisclosertop u thantheconvexhullof p0 p1 p2 p3 Thisisknownasthevariationdiminishingproperty Thepolylinefroml0tol3 r0 tor3isanapproximationtop u Repeatingrecursivelywegetbetterapproximations 11 Angel InteractiveComputerGraphics4E Addison Wesley2005 Equations StartwithBezierequationsp u uTMBp l u mustinterpolatep 0 andp 1 2 l 0 l0 p0l 1 l3 p 1 2 1 8 p0 3p1 3p2 p3 Matchingslopes takingintoaccountthatl u andr u onlygooverhalfthedistanceasp u l 0 3 l1 l0 p 0 3 2 p1 p0 l 1 3 l3 l2 p 1 2 3 8 p0 p1 p2 p3 Symmetricequationsholdforr u 12 Angel InteractiveComputerGraphics4E Addison Wesley2005 EfficientForm l0 p0r3 p3l1 p0 p1 r1 p2 p3 l2 l1 p1 p2 r1 r2 p1 p2 l3 r0 l2 r1 Requiresonlyshiftsandadds 13 Angel InteractiveComputerGraphics4E Addison Wesley2005 EveryCurveisaBezierCurve WecanrenderagivenpolynomialusingtherecursivemethodifwefindcontrolpointsforitsrepresentationasaBeziercurveSupposethatp u isgivenasaninterpolatingcurvewithcontrolpointsqThereexistBeziercontrolpointspsuchthatEquatingandsolving wefindp MB 1MI p u uTMIq p u uTMBp 14 Angel InteractiveComputerGraphics4E Addison Wesley2005 Matrices InterpolatingtoBezier B SplinetoBezier 15 Angel InteractiveComputerGraphics4E Addison Wesley2005 Example ThesethreecurveswereallgeneratedfromthesameoriginaldatausingBezierrecursionbyconvertingallcontrolpointdatatoBeziercontrolpoints Bezier Interpolating BSpline 16 Angel InteractiveComputerGraphics4E Addison Wesley2005 Surfaces CanapplytherecursivemethodtosurfacesifwerecallthatforaBezierpatchcurvesofconstantu orv areBeziercurvesinu orv FirstsubdivideinuProcesscreatesnewpointsSomeoftheoriginalpointsarediscarded originalandkept new originalanddiscarded 17 Angel InteractiveComputerGraphics4E Addison Wesley2005 SecondSubdivision 16finalpointsfor1of4patchescreated 18 Angel InteractiveComputerGraphics4E Addison Wesley2005 Normals ForrenderingweneedthenormalsifwewanttoshadeCancomputefromparametricequationsCanuseverticesofcornerpointstodetermineOpenGLcancomputeautomatically 19 Angel InteractiveComputerGraphics4E Addison Wesley2005 UtahTeapot MostfamousdatasetincomputergraphicsWidelyavailableasalistof3063Dverticesandtheindicesthatdefine32Bezierpatches 20 Angel InteractiveComputerGraphics4E Addison Wesley2005 Quadrics AnyquadriccanbewrittenasthequadraticformpTAp bTp c 0wherep x y z TwithA bandcgivingthecoefficientsRenderbyraycastingIntersectwithparametricrayp a p0 adthatpassesthroughapixelYieldsascalarquadraticequationNosolution raymissesquadricOnesolution raytangenttoquadricTwosolutions entryandexitpoints