3033002 Topics in Computer Graphics Lecture 03 Geometric Modeling New York University NURBS Spline Surfaces and Blossoming in Twodimensions Lecture 03 23 September 2002 Lecturer Prof Denis Zorin Scribe Zhihua ID: 62338
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G22.3033-002:TopicsinComputerGraphics:Lecture#03GeometricModelingNewYorkUniversity NURBS,SplineSurfacesandBlossominginTwo-dimensionsLecture#03:23September2002Lecturer:Prof.DenisZorinScribe:ZhihuaWangInthislecturewegeneralizethepolarformandblossomingandintroduceNURBS(non- G22.3033-002:Lecture#03 12345Figure1:ComparisonofparameterintervalsbetweenuniformandnonuniformB-splines.associatedwithnon-uniformlyspacedknots: .SeeFigure1.Infactsomeoftheknotsmaycoincidebutwewillnotconsiderthiscaseindetail.Theblossomingpyramidfornon-uniformsplinesisshowninFigure2. Figure2:BlossomingfornonuniformB-splines.ThemultiafÞnepropertyleadstothefollowingequationsforlinearinterpolationweights:fromwhichweget: t4t12=tt1 Generally,for titi2=tti WeobservethattheindicesofbaseparametersdifferbythedegreeofB-spline. ÓMulti-Óheremeansthatthepropertyholdsforeveryargument. G22.3033-002:Lecture#031.3Usageofnon-uniformsplinesOneofthemostimportantadvantagesofnon-uniformsplinesispossibilityofknotinsertion.Supposewewanttoaddacontrolpointwithoutchangingthecurve;inthisway,wecanaddsmallfeaturestoacurvegraduallywhilehavingcompletecontrolovertheshape.Foranon-uniformB-spline,theproblemcanbeturnedintoreplacingtheoldsequenceofparameters bynewsequenceofparameters suchthat:whichwasevaluatedononii+1]isexactlythesameaspolynomialsononi ]and[ afterinsertion;seeFigure3.TheusereitherdeÞnesdirectly,or,morecom-monlyclicksonthepointofthecurveandthevalueofiscomputedbysolvinganequation ,whereisthepointselectedbytheuser. Figure3:Addingnewknot.Asaresultofknotinsertion,onenewcontrolpointisadded,and2oldarereplacedbynew;thetwonewpolynomialsegmentsaredeÞnedbythefollowingtwogroupsofcontrolpoints: Thenewcontrolpointsareobtainedusingblossomingasbefore.Eachnewcontrolpointisalinearcombinationoftwooldcontrolpoints;seeFigure41.4RationalsplinesTheÒRÓinNURBSstandsforrational.Rationalfunctionsareratiosoftwopolynomials.Themainreasonforhavingrationalsplinesistheneedtorepresentquadrics,i.e.curvesdeÞnedassolutionsofquadraticequations,e.g.thecircle.Moregenerally,rationalrepresentationaddsweightstothecontrolpoints,sothatsomecontrolpointsÒattractÓthecurvemorethanothers.Nowwediscusshowrationalfunctionshelptorepresentquadrics.Forexample,anarcofacirclecannotberepresentedinparametricformform ()()]ifpolynomialsareusedfor.Indeed,thenwehave,andsolvingforcoefÞcients,foranydegreewecaneasilygetbyinductionthatshouldbeconstants. G22.3033-002:Lecture#03 Afftectbypointsinin34]Figure4:Newcontrolpoints.Rationalfunctionshoweverallowustorepresentacirculararc:Considerthemostobviousparameterizationofacircle.Thiscanbewrittenas)=sin( )=cos( ,wherewereparameterizeusing=tan TheideaApointpointxy]ontheplanecanberepresentedbybyxyw]wherewistheweightandNowwecancomputeathree-dimensionalcurvesascontrolpointsandrepresentthecircleintheplaneas: 2SplineSurfaceSplineSurfacescanbedividedintotwocategories:tensorpatchesandtriangularpatches,accordingtothedomainonwhichitisdeÞned;seeFigure52.1TensorpatchsurfacesTensorpatchessurfaceisthesimplestextensionofcurves.Considertwocurvefunctionswithintervaldomains,respectively,,andasurfaceformedby.ItiscalledatensorproductsurfacewithdomainForexample,atensorBezierpatchisrepresentedas: G22.3033-002:Lecture#03 Figure5:Tensorpatchesandtriangularpatches.Sothereis16controlpointsand16basisfunctions.Wecandoblossominginasimilarway.SeeFigure6 v,uFigure6:BlossomingoftensorBezierpatch.AlthoughcombiningBeziercurvesegmentsintolongercurvesiseasy(justmakesurethatthecontrolpointsarealigned),itismoredifÞculttomatchBezierpatchesin3D.Tensor-productB-splinesurfacecanhavearbitrarynumberofcontrolpointsandtheirpolynomialpatchesareautomaticallyjointedsmoothly(forexampleforbicubicsurfaces,(i.e.cubicin)thepatcheesarejoinedwithcontinuity.Weuse+2)+2)controlpointstomatchedpatches.2.2TriangularpatchesFortriangularpolynomialpatchesthepolarformcanberepresentedasforeach;seeFigure7 G22.3033-002:Lecture#03 Figure7:Interpolationoftriangularpatch.Oneoftheadvantagesoftriangularisitslowerdegreeforgivensmoothness.Forexample,togetcontinuity,foratensor-productsplinesurfacewemustusebidegree3(totaldegree6),whileweonlyneedtotaldegree4fortriangularpatches.