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A blossoming development of splines PDF

108 Pages·2006·0.939 MB·English
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P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 A Blossoming Development of Splines i P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 Copyright©2006byMorgan&Claypool Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin anyformorbyanymeans—electronic,mechanical,photocopy,recording,oranyotherexceptforbriefquotations inprintedreviews,withoutthepriorpermissionofthepublisher. ABlossomingDevelopmentofSplines StephenMann www.morganclaypool.com 1598291165 9781598291162 paperback 1598291173 9781598291179 ebook DOI10.2200/S00041ED1V01200607CGR001 APublicationintheMorgan&ClaypoolPublishersSeries SYNTHESISLECTURESONCOMPUTERGRAPHICSANDANIMATION#1 Lecture#1 SeriesEditor:BrianA.Barsky,UniversityofCalifornia,Berkeley FirstEdition 10987654321 ii P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 A Blossoming Development of Splines StephenMann UniversityofWaterloo Canada SYNTHESISLECTURESINCOMPUTERGRAPHICSANDANIMATION#1 M &C & Morgan Claypool Publishers iii P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 iv ABSTRACT Inthislecture,westudyBe´zierandB-splinecurvesandsurfaces,mathematicalrepresentations for free-form curves and surfaces that are common in CAD systems and are used to design aircraft and automobiles, as well as in modeling packages used by the computer animation industry. Be´zier/B-splines represent polynomials and piecewise polynomials in a geometric mannerusingsetsofcontrolpointsthatdefinetheshapeofthesurface. Theprimaryanalysistoolusedinthislectureisblossoming,whichgivesanelegantlabeling ofthecontrolpointsthatallowsustoanalyzetheirpropertiesgeometrically.Blossomingisused toexplorebothBe´zierandB-splinecurves,andinparticulartoinvestigatecontinuityproperties, changeofbasisalgorithms,forwarddifferencing,B-splineknotmultiplicity,andknotinsertion algorithms.Wealsolookattrianglediagrams(whicharecloselyrelatedtoblossoming),direct manipulationofB-splinecurves,NURBScurves,andtriangularandtensorproductsurfaces. KEYWORDS Be´zier and B-splines curves and surface, Blossoming, Computer-aided geometric design, Splines,Triangularandtensorproductsplinesurfaces P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 v Contents Preface............................................................................ix 1. IntroductionandBackground..................................................1 1.1 MathematicalBackground................................................1 1.1.1 WhyAffineGeometry?............................................3 1.1.2 Exercises.........................................................4 2. PolynomialCurves............................................................5 2.1 Implementations.........................................................7 2.2 BernsteinPolynomialsandBe´zierCurves...................................7 2.2.1 Exercises........................................................12 2.3 Blossoming.............................................................12 2.3.1 deCasteljauRevisited............................................15 2.3.2 DegreeRaising..................................................16 2.3.3 FunctionalBe´zierCurves.........................................17 2.3.4 Exercises........................................................18 2.3.5 Implementations.................................................19 2.4 MultilinearBlossom.....................................................19 2.4.1 Exercises........................................................23 2.5 DerivativesofBe´zierCurves ............................................. 24 2.5.1 Exercises........................................................26 2.6 Continuity..............................................................26 2.6.1 CubicHermiteInterpolation......................................27 2.6.2 C1 ContinuityandtheBlossom...................................27 2.6.3 C2 Continuity...................................................29 2.6.4 Ck Continuity...................................................29 2.6.5 Exercises........................................................30 2.7 ChangeofBasis.........................................................31 2.8 Exercises...............................................................32 2.9 FastEvaluation ......................................................... 32 2.9.1 Exercise.........................................................35 2.9.2 Implementations.................................................35 P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 vi CONTENTS 3. B-Splines....................................................................37 3.1 Implementations........................................................42 3.2 KnotMultiplicity ....................................................... 42 3.2.1 Exercises........................................................46 3.2.2 Implementations.................................................47 3.3 TriangleDiagrams ...................................................... 47 3.3.1 Exercises........................................................49 3.4 KnotInsertion..........................................................49 3.4.1 Implementations.................................................51 3.5 B-splineBasisFunctions.................................................51 3.5.1 Exercise.........................................................55 3.5.2 Implementations.................................................55 3.6 ClosedB-splines........................................................56 3.7 ModelingwithPolynomialandSplineCurves:DirectManipulation ......... 57 3.7.1 Implementations.................................................59 3.8 NURBS................................................................59 4. Surfaces.....................................................................61 4.1 TriangularSurfacePatches...............................................61 4.1.1 Blossoming ..................................................... 65 4.1.2 Exercise.........................................................68 4.1.3 Derivatives......................................................68 4.1.4 ParametricContinuity............................................71 4.1.5 SurfacesAbovethePlane.........................................73 4.1.6 Exercise.........................................................73 4.1.7 StoringtheControlPoints........................................74 4.1.8 EfficientEvaluationataSinglePoint..............................75 4.2 FastEvaluationonaGridofPoints.......................................76 4.2.1 AGridofEvaluationPoints......................................76 4.2.2 Implementations.................................................77 4.2.3 3-to-1Subdivision...............................................77 4.2.4 2-to-1Subdivision...............................................78 4.2.5 4-to-1Subdivision...............................................79 4.2.6 CurveEvaluation................................................81 4.2.7 CrackingProblems...............................................82 4.2.8 Discussion......................................................83 4.2.9 Exercise.........................................................83 P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 CONTENTS vii 4.3 Tensor-ProductSurfacePatches..........................................83 4.3.1 TheBlossomofaTensor-ProductSurface..........................84 4.3.2 Derivatives......................................................85 4.3.3 Continuity......................................................86 4.3.4 Tensor-ProductB-Splines........................................87 4.3.5 SurfacesAbovethePlane.........................................87 4.3.6 GeneralizingtheDimension......................................88 4.3.7 Storage.........................................................88 4.4 AlternativeEvaluationMethodsforTensorProductSurfaces................88 4.4.1 RepeatedBilinearInterpolation...................................88 4.4.2 RepeatedCurveEvaluation:Revisited.............................90 4.4.3 RecursiveSubdivision............................................90 4.4.4 CurveEvaluation................................................91 4.4.5 Discussion......................................................91 4.4.6 Exercise.........................................................92 Bibliograpy........................................................................93 Index.............................................................................95 Biography.........................................................................97 P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 viii P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 ix Preface These are a subset of course notes that I started developing in 1993 for the University of Waterloocourseonsplines.Iwantedablossomdevelopmentofthesplinematerial,andfound no reference adequate for that purpose. Although some possible choices have appeared since then [10, 9], I preferred the material that I had developed. The notes you see here are mostly restrictedtotheblossomingmaterialthatappearsinmycoursenotes,althoughthereareafew sidetripstoemphasizesomeimportantpoints. Awordaboutthestyleandintendedaudience:thislectureisnotaimedatmathematicians. Instead,itisaimedatseniorundergraduatesorfirst-yeargraduatestudentsincomputerscience, whosemathematicsisabitweakorperhapsabitrusty.Exposuretocalculusandlinearalgebra isexpected,butItrytoincludeabriefreminderofthemathematicalideasneededinthetext. The proofs are informal and could be tightened up a lot, but I have tried to write them in a stylethatismoreusefultosomeonethatneedsabitmoreguidance.Further,thereisavarying amount of rigor: sometimes, proofs are omitted or glossed over, at other times, the proofs are doneinmoredetail.Andtheexpositiontendstobeabitchatty.Again,theintentistoprovide alevelofdetailthatwillallowtheintendedaudiencetoabsorbandappreciatethematerial. Twosourceswerethemaininspirationforthesenotes:LyleRamshaw’stechreport,and the papers and talks that were the basis for Ron Goldman’s book, Pyramid Algorithms. Both are excellent supplements to these notes. Because of the way these notes were developed, the referencesareabitskimpy;myapologiestoanyonewhoseworkIshouldhavecited—Iwouldbe interestedinhearingfromyoutoaddanappropriatecitationtofutureversionsofthesenotes. IamindebtedtoLyleRamshawandananonymousreviewer,whosecommentsonadraft of these notes allowed me to expand several portions from short notes to myself (to elaborate oninclass)intoareadabletext. P1:IML/FFX P2:IML/FFX QC:IML/FFX T1:IML MOBK033-FM MOBK033-Smann.cls September26,2006 15:39 x

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