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On Multiplying Points: the Paired Algebras of Forms and Sites

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On Multiplying Points: The Paired Algebras of Forms and Sites Lyle Ramshaw May 1, 2001 Sym(Aˆ)Sym(Aˆ∗) 1 1 Aˆ Aˆ∗ Sym0( ) Sym0( ) C, w, A ξ,η x, y SitesAˆ Aˆ∗ Forms C2, w2, Cξ,Cη, wx, wy, ξ2,ξη,η2 x2,xy,y2 Aˆ Aˆ∗ Sym2( )Sym2( ) C3, w3, C2ξ,C2η, w2x, w2y, Cξ2,Cξη,Cη2, wx2,wxy,wy2, ξ3,ξ2η, ξη2,η3 x3,x2y, xy2,y3 Aˆ Aˆ∗ Sym3( )Sym3( ) c Compaq Computer Corporation 2001 This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission ofthe Systems Research Center ofCompaq Computer Corporation in Palo Alto, California; an acknowledgment of the authors and individual contributors to the work; and all applicable portions ofthe copyright notice. Copying, reproducing, or republishing forany other purpose shall require a license with payment offeeto the Systems Research Center. All rights reserved. Contents Teaser ix Preface xi 1Introduction 1 1.1Multiplyingpoints........................ 1 1.1.1 Question1......................... 1 1.1.2 Question2......................... 2 1.1.3 TheFlaw......................... 2 1.1.4 TheRepair........................ 2 1.2 Labeling B´eziercontrolpoints.................. 3 1.3Addingpoints........................... 6 1.4Linearization........................... 7 1.5Multiplyingvectors........................ 9 1.6Thepairedalgebras........................ 10 1.6.1 Linearization....................... 10 1.6.2 Algebrization....................... 11 1.6.3 Choosingthepairingmaps................ 12 1.7 Piecewise models with smooth joints . ........... 13 1.8 Cubic B´eziertriangles...................... 14 1.9Relatedwork........................... 15 1.9.1 Bernstein bases and B´ezierpoints............ 15 1.9.2 B´ezierpointsaspolarvalues............... 16 1.9.3 Frompolarizationtothepairedalgebras........ 17 1.9.4 Vegterexploitsthecontractionoperators........ 18 1.10Thefourframeworks....................... 19 2 Mathematical Preliminaries 21 2.1Onthewords“affine”and“linear”............... 21 2.2Finitedimensionality....................... 21 2.3Linear-spaceduality....................... 22 2.4Algebras.............................. 23 iii iv CONTENTS 3 The Nested-Spaces Framework 27 3.1ChoosingaCartesiancoordinatesystem............ 27 3.2Picturingthenested-spacesframework............. 28 3.3Thedualspaces.......................... 29 3.4Interpretingelementsofthedualspaces............. 30 3.5Arethedualspacesnested?................... 31 4 The Homogenized Framework 33 4.1 To n-formsviabarycentriccoordinates............. 33 4.2 To n-formsviaaweightcoordinate............... 34 4.3Thelinearizationofanaffinespace............... 35 4.4Anewterm:“anchor”...................... 36 4.5Coanchors............................. 37 4.6Thebenefitsoflinearization................... 39 4.7Thealgebraofforms....................... 41 4.8Thedualspaces.......................... 42 4.9Linearizationrevisited...................... 43 4.9.1 Fixingaframe...................... 43 4.9.2 Defininganequivalencerelation............. 44 4.9.3 Exploitingduality..................... 45 4.9.4 Imposingauniversalmappingcondition........ 46 4.9.5 Sowhatisananchor,really?............... 47 5 The Separate-Algebras Framework 49 5.1Thealgebraofsites........................ 51 5.1.1 Imposingauniversalmappingcondition........ 51 5.1.2 Fixingabasis....................... 52 5.1.3 Defininganequivalencerelation............. 52 5.1.4 Exploitingduality..................... 53 5.2Theweightofasite........................ 54 5.3Thedualspaces.......................... 55 5.4Flavorsofevaluation....................... 56 6 The Veronese Prototypes 57 6.1Quadraticsitesovertheline................... 57 6.2Theprototypicalparabola.................... 60 6.3Themomentcurves........................ 61 6.4Therelationshiptoblossoming.................. 63 6.5TheVeronesesurface....................... 65 6.6Degen’sanalysisofquadraticsurfaces.............. 67 6.6.1 Theplanesinthe4-fold∆=0.............. 69 6.6.2 Thetypicalcompletequadrilateral........... 70 6.6.3 ExtendingDegen’sanalysistocubics.......... 74 CONTENTS v 6.7 Polynomial d-folds of degree at most n ............. 75 6.8Tensor-productsurfaces..................... 77 6.9Tensor-productprototypes.................... 79 7 The Paired-Algebras Framework 81 7.1 Picturing our goal . ...................... 83 7.2Linealsandperfectpowers.................... 84 7.3ThePermanentIdentity..................... 86 7.4Definingthepairing........................ 88 7.5Summingisbetter—trustme.................. 90 7.6 Evaluating an n-form....................... 91 7.6.1 Evaluating the blossom ofan n-form.......... 93 7.7Formulasfordifferentiation.................... 93 7.8Thecontractionoperators.................... 95 7.9Differentiationascontraction.................. 97 7.10Derivationsanddifferentialoperators.............. 98 7.11 Agreement to kth order......................100 8 Exploiting the Pairing 103 8.1Thedualsofpopularmonomialbases..............103 8.1.1 Power-basisformsandTaylor-basissites........103 8.1.2 Bernstein-basis forms and B´ezier-basissites......104 8.2ThedeCasteljauAlgorithm...................105 8.3Degreeraising...........................106 8.4Thegeometryofpointevaluations................107 8.4.1 Evenly n-divided d-simplices...............108 8.4.2 TheDifferencingAlgorithm...............111 8.5Integratingoverasimplex....................114 9 Universal Mapping Conditions 119 9.1Linearizationviaauniversalcondition.............119 9.2Algebrizationviaauniversalcondition.............122 9.3Tensors..............................124 9.3.1 Thetensoralgebra....................125 9.3.2 Thealternatingalgebra..................126 9.3.3 TheCliffordalgebra...................129 A Some Category Theory 131 A.1Fixingsometypeerrors......................131 A.2Linearizationasafunctor....................133 A.3Leftadjoints............................133 A.4Behavingnaturally........................135 A.5Aladderofadjunctions......................135 vi CONTENTS A.6Injectiveunits...........................138 A.7Rightadjoints...........................138 A.8Tensor-productsurfacesrevisited................139 B To Sum or to Average? 141 B.1Searchingforprettyformulas..................141 B.2Otheroptionsforavoidingannoyance..............145 C More Math Remarks 147 C.1Thoughtsaboutfunctionalnotations..............147 C.2Pairedalgebrasinwildercontexts................148 C.2.1Fieldsofcharacteristiczero...............148 C.2.2Fieldsofprimecharacteristic..............149 C.2.3Finitefields........................150 C.2.4 Linear spaces ofinfinite dimension ...........151 C.2.5Modulesovercommutativerings.............151 C.2.6Thedivisionringofquaternions.............151 Bibliography 153 Index 157 List of Figures 1.1 A quadratic B´eziertriangle.................... 4 1.2 Computing a point on a quadratic B´eziertriangle....... 5 1.3 Computing a point on a cubic B´eziersegment......... 5 1.4 A point P intheaffineplaneA ................. 6 1.5 The linearization Aˆ oftheaffineplaneA ............ 7 1.6Acubicsplinecurvewithfoursegments............ 14 3.1Thenested-spacesframework.................. 29 4.1 The linearization Aˆ oftheaffineplaneA ............ 35 4.2Thehomogenizedframework................... 41 4.3Thehomogenizedframeworkwithabstractlabels....... 48 5.1Theseparate-algebrasframework................ 50 6.1 The plane ofunit-weight 2-sites over the line L. ........ 58 6.2 The de Casteljau Algorithm on the prototypical cubic κ3 ... 64 6.3 The complete quadrilateral from a typical center line λ .... 72 6.4Theplaneprojectorthatcorrespondstothetriplepoint.... 73 7.1Thepaired-algebrasframework................. 82 8.1Anevenly4-divided2-simplex..................109 8.2ProvingProposition8.4-1forbivariatecubics.........111 A.1Aladderofadjunctions......................136 vii Teaser There is a multiplication operation on points that your teachers failed to tell you about, either because they didn’t know about it or because they judged it to be unimportant. But that multiplication turns out to have important applications in computer-aided geometric design (CAGD). Among other things, it provides the best labels for B´ezier control points — better even than the labels provided by polar forms (a.k.a. blossoms). Let V be a finite-dimensional vector space. Everyone understands that it makes sense to multiply covectors, the elements ofthe dual space V ∗ = Lin(V,R). For example, if x, y,andz are covectors, then the expression x2 − 5yz denotes a quadratic form on the space V . Forms have lots of applications; for example, to put a Euclidean metric on V ,wewouldchoose a positive definite quadratic form as our measure of squared length. But most people don’t yet realize that it also makes sense to multiply vectors, the elements of V itself. If ρ, σ,andτ are vectors, then the expression ρ2 − 5στ denotes an object that is the dual analog ofa quadratic form. Let’s call such an object a quadratic site over V .ThesitesoverV ofall degrees form an algebra, dual to the well-known algebra of forms on V . What are sites good for? Consider, say, a cubic B´ezier curve segment. It is the image, under a cubic function, of a closed interval on the parameter line, say the interval [R..S]. The best labels for the B´ezier points ofthat cubic segment are the cubic sites R3, R2S, RS2,andS3. ix Preface In computer-aided geometric design (CAGD), a beautiful technology
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