Approximate methods for change of representation and their applications in CAGD

137 Pages·2011·2.5 MB·English

Approximate methods for change of representation
and their applications in CAGD
Oliver J. D. Barrowclough
2012

Preface
The work that comprises this thesis has been performed as part of the EU-project Shapes, Ge-
ometry and Algebra (SAGA) under grant agreement n◦ PITN-GA-2008-214584. The SAGA
networkconsistsof10partnersincludingindustrialcompanies,researchinstitutesanduniversi-
ties,basedthroughoutEurope. Thebroadaimsoftheprojectaretoexploitmathematicalresults
from fields such as Algebraic Geometry, Numerical Analysis and Computer Algebra, in the
applied field of Computer Aided Design and Manufacturing (CAD/CAM). The majority of the
workinthisthesisisrelatedtoWorkPackageIintheSAGAproject,whoseaimistoinvestigate
methodsforchangeofrepresentation.
Acknowledgements
I would first like to express my thanks to the main source of funding for this project, which
was through my early-stage researcher (ESR) fellowship in the SAGA project. I would also
like to thank the Research Council of Norway for the extra funding they provided (IS-TOPP,
Projectnumber201280). IwanttoexpressmygratitudetoSINTEFforprovidingastimulating
workplace for me for the past three years. I would also like to thank the other institutions
I have been involved with including Center of Mathematics for Applications (CMA) and the
Department of Informatics at University of Oslo, and the Institute of Applied Geometry at
JohannesKeplerUniversity,Linz.
IwouldliketothankmymainsupervisorDr. TorDokkenwhosepreviousworkinspiredthe
thesisandwhoprovidedcontinualsupportandinsightduringtheproject. Heshouldalsoreceive
greatcreditfortheeffortheputintocoordinatetheSAGAproject. ThanksalsogotoProfessor
Bert Jüttler who guided me well during my time in Linz, and with whom I collaborated on a
paperalongwithDr. TinoSchulz.
AtSINTEF,IamgratefultoallthoseintheDepartmentofAppliedMathematics. Particular
thanks go to Johan Seland for his help in installing his software for algebraic surface visuali-
sation, which has been useful in my research. Also to those in the Geometry group including
JanThomassen andVibeke Skytt whosehelp has beenvaluable throughout. During my time at
SINTEF I have also had the pleasure of working alongside several colleagues from the SAGA
projectincludingJayasimhaBagalkote,ThienNguyen,PeterNørtoft,DangManhNguyen,Tat-
janaKalinkaandHeidiDahl,whohaveallcontributed toabrighter workplace.
There are many other people I would like to thank for making the whole experience more
enjoyable. At CMA, everyone who attended the Geometry seminar, where I learnt a lot about
many different topics. Thanks to everyone who was involved in the SAGA project; the SAGA
meetingshavealwaysbeenfantasticexperiencesbothprofessionallyandsocially. Abigthanks
iii
alsotoeveryoneinLinzwhomademefeelsowelcomeforthethreemonthsIspentthere.
Lastbutnotleast,Iwouldliketothankmyfamilyfortheirsupportandencouragement,and
forputtingupwithmelivingabroadforsolong. AndmygreatestthanksgotoElisabethforher
constantsupport andforbeingthereformethroughout.
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Abstract: In modern computer aided design (CAD) systems, there are two main representa-
tions of curves and surfaces: parametric and implicit representations. The two representations
complementeachotherinthesensethattheirpropertiescanbeusedtosolvedifferentgeometric
problems. Parametric curves are well suited to point generation, whereas implicit representa-
tions efficiently determine whether or not a point lies on a curve or surface. The availability
of both representations is of great advantage in answering geometric queries, such as intersec-
tion and trimming problems. Methods for change of representation are thus of great practical
importance. In this thesis we present several methods for implicitization; the process of chang-
ing from the parametric to the implicit representation. The emphasis is on methods which are
numerically stable and computationally efficient on modern hardware. We present a variety of
approaches to approximate implicitization using linear algebra. In particular, we explore the
implicitization of both rational parametric representations and envelopes - a type of curve or
surface which often has no simple parametric representation. We also introduce a new method
fortheimplicitrepresentationofrationalcubicBéziercurves,whichcanbeimplementedusing
explicitformulas.
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Contents
I Introduction 1
1 Background 3
1 Freeformcurveandsurfacerepresentation . . . . . . . . . . . . . . . . . . . . 4
2 Methodsforimplicitization ofrationalcurvesandsurfaces . . . . . . . . . . . 10
2 Summaryofpapers 21
PaperI:Approximate Implicitization ofTriangularBézierSurfaces . . . . . . . . . . 21
PaperII:Approximate implicitization usinglinearalgebra . . . . . . . . . . . . . . . 22
PaperIII:FastapproximateimplicitizationofenvelopecurvesusingChebyshevpoly-
nomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
PaperIV:Abasisfortheimplicit representationofrational cubicBéziercurves . . . 26
3 Applicationsofmethodsforchangeofrepresentation 29
1 Intersectionalgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Renderingofcurvesandsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Bibliography 39
II Scientific papers 45
PaperI:ApproximateImplicitization ofTriangularBézierSurfaces 47
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 TriangularBézierSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 ApproximateImplicitization . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 ExamplesofImplicitization ofBézierTriangles . . . . . . . . . . . . . . . . . 57
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
PaperII:Approximateimplicitization usinglinearalgebra 65
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Approximateimplicitization -theoriginalapproach . . . . . . . . . . . . . . . 67
4 Weakapproximate implicitization . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Approximateimplicitization usingorthonormal bases . . . . . . . . . . . . . . 70
vii
6 Examplesoftheoriginalapproachwithdifferentbases . . . . . . . . . . . . . 72
7 Comparisonofthealgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8 Approximateimplicitization ofsurfaces . . . . . . . . . . . . . . . . . . . . . 83
9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Paper III: Fast approximate implicitization of envelope curves using Chebyshev poly-
nomials 91
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2 EnvelopesofRationalFamiliesofCurves . . . . . . . . . . . . . . . . . . . . 92
3 FastApproximate Implicitization ofEnvelopeCurves . . . . . . . . . . . . . . 93
4 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
PaperIV:AbasisfortheimplicitrepresentationofrationalcubicBéziercurves 101
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2 AbasisforrepresentingrationalcubicBéziercurvesimplicitly . . . . . . . . . 103
3 Doublepointsoncubiccurves . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4 Degenerationtoconicsections . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5 Collinearpoints,zeroweightsandnumericalstability . . . . . . . . . . . . . . 116
6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7 Discussionandconclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Appendix4.A Somegeometricproperties . . . . . . . . . . . . . . . . . . . . . . 124
Appendix4.B LinearindependenceandproofsofTheorems . . . . . . . . . . . . . 125
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Part I
Introduction
1