SAGA – Advances in ShApes, Geometry, and Algebra: Results from the Marie Curie Initial Training Network

324 Pages·2014·7.001 MB·English

by  Tor Dokken, Georg Muntingh (eds.)

Geometry and Computing 10
Tor Dokken
Georg Muntingh
Editors
SAGA –
Advances in
ShApes, Geometry,
and Algebra
Results from the Marie Curie
Initial Training Network
Geometry and Computing
10
SeriesEditors
HerbertEdelsbrunner
LeifKobbelt
KonradPolthier
EditorialAdvisoryBoard
Jean-DanielBoissonnat
GunnarCarlsson
BernardChazelle
Xiao-ShanGao
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LeoGuibas
Myung-SooKim
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Moreinformationaboutthisseriesat
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Tor Dokken (cid:129) Georg Muntingh
Editors
SAGA – Advances
in ShApes, Geometry,
and Algebra
Results from the Marie Curie
Initial Training Network
123
Editors
TorDokken
GeorgMuntingh
SINTEFICT
Oslo
Norway
ISSN1866-6795 ISSN1866-6809(electronic)
ISBN978-3-319-08634-7 ISBN978-3-319-08635-4(eBook)
DOI10.1007/978-3-319-08635-4
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014953535
MathematicsSubjectClassification(2010):13P25,14Q10,65D17,65D18,65D07,(13D02),41A10,
49M99,65M60,65N30
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Preface
SAGA – ShApes, Geometry and Algebra – was a Marie Curie Initial Training
Network (ITN) organizedfrom 2008 to 2012 and intended to offer researchers in
thefirst fiveyearsoftheircareerstheopportunityto improvetheir researchskills,
joinestablishedresearchteamsandenhancetheircareerprospects.
As a common requirement in an ITN, researchers were usually expected to
exhibittransnationalmobility,i.e.,movefromonecountrytoanother.Inadditionto
crossing geographicborders, SAGA offered mobility between the closely related,
but nearly disjoint, scientific communities of Geometric Modeling and Algebraic
Geometry. Projects funded by the European Union have played a central part in
bridgingthe gap betweenthese communities,with the first steps taken from 2001
to 2005in the Fifth FrameworkFutureand EmergingTechnologiesprojectGAIA
– Intersection Algorithms for Geometry-based IT, which utilized approximate
algebraicmethods.
Foryoungandseasonedresearchersalike,SAGAhasofferedaninspiringvenue
for exchanging ideas and forming new connections for cooperation. The young
researchers of SAGA have not only established relations with researchers at their
own institutions, but also with researchers at other institutions working in their
fieldofinterest,throughthesecondmentstootherpartnersandparticipationinthe
workshops and events. That being said, the most important aspect for the future
careers of these young researchers is most likely the basis for future cooperation
andresearchtheyhaveestablishedamongthemselves.
This book provides insights into research conducted in the SAGA ITN and
represents its long-term documentation. It consists of an introduction and 14
chapters,dividedintofourpartsaccordingtotheworkpackagesintheSAGAITN,
andwrittenbyacombinationofyoungandestablishedresearchers.
Theeditorsaregratefultothereviewersfortheirreports,whichmadeitpossible
to select chapters suitable for publication and to improve them considerably. In
addition,wewouldliketothanktheleadscientistsfortheircontinuingdevotionto
SAGA.SpecialthanksgotoEwaldQuak,forhiscentralroleintheorganizationof
eventsandensuringqualitythroughout.Wewouldalsoliketoexpressourgratitude
v
vi Preface
toallexternalcontributors,amongwhomTomGrandinedeservesspecialmention
forhisexcellenttalksattheeventsconnectingmathematicstotheindustry.Finally,
we would like to thank Springer, in particular Martin Peters and Ruth Allewelt,
for the pleasant and smooth cooperation in preparing this volume. The research
leadingtothesefindingswassupportedbyfundingfromtheEuropeanCommunity’s
Seventh Framework Program under grant agreement no. PITN-GA-2008-214584-
SAGA.
Oslo,Norway TorDokken
May2014 GeorgMuntingh
Contents
1 IntroductiontoShApes,Geometry,andAlgebra ....................... 1
TorDokkenandGeorgMuntingh
1.1 Motivation............................................................ 1
1.2 TheSAGAInitialTrainingNetwork................................ 2
1.3 APreviewofThisBook ............................................. 4
References.................................................................... 7
PartI ChangeofRepresentation
2 NumericalMethodsforImplicitisationandTheirApplications ...... 11
OliverJ.D.Barrowclough
2.1 Introduction........................................................... 11
2.2 Implicitisation Methodsfor Low Degree Planar
RationalCurves ...................................................... 13
2.2.1 LinesinthePlane........................................... 13
2.2.2 RationalQuadraticBézierCurvesinthePlane ........... 14
2.2.3 RationalCubicBézierCurvesinthePlane................ 15
2.2.4 RationalBézierCurvesofHigherDegree................. 16
2.3 NumericalMethodsforImplicitisationof Higher
DegreeCurves,SurfacesandHypersurfaces........................ 18
2.3.1 A General Method for Implicitisation
andApproximationUsingLinearAlgebra................ 19
2.3.2 ApproximateImplicitisation............................... 20
2.3.3 InterpolationandApproximationofPointData........... 22
2.3.4 SparseImplicitisation ...................................... 24
2.4 ImplicitisationofEnvelopeCurves,Surfaces
andHypersurfaces.................................................... 25
vii
viii Contents
2.5 ApplicationsofMethodsforImplicitisation........................ 26
2.5.1 IntersectionAlgorithmsinComputerAidedDesign...... 26
2.5.2 RenderingCurvesandSurfaces............................ 30
2.5.3 Robotics..................................................... 33
References.................................................................... 35
3 SparseImplicitizationviaInterpolation ................................. 39
IoannisZ.Emiris,TatjanaKalinka,andChristosKonaxis
3.1 Introduction........................................................... 39
3.2 ImplicitizationReducedtoElimination............................. 41
3.3 AlgorithmandImplementation...................................... 44
3.3.1 MultidimensionalKernel................................... 46
3.3.2 BernsteinBasis ............................................. 47
3.3.3 ComparisonstoOtherMethods............................ 48
ConclusionandFutureWork................................................ 49
References.................................................................... 50
4 The Intersection Problems of Parametric Curves
andSurfacesbyMeansofMatrix-BasedImplicitRepresentations... 53
ThangLuuBa
4.1 Introduction........................................................... 53
4.2 MatrixBasedImplicitRepresentationsofParametricSurfaces.... 54
4.2.1 ConstructionofMatrixRepresentations................... 54
4.2.2 PointsonSurfaceandInversionProblem ................. 57
4.3 Curve/SurfaceIntersection........................................... 58
4.3.1 Linearization of a PolynomialMatrix
intheMonomialBasis ..................................... 59
4.3.2 ExtractingtheRegularPartofaNon-square
PencilofMatrices .......................................... 60
4.4 Surface/SurfaceIntersection......................................... 63
4.5 Matrix-BasedImplicitRepresentationsofParametric
CurvesinSpace ...................................................... 67
4.5.1 ConstructionoftheRepresentationMatrix................ 67
4.5.2 PointsonCurvesandInversionProblems................. 69
4.5.3 RankofaRepresentationMatrixataSingularPoint..... 71
4.5.4 Curve/CurveIntersection................................... 72
Conclusion ................................................................... 72
References.................................................................... 73
PartII GeometricComputing:AlgebraicTools
5 SingularZerosofPolynomialSystems................................... 77
AngelosMantzaflarisandBernardMourrain
5.1 Introduction........................................................... 77
5.2 PreliminaryConsiderations.......................................... 79
5.2.1 IsolatedPointsandDifferentials........................... 80
5.2.2 QuotientRingandDualStructure ......................... 82
Contents ix
5.3 ComputingLocalRingStructure.................................... 85
5.3.1 Macaulay’sDialyticMatrices.............................. 86
5.3.2 IntegrationMethod ......................................... 87
5.3.3 ComputingaPrimal-DualPair............................. 88
5.4 DeflationofaSingularPoint ........................................ 92
5.4.1 TheUnivariateCase........................................ 92
5.4.2 DeflationUsingtheDialyticApproach.................... 93
5.4.3 DeflationUsingtheInverseSystem ....................... 95
5.5 ApproximateMultiplePoint......................................... 98
5.6 Experimentation...................................................... 100
References.................................................................... 102
6 PlaneMixedDiscriminantsandToricJacobians ....................... 105
AliciaDickenstein,IoannisZ.Emiris,andAnnaKarasoulou
6.1 Introduction........................................................... 105
6.2 PreviousWorkandNotation......................................... 108
6.3 AGeneralFormula................................................... 112
6.4 TheMultiplicativityoftheMixedDiscriminant.................... 118
ConclusionandFutureWork................................................ 120
References.................................................................... 120
7 Topology of the Intersection of Two Parameterized
Surfaces,UsingComputationsin4DSpace ............................. 123
StéphaneChauandAndréGalligo
7.1 Introduction........................................................... 123
7.1.1 InterestoftheProblem..................................... 123
7.1.2 An Example of Biquadratic Meshing
ofaProceduralSurface..................................... 124
7.1.3 OrganizationofthePaper .................................. 126
7.2 PreviousWorkonTopologyComputationofaCurve.............. 126
7.2.1 IsotopicCurve .............................................. 126
7.2.2 RegularityTestandSubdivisionMethod,
theCaseofaPlaneCurve.................................. 127
7.2.3 NoLoopandSingleComponent........................... 128
7.3 Topologyof(Parameterized)Surface/SurfaceIntersection ........ 128
7.3.1 Equations.................................................... 128
7.3.2 Topologyofa4DimensionalImplicitCurve
(RegularityCriterion) ...................................... 129
7.4 AlgorithmsandDataStructures..................................... 135
7.4.1 HextreeDataStructureandTopology ..................... 135
7.4.2 SubdivisionAlgorithm ..................................... 136
7.4.3 ConnectedComponentsandLoops........................ 137
7.5 TopologyinR3 ....................................................... 138
7.5.1 OneCurveBox ............................................. 139
7.5.2 NodeandDiscretization.................................... 140
7.5.3 Examples.................................................... 142