Approximation and Modeling with B-Splines

221 Pages·2014·10.838 MB·English

by  Klaus Höllig, Jörg Hörner

Approximation
and Modeling
with B-Splines

Approximation
and Modeling
with B-Splines
Klaus Höllig
Universität Stuttgart
Stuttgart, Germany
Jörg Hörner
Universität Stuttgart
Stuttgart, Germany
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2013 by the Society for Industrial and Applied Mathematics
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Library of Congress Cataloging-in-Publication Data

Höllig, Klaus, author.
Approximation and modeling with B-splines / Klaus Höllig, Universität Stuttgart, Stuttgart,
Germany; Jörg Hörner, Universität Stuttgart, Stuttgart, Germany.
pages cm. -- (Applied mathematics)
Includes bibliographical references and index.
ISBN 978-1-611972-94-8 (alk. paper)
1. Spline theory. 2. Approximation theory. 3. Numerical analysis. 4. Mathematical models.
5. Engineering--Mathematical models. 6. Computer science--Mathematics. 7. Algorithms. 8.
Spline theory--Industrial applications. I. Hörner, Jörg, author. II. Title.
QA224.H645 2013
511’.4223--dc23
2013029695
is a registered trademark.
Contents
Preface vii
Introduction ix
Notation andSymbols xiii
1 Polynomials 1
1.1 MonomialForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TaylorApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 BernsteinPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 PropertiesofBernsteinPolynomials . . . . . . . . . . . . . . . . . . . . . 11
1.6 HermiteInterpolant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 ApproximationofContinuousFunctions. . . . . . . . . . . . . . . . . . 16
2 BézierCurves 19
2.1 ControlPolygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 PropertiesofBézierCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 AlgorithmofdeCasteljau . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 GeometricHermiteInterpolation. . . . . . . . . . . . . . . . . . . . . . . 33
3 RationalBézierCurves 37
3.1 ControlPolygonandWeights . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 BasicProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 ConicSections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 B-Splines 51
4.1 RecurrenceRelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 RepresentationofPolynomials. . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 EvaluationandDifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6 PeriodicSplines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
v
vi Contents
5 Approximation 77
5.1 Schoenberg’sScheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Quasi-Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 AccuracyofQuasi-Interpolation . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.5 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.6 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6 SplineCurves 105
6.1 ControlPolygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 BasicProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 MultivariateSplines 133
7.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 PolynomialApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.3 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 ApproximationMethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.6 HierarchicalBases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8 SurfacesandSolids 155
8.1 BézierSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.2 SplineSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.3 SubdivisionSurfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.4 Blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.5 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9 FiniteElements 173
9.1 Ritz–GalerkinApproximation . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.2 WeightedB-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.3 IsogeometricElements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
9.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.5 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
NotesandComments 193
Appendix 197
Bibliography 205
Index 213
Preface
B-splinesarefundamentaltoapproximation anddatafitting,geometricmodeling, auto-
mated manufacturing, computer graphics, and numerical simulations. Because of their
computationalefficiency,flexibility,andelegance,B-splinetechniquesaremuchsuperior
toother,moreelementary,piecewisepolynomialrepresentations. Asaconsequence,they
havebecomethemethodofchoiceinnumerousbranchesofappliedmathematics,com-
puterscience,andengineering.
InthisbookwegiveanintroductiontothebasicB-splinetheory,describingapproxi-
mationmethodsandalgorithmsaswellasmodelinganddesigntechniques. Wethinkthat
onlyasolidknowledgeinalltheseareasprovidesanoptimalbasisforinterdisciplinaryre-
searchandhandlingofcomplexnovelapplications. Thenewfiniteelementschemeswith
B-splinesprovideaperfectexampleofasuccessfulsynthesisofmethodsfromthesediffer-
entfields.
TopicsdiscussedinourbookincludetheBézierform,computingwithB-splines,ap-
proximationandinterpolation,splinerepresentationsofcurves, surfaces,andsolids,hi-
erarchicalbases,andfiniteelementsimulation. Wedonotaimforcompletenessasmore
comprehensive and specialized texts do. Instead, we focus on key results and methods
whicharemostwidelyusedinpractice. Inthisway,everyimportantaspectofB-spline
theoryisdescribedinarelativelyshortmonograph,leadingfromelementarybasicmate-
rialtoadvancedtopicswhicharesubjectofcurrentresearch.
Thematerialofthebookcanbealmostcoveredinaone-semestermathematicsorcom-
putersciencegraduatecourse. Thecombinationofmathematics,programming, model-
ing,andgraphicsmakesthesubjectfascinatingtoteach. Thereisanever-endingsupplyof
interestingthesistopics,typicallyprovidedbynewindustrialapplications. Furtherstim-
ulatingtheenthusiasmforB-splinesamongstudentsandassistinginteachingaresomeof
ourgoals. Togetherwiththisbookweplantoprovide
• acollectionofproblems,partiallywithsolutions;
• slidesforlectures;
• programsanddemos.
Thissupplementarymaterialwillbemadeavailableonthewebsiteforourbook(http://
www.siam.org/books/ot132).
Thebookisessentiallyself-contained. Somebasicfactsfromlinearalgebra,analysis,
aswellaselementarydifferentialgeometry andfunctionalanalysis, whicharerequired,
arelistedinanappendix. Hence,thematerialiseasilyaccessiblenotonlyformathematics
andcomputersciencestudentsbutalsoforbeginninggraduatesinengineering.
Wehaveverymuchenjoyedwritingthisbook. TheNumericalAnalysisandGeomet-
ricModelinggroup inStuttgart provided avery creative atmosphere. Inparticular, we
gratefullyacknowledgetheexcellentcooperation withUlrichReifandJoachimWipper
vii
viii Preface
onfiniteelementprojects. SpecialthanksalsotoElisabethHölligandIrmgardWalterfor
helpingtoproofreadvariousdraftsofourmanuscipt.
Stuttgart,December2012
KlausHölligandJörgHörner
Introduction
B-splinesplayanimportantroleinmanyareasofappliedmathematics,computerscience,
andengineering. Typical applicationsariseinapproximation offunctionsanddata, au-
tomateddesignandmanufacturing,computergraphics,medicalimaging,andnumerical
simulation. Therearealsobeautifulresultsinpuremathematics,inparticularonn-width
[88]andinconnectionwithbox-splinetheory[22]. Thisdiversityofareasandtechniques
involvedmakesB-splinesafascinatingresearchtopicwhichhasattractedagrowingnum-
berofscientistsinuniversitiesandindustry.
Polynomialandpiecewisepolynomialapproximationshavebeenusedinanumberof
different contexts for a very long time. But perhaps it isfairto say that the systematic
analysisofsplinesbeganwithSchoenberg’spaperonapproximation ofequidistantdata
in1946[105]. Hiswork initiatedvery activeresearch onapproximation methods, asis
documented, e.g., in theearly books on splines by Ahlberg, Nilson, and Walsh [1], de
Boor[19],andSchumaker[111]. ThefullpotentialofB-splinesfornumericalmethods
wasrealized by deBoor. Hisalgorithmsforcomputing withlinearcombinationsofB-
splines remain basic toolsfor any splinesoftware; see [15, 18] and APractical Guide to
Splines[19]. Onthetheoreticalside,deBoor’sdefinitionofamultivariateB-spline[17]
was the starting point of a genuine multivariate spline theory with key contributions
byDahmenandMicchelli(cf.,e.g., [36]forasmallsampleofintriguingresultsonbox-
splines),deBoor,DeVore,Riemenschneider,Schumaker,andmanyothers.
Paralleltothemathematicalresearchonsplines,engineersdevelopedpiecewisepoly-
nomialrepresentationsforautomateddesignandmanufacturing[46]. AtBoeing,Fergu-
son [53]used splineinterpolation forcurve andsurfacemodeling. AtGeneral Motors,
Coons [33]invented the well-knownblending scheme forcurve networks. AtRenault
andCitroën,afundamentalsteptowardsmodernmodelingtechniqueswasmadebyBézier
[7,8,9,10]anddeCasteljau[25,26]. Theyindependentlydiscoveredthefavorablegeo-
metricandalgorithmicpropertiesoftheBernsteinbasis,introducingcontrolpolygonsas
anintuitivetoolfordesigners.
TheconferenceinUtah[3],organizedbyBarnhillandRiesenfeldin1974,helpedto
bridgethegapbetween themathematical andengineeringcommunity. Anewresearch
area was founded: Computer Aided Geometric Design (CAGD). The famous knot in-
sertion and subdivision algorithms forcontrol polygons developed by Boehm [11]and
Cohen,Lyche,andRiesenfeld[30]illustratethebeautyoftheinterplaybetweenmodel-
ing(engineering)andapproximation(mathematics). Farin’sbook[49]givesanexcellent
description of the emerging geometric spline theory (cf. also the recent books by Co-
hen,Riesenfeld,andElber[31]andPrautzsch, Boehm, andPaluszny[90]aswellasthe
HandbookofComputerAidedGeometricDesign[52]and[48]).
B-splinesalsobegantoplayanimportantroleincomputergraphics. Chaikin’salgo-
rithm[29,100]wasthefirstexampleofafastspline-based renderingschemeforcurves.
Very well known are the algorithms of Catmull/Clark [27] and Doo/Sabin [43] for
ix
x Introduction
refinement of surface meshes with arbitrary topology (cf. the book by Peters and Reif
[86]forthe fascinatingmathematical theory behind such seeminglysimple subdivision
strategies). Step by step, B-splines entered almost all engineering areas. Most recently,
B-splines were used as finite elements. Weighted approximations, proposed by Höllig,
Reif,andWipper[69],andisogeometricmethods,introducedbyHughes,Cottrell,and
Bazilevs[71],haveprovedtobequitesuccessful.
Λsplines,β-splines,ν-splines,ω-splines,τ-splines,A-splines,ARMAsplines,B-splines,Bernoullisplines,
BM-splines,box-splines,cardinalsplines,Catmull-Romsplines,Dm-splines,Dirichletsplines,discretesplines,
E-splines,ellipticsplines,exponentialEulersplines,exponentialboxsplines,fundamentalsplines,g-splines,
Gibbs-Wilbrahamsplines,Hm,p-splines,harmonicsplines,Helixsplines,Hermitesplines,Hermite-Birkhoff
splines,histosplines,hyperbolicsplines,Inf-convolutionsplines,K-splines,L-monosplines,L-splines,Lagrange
splines,LB-splines,Legendresplines,Lg-splines,M-splines,metaharmonicsplines,minimal-energysplines,
monosplines,naturalsplines,NBV-splines,NURBS,ODRsplines,perfectsplines,polyharmonicsplines,
Powell-Sabinsplines,pseudosplines,Q-splines,Schoenbergsplines,simplexsplines,smoothingsplines,super
splines,thin-platesplines,triangularsplines,trigonometricsplines,Tschebyscheffsplines,TURBS,v-splines,
variationdiminishingsplines,vertexsplines,VP-splines,web-splines,Wilson-Fowlersplines,X-splines,...
DifferentTypesofSplinesandSplineTechniques
Todatemorethan8000articlesandbooksonsplineshavebeenpublished. Theabove
listing gives the impression that there is a spline for almost any application, and there
is! Our book covers only a very limited selection—it serves as a basic introduction to
B-splinetheory. Inadditiontoessentialbasicmaterial,wehavegivenprioritytothede-
scriptionoftopicswhichareofprimaryimportanceinpracticeand/orwhicharesubject
ofcontinuingresearch.
OutlineoftheText
After some preliminary discussion of basic facts about polynomials in Chapter 1,
Chapters2and3aredevotedtotheBernstein–Bézierform. Thisimportantspecialcase
allowsustoexplain manykeyfeaturesofB-splinerepresentationsinarelativelysimple
setting. UnivariateB-splinesaredefinedinChapter4. Wetakeanalgorithmicapproach,
derivingpropertiesfromthefundamentalrecurrencerelations. Chapter5discussesvar-
iousapproximation methodssuchasinterpolation, quasi-interpolation, andsmoothing.
Then in Chapter 6, we describe modeling techniques for spline curves. Fundamental
geometricalgorithmsareknotinsertionanduniformsubdivisionwithnumerousappli-
cations. Chapters 7and8 aredevotedtothemultivariatetheory ofapproximation and
modelingbased onthetensorproduct formalism. Thissimplestandmostefficient rep-
resentationdoesalsopossesssufficientlocalflexibilityincombinationwithhierarchical
refinement. Thelastchaptergivesabriefintroductiontofiniteelementsimulationwith
weightedandisogeometricB-splineelements(cf.thebooks[65,34]foracomprehensive
treatmentofthesetechniques).
NotationandConventions
Throughoutthebook,linearcombinationsofB-splines,
(cid:2)
p= c bn ,
k k,ξ
k
play thefundamental role. Theletter“b”isacanonical choice for basisfunctions, and
“c” stands for coefficients or control points. For simplicity, we will often suppress de-
pendencies on parameters if they are fixed or implicitly understood while discussing a