Subdivision Surfaces

212 Pages·2007·3.14 MB·English

by  Jörg Peters

Geometry and Computing
SeriesEditors
HerbertEdelsbrunner
LeifKobbelt
KonradPolthier
EditorialAdvisoryBoard
Jean-DanielBoissonnat
GunnarCarlsson
BernardChazelle
Xiao-ShanGao
CraigGotsman
LeoGuibas
Myung-SooKim
TakaoNishizeki
HelmutPottmann
RobertoScopigno
Hans-PeterSeidel
SteveSmale
PeterSchro¨der
DietrichStoyan
•
Jo¨rg Peters Ulrich Reif
Subdivision Surfaces
With 52 Figures
123
Jo¨rgPeters UlrichReif
UniversityofFlorida TUDarmstadt
Dept.Computer&InformationScience FBMathematik
GainesvilleFL32611-6120 Schloßgartenstr.7
USA 64289Darmstadt
[email protected]fl.edu Germany
[email protected]
ISBN978-3-540-76405-2 e-ISBN978-3-540-76406-9
SpringerSeriesinGeometryandComputing
LibraryofCongressControlNumber:2008922290
MathematicsSubjectsClassification(2000):65D17,65D07,53A05,68U05
(cid:1)c 2008Springer-VerlagBerlinHeidelberg
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Preface
AkintoB-splines,theappealofsubdivisionsurfacesreachesacrossdisciplinesfrom
mathematics to computer science and engineering. In particular, subdivision sur-
faces have had a dramatic impact on computer graphics and animation over the
last 10 years: the results of a development that started three decades ago can be
viewed today at movie theaters, where feature length movies cast synthetic char-
acters ‘skinned’ with subdivision surfaces. Correspondingly, there is a rich, ever-
growingliteratureonitsfundamentalsandapplications.
Yet,aswitheveryvibrantnewfield,thelackofauniformnotationandstandard
analysistoolshasaddedunnecessary,attimesinconsistent,repetitionthatobscures
thesimplicityandbeautyoftheunderlyingstructures.Onegoalinwritingthisbook
istohelpshortenintroductorysectionsandsimplifyproofsbyproposingastandard
setofconceptsandnotation.
Whenwestartedwritingthisbookin2001,wefeltthatthefieldhadsufficiently
settled for standardization. After all, Cavaretta, Dahmen, and Micchelli’s mono-
graph [CDM91] had appeared 10 years earlier and we could build on the habilita-
tionofthesecondauthoraswellasanumberofjointpapers.Butitwasonlyinthe
processofwritingandseeingtheissuesinconjunction,thatstructuresandnotation
becameclearer.Infact,thelengthofthebookrepeatedlyincreasedanddecreased,
askeyconceptsandstructuresemerged.
Chapter2 ,forexample,wasalateaddition,asitbecameclearthatthedifferen-
/15
tial geometry for singular parameterizations, of continuity, smoothness, curvature,
and injectivity, must be established upfront and in generality to simplify the ex-
positionandlaterproofs.Bycontrast,thekeydefinitionofsubdivisionsurfacesas
splineswithsingularities,inChap.3 ,wasapartofthefoundationsfromtheoutset.
/39
This point of view implies a radical departure from any focus on control nets and
insteadplacesthemainemphasisonnestedsurfacerings,asexplainedinChap.4 .
/57
Carefulexaminationofexistingproofsledtotheexplicitformulationofanumber
of assumptions, in Chap.5 , that must hold when discussing subdivision surfaces
/83
ingenerality.Conversely,placingthesekeyassumptionsupfront,shortenedthepre-
sentationconsiderably.Therefore,thestandardexamplesofsubdivisionalgorithms
vii
viii Preface
reviewed in Chap.6 are presented with a new, shorter and simpler analysis than
/109
in earlier publications. Chapters 7 and 8 were triggered by very recent, new
/125 /157
insights and results and partly contain unpublished material. The suitability of the
majorknownsubdivisionalgorithmsforengineeringdesignwasattheheartofthe
investigationsintotheshapeofsubdivisionsurfacesinChap.7 .Theshortcomings
/125
of the standard subdivision algorithms discovered inthe process forced arenewed
search for an approach to subdivision capable of meeting shape and higher-order
continuity requirements. Guided subdivision was devised in response. The second
partofChap.7 recaststhisclassofsubdivisionalgorithmsinamoreabstractform
/125
that may be used as a prototype for a number of new curvature continuous subdi-
vision algorithms. The first part of Chap.8 received a renewed impetus from a
/157
recentstreamofpublicationsaimedatpredictingthedistanceofasubdivisionsur-
face from their geometric control structures after some m refinement steps. The
introduction of proxy surfaces and the distance to the corresponding subdivision
surface subsumes this set of questions and provides a framework for algorithm-
specificoptimalestimates.ThesecondpartofChap.8 grewoutofthesurprising
/157
observationthattheCatmull–Clarksubdivisioncanrepresentthesamesphere-like
objectstartingfromanymemberofawholefamilyofinitialcontrolconfigurations.
The final chapter, Chap.9 , shows that a large variety of subdivision algorithms
/175
is fully covered by the exposition in the book. But it also outlines the limits of
ourcurrentknowledgeandopensawindowtothefascinatingformsofsubdivision
currently beyond the canonical theory and to the many approaches still awaiting
discovery.
As a monograph, the book is primarily targeted at the subdivision community,
including not only researchers in academia, but also practitioners in industry with
aninterestinthetheoreticalfoundationsoftheirtools.Itisnotintendedasacourse
textbookandcontainsnoexercises,butanumberofworkedoutexamples.However,
weaimedatanexpositionthatisasself-containedaspossible,requiring,wethink,
onlybasicknowledgeoflinearalgebra,analysisorelementarydifferentialgeometry.
The book should therefore allow for independent reading by graduate students in
mathematics,computerscience,andengineeringlookingforadeeperunderstanding
ofsubdivisionsurfacesorstartingresearchinthefield.
Two valuable sources that complement the formal analysis of this book are the
SIGGRAPH course notes [ZS00] compiled by Schro¨der and Zorin, and the book
‘SubdivisionMethodsforGeometricdesign’byWarrenandWeimer[WW02].The
notes offer the graphics practitioner a quick introduction to algorithms and their
implementation and the book covers a variety of interesting aspects outside our
focus; for example, a connection to fractals, details of the analysis of univariate
algorithms,variationalalgorithmsbasedondifferentialoperatorsandobservations
thatcansimplifyimplementation.
Weaimedatunifyingthepresentation,placingforexamplebibliographicalnotes
attheendofeachcorechaptertopointoutrelevantandoriginalreferences.Inaddi-
tiontothese,weincludedalargenumberofpublicationsonsubdivisionsurfacesin
Preface ix
thereferencesection.Ofcourse,giventheongoinggrowthofthefield,thesenotes
cannotclaimcompleteness.Wethereforereservedtheinternetsite
www.subdivision-surface.org
for future pointers and additions to the literature and theme of the book, and, just
possibly,tomitigateanydamageofinsufficientproofreadingonourpart.
Itisourpleasuretothankatthispointourcolleaguesandstudentsfortheirsup-
port: Jianhua Fan, Ingo Ginkel, Jan Hakenberg, Rene´ Hartmann, Ke¸stutis
Karcˇiauskas,MinhoKim,AshishMyles,TianyunLisaNi,AndyLeJengShiue,
GeorgUmlauf,andXiaobinWuwhoworkedwithusonsubdivisionsurfacesover
many years. JanHakenberg,Rene´Hartmann,MalcolmSabin,NeilStewart, and
GeorgUmlaufhelpedtoenhancethemanuscriptbycarefulproof-readingandpro-
viding constructive feedback. MalcolmSabinand GeorgUmlaufadded valuable
materialforthebibliographicalnotes.NiraDynandMalcolmSabinwillinglycon-
tributed two sections to the introductory chapter, and it was again Malcolm Sabin
whosharedhisextensivelistofreferencesonsubdivisionwhichformedthestarting
pointofourbibliography.ChandrajitBajajgenerouslyhostedaretreatoftheauthors
thatbroughtaboutthefinalstructureofthebook.Manythankstoyouall!Thework
wassupportedbytheNSFgrantsCCF-0430891andDMI-0400214.
Ourfinalthanksarereservedforourfamiliesfortheirsupportandtheirpatience.
Youkeptusinspired.
Contents
1 IntroductionandOverview...................................... 1
1.1 RefinedPolyhedra.......................................... 1
1.2 ControlNets............................................... 2
1.3 SplineswithSingularities.................................... 4
1.4 FocusandScope ........................................... 6
1.5 Overview ................................................. 7
1.6 Notation .................................................. 7
1.7 AnalysisintheShift-InvariantSetting ......................... 8
1.8 HistoricalNotesonSubdivisiononIrregularMeshes............. 11
2 GeometryNearSingularities .................................... 15
2.1 DotandCrossProducts ..................................... 16
2.2 RegularSurfaces ........................................... 17
2.3 SurfaceswithaSingularPoint................................ 23
2.4 CriteriaforInjectivity....................................... 31
3 GeneralizedSplines ............................................ 39
3.1 AnAlternativeViewofSplineCurves ......................... 40
3.2 ContinuousBivariateSplines................................. 41
3.3 Ck-Splines................................................ 44
3.4 Ck-Splines................................................ 47
r
3.5 ABicubicIllustration ....................................... 53
4 SubdivisionSurfaces ........................................... 57
4.1 Refinability ............................................... 58
4.2 SegmentsandRings ........................................ 59
4.3 SplinesinFinite-DimensionalSubspaces....................... 65
4.4 SubdivisionAlgorithms ..................................... 67
4.5 AsymptoticExpansionofSequences .......................... 71
4.6 JordanDecomposition ...................................... 72
4.7 TheSubdivisionMatrix ..................................... 75
xi
xii Contents
5 Ck-SubdivisionAlgorithms ..................................... 83
1
5.1 GenericInitialData......................................... 84
5.2 StandardAlgorithms........................................ 84
5.3 GeneralAlgorithms......................................... 89
5.4 ShiftInvariantAlgorithms ................................... 95
5.5 SymmetricAlgorithms ......................................103
6 CaseStudiesofCk-SubdivisionAlgorithms .......................109
1
6.1 Catmull–ClarkAlgorithmandVariants.........................109
6.2 Doo–SabinAlgorithmandVariants............................116
6.3 SimplestSubdivision .......................................120
7 ShapeAnalysisandCk-Algorithms ..............................125
2
7.1 HigherOrderAsymptoticExpansions .........................126
7.2 ShapeAssessment..........................................134
7.3 ConditionsforCk-Algorithms................................140
2
7.4 AFrameworkforCk-Algorithms .............................145
2
7.5 GuidedSubdivision.........................................149
8 ApproximationandLinearIndependence.........................157
8.1 ProxySplines..............................................157
8.2 LocalandGlobalLinearIndependence ........................169
9 Conclusion ....................................................175
9.1 FunctionSpaces............................................176
9.2 Recursion .................................................177
9.3 CombinatorialStructure .....................................178
References.........................................................183
Index .............................................................199