Table Of ContentGeometric Modeling of Fractal Forms for CAD
Geometric Modeling and Applications Set
coordinated by
Marc Daniel
Volume 5
Geometric Modeling of
Fractal Forms for CAD
Christian Gentil
Gilles Gouaty
Dmitry Sokolov
First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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ISTE Ltd John Wiley & Sons, Inc.
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Library of Congress Control Number: 2021932086
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ISBN 978-1-78630-040-9
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter1.TheBC-IFSModel . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.Self-similarityandIFS . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1.Mathematicaldefinitionsandreminders. . . . . . . . . . . . . . . . 4
1.1.2.Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.3.Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.4.Iteratedfunctionsystems . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.5.Visualizationandapproximation . . . . . . . . . . . . . . . . . . . . 18
1.1.6.Computerimplementation . . . . . . . . . . . . . . . . . . . . . . . 19
1.1.7.Notionsofaddressandparameterization . . . . . . . . . . . . . . . 23
1.1.8.Finiteaddress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.1.9.IFSmorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.1.10.Exampleofacurve. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.1.11.TheexampleoftheSierpinskitriangle . . . . . . . . . . . . . . . . 28
1.2.Controllediteratedfunctionsystem . . . . . . . . . . . . . . . . . . . . 30
1.2.1.Formalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.2.2.Visualizationandapproximation . . . . . . . . . . . . . . . . . . . . 36
1.2.3.Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.Boundarycontrollediteratedfunctionsystem . . . . . . . . . . . . . . 38
1.3.1.Attractorsdefinedfromasetofcontrolpoints . . . . . . . . . . . . 43
1.3.2.Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.3.3.Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.3.4.Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.3.5.Topologycontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.3.6.Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.3.7.Evaluationtreeandquotientgraph. . . . . . . . . . . . . . . . . . . 62
vi GeometricModelingofFractalFormsforCAD
1.3.8.Constraintresolution . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.3.9.Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
1.3.10.Controlofthelocaltopology . . . . . . . . . . . . . . . . . . . . . 77
1.3.11.Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.3.12.Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
1.3.13.Choiceofthedisplayprimitive . . . . . . . . . . . . . . . . . . . . 82
1.3.14.Choiceofdimension . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1.3.15.Choiceofgeometry . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1.3.16.Primitiveproperty . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
1.3.17.BC-IFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Chapter2.DesignExamples . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.1.Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.2.Wiredstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.3.Surfacesandlaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.3.1.Quadrangularsubdivisions . . . . . . . . . . . . . . . . . . . . . . . 98
2.3.2.Triangularsubdivisions . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.3.3.Otherregularpolygonalsubdivisions . . . . . . . . . . . . . . . . . 109
2.3.4.TheSierpinskitriangle . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.3.5.Penrosetilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.4.Volumesandlacunarobjects . . . . . . . . . . . . . . . . . . . . . . . . 116
2.5.Treestructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.5.1.Basicexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.5.2.Basicexamplewithanundividedtrunk . . . . . . . . . . . . . . . . 120
2.5.3.Examplebytilingthespace . . . . . . . . . . . . . . . . . . . . . . 122
2.6.Formassembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Chapter3.SurfaceNURBS,SubdivisionSurfacesandBC-IFS . . . 129
3.1.Beziercurvesandsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.2.UniformB-splinecurvesandsurfaces . . . . . . . . . . . . . . . . . . . 132
3.3.Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.3.1.Polynomialcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.3.2.Rationalcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.4.NURBScurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.4.1.Knot-doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.4.2.Curvesubdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.4.3.Surfacesubdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.5.Subdivisioncurvesandsurfaces . . . . . . . . . . . . . . . . . . . . . . 151
3.5.1.Subdivisioncurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3.5.2.Subdivisionsurfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Contents vii
Chapter4.BuildingOperations,AssistancetoDesign
andApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.1.Topologicalconsistencyandsymmetryconstraints . . . . . . . . . . . 168
4.1.1.Orientationconstraints . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.1.2.Permutationoperatorsandsymmetryconstraints. . . . . . . . . . . 170
4.2.Topologicalcombination . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.2.1.Topologicaltensorproduct . . . . . . . . . . . . . . . . . . . . . . . 179
4.2.2.Exampleofsurfacegenerationfromcurves. . . . . . . . . . . . . . 184
4.2.3.Volumegenerationsfromcurves . . . . . . . . . . . . . . . . . . . . 188
4.2.4.Generationofthetreestructure . . . . . . . . . . . . . . . . . . . . 188
4.3.Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.3.1.Interactionbetweenrepresentationmodels . . . . . . . . . . . . . . 193
4.3.2.Structureoptimization. . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.3.3.Roughnessandspacerefilling . . . . . . . . . . . . . . . . . . . . . 208
4.3.4.Thedesignofaheatexchanger. . . . . . . . . . . . . . . . . . . . . 210
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Preface
This work introduces a model of geometric representation for describing and
manipulating complex non-standard shapes such as rough surfaces or porous
volumes. It is aimed at students in scientific education (mathematicians, computer
scientists, physicists, etc.), engineers, researchers or anyone familiar with the
mathematical concepts addressed at early stages of the graduate level. However,
many parts are accessible to all, in particular, all introductory sections that present
ideas with examples. People with no prior background, whether they are artists,
designers or simply curious, will be able to understand the philosophy of our
approach,anddiscoveranewuniverseofunsuspectedandexcitingforms.
Geometric representation models are mathematical tools integrated into
computer-aided geometric design (CAGD) software. They make the production of
numerical representations of forms possible. By means of graphical interfaces or
programming tools, users can draw and/or manipulate these shapes. They can also
testorevaluatetheirphysicalproperties(mechanical,electro-magnetic,acoustic,etc.)
by communicating geometric descriptions to further specific numerical simulation
software.
The geometric representation model we present here is based on the fractal
geometry paradigm. The principle behind this consists of studying the properties
(signal, geometry, phenomena, etc.) at different scales and identifying the invariants
fromthere.Theobjectsaredescribedasself-referentialbetweentwoscales: eachof
the object features (namely, the lower scale level) is described as a reference to the
object itself (namely, the higher scale). This approach is not conventional and often
confusingatfirst.Wecometoperceiveitsrichnessandpowerveryquickly,however.
The universe of forms that can possibly be created is infinite and has still only
partiallybeenexplored.
In this book, we present the mathematical foundations, so that the reader can
accessalltheinformationtounderstand,testandmakeuseofthismodel.Properties,
x GeometricModelingofFractalFormsforCAD
theoremsandconstructionmethodsaresupplementedwithalgorithmsandnumerous
examples and illustrations. Concerning the formalization, we have chosen to use
precise and rigorous mathematical notations to remove any ambiguity and make
understandingeasier.
Readersunwillingtobeconcernedwithmathematicalformalismscangettogrips
with the philosophy of our approach by focusing on the sections found at the
beginnings of the chapters, in which ideas and principles are intuitively presented,
basedonexamples.
This book is the result of 25 years of research carried out mainly in the LIRIS
laboratories of the University of Lyon I and LIB of the University of Burgundy
Franche-Comté. This research was initiated by Eric Tosan, who was instrumental at
theoriginofthisformalismandtowhomwededicatethiswork.
ChristianGENTIL
GillesGOUATY
DmitrySOKOLOV
February2021
Introduction
I.1. Fractalsforindustry: whatfor?
This book shows our first steps toward the fundamental and applied aspects of
geometric modeling. This area of research addresses the acquisition, analysis and
optimizationofthenumericalrepresentationof3Dobjects.
a) 3Dmodel b) 3Dprinting
FigureI.1.3Dtreebuiltbyiterativemodeling(source:
projectMODITEREno.ANR-09-COSI-014)
Figure I.1 shows an example of a structure that admits high vertical loads, while
minimizing the transfer of heat between the top and bottom of the part. Additive
manufacturing (3D printing) allows, for the first time, the creation of such complex
objects, even in metal (here with a high-end laser printer EOS M270). This type of
technologywillhaveahighsocietalandeconomicimpact,enablingbettersystemsto
be created (engines, cars, airplanes, etc.), designed and adapted numerically for
optimalfunctionality,thusconsuminglessrawmaterial,fortheirmanufacturing,and
energy,whenused.
