Implicit objects in computer graphics

211 Pages·2002·1.744 MB·English

by  Luiz Velho, Jonas Gomes, Luiz H. de Figueiredo

Implicit Objects in
Computer Graphics
Luiz Velho
Jonas Gomes
Luiz Henrique de Figueiredo
Springer
Implicit Objects in Computer Graphics

Luiz Velho Jonas Gomes
Luiz Henrique de Figueiredo
Implicit Objects in
Computer Graphics
With 65 Figures
LuizVelho
JonasGomes
LuizHenriquedeFigueiredo
InstitutodeMatematicaPuraeAplicada
RiodeJaneiro,RJ22460-320
Brazil
LibraryofCongressCataloging-in-PublicationData
Velho,Luiz.
Implicitobjectsincomputergraphics/LuizVelho,JonasGomes,LuizHenriquede
Figueiredo.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN0-387-98424-0(alk.paper)
1.Computergraphics. I.Gomes,Jonas. II.Figueiredo,LuizHenriquede.
III.Title.
TS385.V452002
006.6—dc21 2002016005
ISBN0-387-98424-0 Printedonacid-freepaper.
2002Springer-VerlagNewYork,Inc.
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Preface
Overview
In this book we discuss the role of implicitly defined objects in com-
puter graphics, within a coherent conceptual framework. We are con-
cerned with the mathematical definition of shapes using an implicit
form, as well as with its applications to geometric modeling, visualiza-
tion and animation.
Organization and Features
The book is divided into five parts: mathematical foundations of ge-
ometric models, implicit formulations for the specification of shapes,
implicit primitives, techniques for constructing and manipulating im-
plicit objects, modeling, rendering and animation implicit objects.
Audience
The book is useful to researchers and graduate students in computer
graphicsandgeometricmodeling,andalsotoprofessionalsinthefields
of CAD/CAM and special effects.
Acknowledgments
We wish to thank our colleagues of the VISGRAF Laboratory for
providing a great research enviroment that stimulated our work in the
area of computer graphics. We also thank our students from IMPA,
PUC-Rio, and UFRJ, that participated in many activities related to
the development of this book.
iv
Rio de Janeiro, November 2001
Luiz Velho
Jonas Gomes
Luiz Henrique Figureiredo
Contents
1. Introduction ........................................ 1
1.1 Parametric versus Implicit.......................... 1
1.2 Motivation ....................................... 2
1.3 Scope and Overview ............................... 3
2. Manifolds ........................................... 5
2.1 Definition ........................................ 5
2.2 Local Charts ..................................... 6
2.3 Atlases and Structures ............................. 6
2.4 Calculus on Manifolds ............................. 8
2.5 Immersions and Embeddings........................ 9
2.6 Submanifolds ..................................... 9
2.7 Realizations of a Manifold.......................... 10
2.8 Manifolds with Boundary .......................... 11
2.9 Orientability...................................... 12
2.10 Classification of Manifolds.......................... 13
2.11 A Suitable Model ................................. 13
3. Parametric and Implicit Manifolds .................. 15
3.1 Parametrizations .................................. 15
3.2 Stratifications .................................... 16
3.3 Piecewise Descriptions ............................. 17
3.4 Implicit Description ............................... 19
3.5 Regularity and Transversality....................... 19
3.6 Implicit Manifolds................................. 20
3.7 Geometric Interpretation........................... 21
3.8 Algebraic Varieties ................................ 23
3.9 Algebraic Interpretation............................ 23
3.10 Parametric versus Implicit.......................... 24
vi Contents
4. Space Decompositions............................... 27
4.1 Types of Space Decompositions ..................... 27
4.1.1 Space Partitions ............................. 28
4.1.2 Cell Decompositions ......................... 28
4.1.3 Affine Cell Decompositions.................... 29
4.1.4 Simplicial Decompositions .................... 29
4.2 Properties of Space Decompositions ................. 32
4.2.1 Invariance .................................. 32
4.2.2 Uniqueness ................................. 33
4.2.3 Minimality.................................. 33
4.2.4 Finiteness and Local Finiteness................ 33
4.2.5 Regularity .................................. 33
4.2.6 Boundary Condition ......................... 33
4.2.7 Refinement ................................. 33
4.3 Algebraic Structure of Space Decompositions ......... 34
4.4 Spatial Data Structures ............................ 34
4.4.1 Topological Graphs .......................... 35
4.4.2 Trees....................................... 35
4.4.3 N-dimensional Arrays........................ 36
5. Shape and Space .................................... 39
5.1 Tubular Neighborhoods ............................ 39
5.1.1 Definitions.................................. 39
5.1.2 The Projection on the Surface................. 40
5.1.3 The Maximal Tubular Neighborhood ........... 41
5.2 Medial Axes ...................................... 41
5.2.1 Definitions.................................. 42
5.2.2 Intuition ................................... 42
5.2.3 Characteristics .............................. 43
5.3 Morse Theory .................................... 43
5.3.1 Critical Points and the Hessian ................ 44
5.3.2 Morse Function.............................. 44
5.3.3 The CW-Complex ........................... 45
5.3.4 Distance Fields as Morse Functions ............ 46
5.3.5 Topology of Implicit Shapes................... 47
5.4 Surfaces in Space.................................. 48
5.4.1 Surfaces Structuring Space.................... 48
5.4.2 What is a Good Implicit Model?............... 48
5.5 Universal Representation........................... 49
Contents vii
5.5.1 Medial Axis Models.......................... 49
5.5.2 Distance Function Models .................... 50
5.6 Summary ........................................ 50
6. Implicit Objects .................................... 51
6.1 Definition of an Implict Object ..................... 51
6.2 Mathematical Elements ............................ 53
6.2.1 The Function f ............................. 53
6.2.2 The Domain of f ............................ 54
6.2.3 The Characteristic Function .................. 54
6.2.4 The Gradient of f ........................... 54
6.2.5 The Hessian of f ............................ 55
6.3 Geometrical Characterization ....................... 55
6.3.1 Local Parametrization........................ 55
6.3.2 Orientation and Surface Normal ............... 56
6.4 Differentiable Attributes ........................... 56
6.4.1 Geodesics................................... 56
6.4.2 The Gauss Map ............................. 57
6.4.3 The Fundamental Forms...................... 58
6.4.4 Surface Curvature ........................... 59
6.5 Computational Attributes .......................... 60
6.5.1 Object Oriented Approach .................... 60
6.5.2 Basic Functions ............................. 61
7. Manipulating Implicit Objects ...................... 63
7.1 The Implicit Function as a Metric ................... 63
7.2 Properties of the Implicit Function .................. 65
7.3 Operations on the Range of F ...................... 66
7.3.1 Density Change ............................. 66
7.3.2 Mapping between Canonical Forms ............ 67
7.3.3 Complement ................................ 67
7.3.4 Dilations and Erosions ....................... 68
7.4 Mappings of the Embedding Space .................. 68
7.4.1 Affine Transformations ....................... 70
7.4.2 Deformations ............................... 70
7.5 Operations on the Domain of F ..................... 71
7.5.1 Analytical Transformation of F................ 71
7.5.2 Transformation of Points ..................... 71
7.5.3 Transformation of the Tangent Plane........... 72