Agoston Computer Graphics and Geometric Modelling

Mathematics
1. Auflage 2005
ISBN: 978-1-84628-122-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 973 Seiten

Possibly the most comprehensive overview of computer graphics as seen in the context of geometric modelling, this two volume work covers implementation and theory in a thorough and systematic fashion. Computer Graphics and Geometric Modelling: Mathematics, contains the mathematical background needed for the geometric modeling topics in computer graphics covered in the first volume.

This volume begins with material from linear algebra and a discussion of the transformations in affine & projective geometry, followed by topics from advanced calculus & chapters on general topology, combinatorial topology, algebraic topology, differential topology, differential geometry, and finally algebraic geometry. Two important goals throughout were to explain the material thoroughly, and to make it self-contained.

This volume by itself would make a good mathematics reference book, in particular for practitioners in the field of geometric modelling. Due to its broad coverage and emphasis on explanation it could be used as a text for introductory mathematics courses on some of the covered topics, such as topology (general, combinatorial, algebraic, and differential) and geometry (differential & algebraic).

Autoren/Hrsg.

Agoston, Max K.

Inhaltsverzeichnis

1;Preface;5
2;Contents;9
3;1 Linear Algebra Topics;15
3.1;1.1 Introduction;15
3.2;1.1 Introduction;15
3.3;1.2 Lines;16
3.4;1.3 Angles;19
3.5;1.4 Inner Product Spaces: Orthonormal Bases;21
3.6;1.5 Planes;28
3.7;1.6 Orientation;36
3.8;1.7 Convex Sets;44
3.9;1.8 Principal Axes Theorems;51
3.10;1.9 Bilinear and Quadratic Maps;58
3.11;1.10 The Cross Product Reexamined;64
3.12;1.11 The Generalized Inverse Matrix;67
3.13;1.12 EXERCISES;72
4;2 Affine Geometry;77
4.1;2.1 Overview;77
4.2;2.2 Motions;78
4.2.1;2.2.1 Translations;81
4.2.2;2.2.2 Rotations in the Plane;82
4.2.3;2.2.3 Re.ections in the Plane;86
4.2.4;2.2.4 Motions Preserve the Dot Product;90
4.2.5;2.2.5 Some Existence and Uniqueness Results;93
4.2.6;2.2.6 Rigid Motions in the Plane;96
4.2.7;2.2.7 Summary for Motions in the Plane;99
4.2.8;2.2.8 Frames in the Plane;101
4.3;2.3 Similarities;108
4.4;2.4 Affine Transformations;109
4.4.1;2.4.1 Parallel Projections;116
4.5;2.5 Beyond the Plane;119
4.5.1;2.5.1 Motions in 3-Space;126
4.5.2;2.5.2 Frames Revisited;132
4.6;2.6 EXERCISES;135
5;3 Projective Geometry;140
5.1;3.1 Overview;140
5.2;3.2 Central Projections and Perspectivities;141
5.3;3.3 Homogeneous Coordinates;150
5.4;3.4 The Projective Plane;153
5.4.1;3.4.1 Analytic Properties of the Projective Plane;157
5.4.2;3.4.2 Two-dimensional Projective Transformations;166
5.4.3;3.4.3 Planar Maps and Homogeneous Coordinates;168
5.5;3.5 Beyond the Plane;172
5.5.1;3.5.1 Homogeneous Coordinates and Maps in 3-Space;175
5.6;3.6 Conic Sections;180
5.6.1;3.6.1 Projective Properties of Conics;194
5.7;3.7 Quadric Surfaces;204
5.8;3.8 Generalized Central Projections;210
5.9;3.9 The Theorems of Pascal and Brianchon;213
5.10;3.10 The Stereographic Projection;215
5.11;3.11 EXERCISES;219
6;4 Advanced Calculus Topics;222
6.1;4.1 Introduction;222
6.2;4.2 The Topology of Euclidean Space;222
6.3;4.3 Derivatives;232
6.4;4.4 The Inverse and Implicit Function Theorem;246
6.5;4.5 Critical Points;254
6.6;4.6 Morse Theory;263
6.7;4.7 Zeros of Functions;266
6.8;4.8 Integration;270
6.9;4.9 Differential Forms;278
6.9.1;4.9.1 Differential Forms and Integration;287
6.10;4.10 EXERCISES;291
7;5 Point Set Topology;295
7.1;5.1 Introduction;295
7.2;5.2 Metric Spaces;296
7.3;5.3 Topological Spaces;303
7.4;5.4 Constructing New Topological Spaces;312
7.5;5.5 Compactness;318
7.6;5.6 Connectedness;322
7.7;5.7 Homotopy;323
7.8;5.8 Constructing Continuous Functions;327
7.9;5.9 The Topology of Pn;329
7.10;5.10 EXERCISES;332
8;6 Combinatorial Topology;335
8.1;6.1 Introduction;335
8.2;6.2 What Is Topology?;340
8.3;6.3 Simplicial Complexes;342
8.4;6.4 Cutting and Pasting;347
8.5;6.5 The Classification of Surfaces;352
8.6;6.6 Bordered and Noncompact Surfaces;367
8.7;6.7 EXERCISES;369
9;7 Algebraic Topology;372
9.1;7.1 Introduction;372
9.2;7.2 Homology Theory;373
9.2.1;7.2.1 Homology Groups;373
9.2.2;7.2.2 Induced Maps;389
9.2.3;7.2.3 Applications of Homology Theory;398
9.2.4;7.2.4 Cell Complexes;403
9.2.5;7.2.5 Incidence Matrices;413
9.2.6;7.2.6 The Mod 2 Homology Groups;419
9.3;7.3 Cohomology Groups;423
9.4;7.4 Homotopy Theory;426
9.4.1;7.4.1 The Fundamental Group;426
9.4.2;7.4.2 Covering Spaces;436
9.4.3;7.4.3 Higher Homotopy Groups;448
9.5;7.5 Pseudomanifolds;452
9.5.1;7.5.1 The Degree of a Map and Applications;457
9.5.2;7.5.2 Manifolds and Poincaré Duality;460
9.6;7.6 Where to Next: What We Left Out;463
9.7;7.7 The CW Complex Pn;467
9.8;7.8 EXERCISES;470
10;8 Differential Topology;473
10.1;8.1 Introduction;473
10.2;8.2 Parameterizing Spaces;474
10.3;8.3 Manifolds in Rn;479
10.4;8.4 Tangent Vectors and Spaces;488
10.5;8.5 Oriented Manifolds;497
10.6;8.6 Handle Decompositions;503
10.7;8.7 Spherical Modi.cations;511
10.8;8.8 Abstract Manifolds;514
10.9;8.9 Vector Bundles;523
10.10;8.10 The Tangent and Normal Bundles;533
10.11;8.11 Transversality;542
10.12;8.12 Differential Forms and Integration;549
10.13;8.13 The Manifold Pn;562
10.14;8.14 The Grassmann Manifolds;564
10.15;8.15 EXERCISES;566
11;9 Differential Geometry;571
11.1;9.1 Introduction;571
11.2;9.2 Curve Length;572
11.3;9.3 The Geometry of Plane Curves;577
11.4;9.4 The Geometry of Space Curves;587
11.5;9.5 Envelopes of Curves;593
11.6;9.6 Involutes and Evolutes of Curves;597
11.7;9.7 Parallel Curves;600
11.8;9.8 Metric Properties of Surfaces;603
11.9;9.9 The Geometry of Surfaces;612
11.10;9.10 Geodesics;634
11.11;9.11 Envelopes of Surfaces;652
11.12;9.12 Canal Surfaces;652
11.13;9.13 Involutes and Evolutes of Surfaces;654
11.14;9.14 Parallel Surfaces;657
11.15;9.15 Ruled Surfaces;659
11.16;9.16 The Cartan Approach: Moving Frames;663
11.17;9.17 Where to Next?;673
11.18;9.18 Summary of Curve Formulas;679
11.19;9.19 Summary of Surface Formulas;681
11.20;9.20 EXERCISES;683
12;10 Algebraic Geometry;688
12.1;10.1 Introduction;688
12.2;10.2 Plane Curves: There Is More than Meets the Eye;691
12.3;10.3 More on Projective Space;698
12.4;10.4 Resultants;704
12.5;10.5 More Polynomial Preliminaries;709
12.6;10.6 Singularities and Tangents of Plane Curves;716
12.7;10.7 Intersections of Plane Curves;724
12.8;10.8 Some Commutative Algebra;729
12.9;10.9 Defining Parameterized Curves Implicitly;738
12.10;10.10 Gröbner Bases;742
12.11;10.11 Elimination Theory;759
12.12;10.12 Places of a Curve;761
12.13;10.13 Rational and Birational Maps;778
12.14;10.14 Space Curves;796
12.15;10.15 Parameterizing Implicit Curves;800
12.16;10.16 The Dimension of a Variety;804
12.17;10.17 The Grassmann Varieties;810
12.18;10.18 N-Dimensional Varieties;811
12.19;10.19 EXERCISES;819
13;Appendix A Notation;827
14;Appendix B Basic Algebra;831
14.1;B.1 Number Theoretic Basics;831
14.2;B.2 Set Theoretic Basics;832
14.3;B.3 Permutations;835
14.4;B.4 Groups;837
14.5;B.5 Abelian Groups;845
14.6;B.6 Rings;849
14.7;B.7 Polynomial Rings;854
14.8;B.8 Fields;861
14.9;B.9 The Complex Numbers;864
14.10;B.10 Vector Spaces;865
14.11;B.11 Extension Fields;869
14.12;B.12 Algebras;873
15;Appendix C Basic Linear Algebra;874
15.1;C.1 More on Linear Independence;874
15.2;C.2 Inner Products;876
15.3;C.3 Matrices of Linear Transformations;879
15.4;C.4 Eigenvalues and Eigenvectors;884
15.5;C.5 The Dual Space;887
15.6;C.6 The Tensor and Exterior Algebra;889
16;Appendix D Basic Calculus and Analysis;903
16.1;D.1 Miscellaneous Facts;903
16.2;D.2 Series;906
16.3;D.3 Differential Equations;908
16.4;D.4 The Lebesgue Integral;910
17;Appendix E Basic Complex Analysis;912
17.1;E.1 Basic Facts;912
17.2;E.2 Analytic Functions;913
17.3;E.3 Complex Integration;916
17.4;E.4 More on Complex Series;917
17.5;E.5 Miscellaneous Facts;919
18;Appendix F A Bit of Numerical Analysis;921
18.1;F.1 The Condition Number of a Matrix;921
18.2;F.2 Approximation and Numerical Integration;922
19;Bibliography;929
19.1;Abbreviations;929
19.2;Abstract Algebra;929
19.3;Advanced Calculus;929
19.4;Algebraic Curves and Surfaces;929
19.5;Algebraic Geometry;930
19.6;Algebraic Topology;930
19.7;Analytic Geometry;931
19.8;Complex Analysis;931
19.9;Conics;931
19.10;Cyclides;931
19.11;Differential Geometry;932
19.12;Differential Topology;932
19.13;Geodesics;933
19.14;Geometric Modeling;933
19.15;Linear Algebra;933
19.16;Miscellaneous;933
19.17;Numerical Methods;933
19.18;Offset Curves and Surfaces;934
19.19;Projective Geometry and Transformations;934
19.20;Quadrics;934
19.21;Real Analysis;934
19.22;Topology;934
20;Index;935
21;More eBooks at www.ciando.com;0

Leseproben

Chapter 7
Algebraic Topology (p. 358)

7.1 Introduction

The central problem of algebraic topology is to classify spaces up to homeomorphism by means of computable algebraic invariants. In the last chapter we showed how two invariants, namely, the Euler characteristic and orientability, gave a complete classi- .cation of surfaces. Unfortunately, these invariants are quite inadequate to classify higher-dimensional spaces. However, they are simple examples of the much more general invariants that we shall discuss in this chapter.

The heart of this chapter is its introduction to homology theory. Section 7.2.1 de.nes the homology groups for simplicial complexes and polyhedra, and Section 7.2.2 shows how continuous maps induce homomorphisms of these groups. Section 7.2.3 describes a few immediate applications. In Section 7.2.4 we indicate how homology theory can be extended to cell complexes and how this can greatly simplify some computations dealing with homology groups. Along the way we de.ne CW complexes, which are really the spaces of choice in algebraic topology because one can get the most convenient description of a space with them. Section 7.2.5 de.nes the incidence matrices for simplicial complexes.

These are a fundamental tool for computing homology groups with a computer. Section 7.2.6 describes a useful extension of homology groups where one uses an arbitrary coef.cient group, in particular, Z2. After this overview of homology theory we move on to de.ne cohomology in Section 7.3. The cohomology groups are a kind of dual to the homology groups.

We then come to the other major classical topic in algebraic topology, namely, homotopy theory. We start in Sections 7.4.1 and 7.4.2 with a discussion of the fundamental group of a topological space and covering spaces. These topics have their roots in complex analysis. Section 7.4.3 sketches the de.nition of the higher-dimensional homotopy groups and concludes with some major theorems from homotopy theory. Section 7.5 is devoted to pseudomanifolds, the degree of a map, manifolds, and Poincaré duality (probably the single most important algebraic property of manifolds and the property that sets manifolds apart from other spaces).

We wrap up our overview of algebraic topology in Section 7.6 by telling the reader brie.y about important aspects that we did not have time for and indicate further topics to pursue. Finally, as one last example, Section 7.7 applies the theory developed in this chapter to our ever-interesting space Pn.

The reader is warned that this chapter may be especially hard going if he/she has not previously studied some abstract algebra. We shall not be using any really advanced ideas from abstract algebra, but if the reader is new to it and has no one for a guide, then, as usual, it will take a certain amount of time to get accustomed to thinking along these lines. Groups and homomorphism are quite a bit different from topics in calculus and basic linear algebra.

The author hopes the reader will persevere because in the end one will be rewarded with some beautiful theories. The next chapter will make essential use of what is developed here and apply it to the study of manifolds. Manifolds are the natural spaces for geometric modeling and getting an understanding of our universe.