Computer Graphics and Geometric Modelling (eBook)

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