Oriented Projective Geometry (eBook)

A Framework for Geometric Computations

Jorge Stolfi (Autor)

eBook Download: PDF

2014 | 1. Auflage
246 Seiten
Elsevier Science (Verlag)
978-1-4832-6519-3 (ISBN)

Infos

Oriented Projective Geometry: A Framework for Geometric Computations proposes that oriented projective geometry is a better framework for geometric computations than classical projective geometry. The aim of the book is to stress the value of oriented projective geometry for practical computing and develop it as a rich, consistent, and effective tool for computer programmers. The monograph is comprised of 20 chapters. Chapter 1 gives a quick overview of classical and oriented projective geometry on the plane, and discusses their advantages and disadvantages as computational models. Chapters 2 through 7 define the canonical oriented projective spaces of arbitrary dimension, the operations of join and meet, and the concept of relative orientation. Chapter 8 defines projective maps, the space transformations that preserve incidence and orientation; these maps are used in chapter 9 to define abstract oriented projective spaces. Chapter 10 introduces the notion of projective duality. Chapters 11, 12, and 13 deal with projective functions, projective frames, relative coordinates, and cross-ratio. Chapter 14 tells about convexity in oriented projective spaces. Chapters 15, 16, and 17 show how the affine, Euclidean, and linear vector spaces can be emulated with the oriented projective space. Finally, chapters 18 through 20 discuss the computer representation and manipulation of lines, planes, and other subspaces. Computer scientists and programmers will find this text invaluable.

Inhalt

Front Cover 1
Oriented Projective Geometry: A Framework for Geometric Computations 4
Copyright Page 5
Table of Contents 6
Chapter 0. Introduction 10
Chapter 1. Projective geometry 12
1. The classic projective plane 12
2. Advantages of projective geometry 15
3. Drawbacks of classical projective geometry 17
4. Oriented projective geometry 19
5. Related work 20
Chapter 2. Oriented projective spaces 22
1. Models of two-sided space 22
2. Central projection 25
Chapter 3. Flats 28
1. Definition 28
2. Points 28
3. Lines 29
4. Planes 32
5. Three-spaces 35
6. Ranks 36
7. Incidence and independence 36
Chapter 4. Simplices and orientation 38
1. Simplices 38
2. Simplex equivalence 39
3. Point location relative to a simplex 43
4. The vector space model 46
Chapter 5. The join operation 48
1. The join of two points 48
2. The join of a point and a line 50
3. The join of two arbitrary flats 51
4. Properties of join 52
5. Null objects 53
6. Complementary flats 54
Chapter 6. The meet operation 56
1. The meeting point of two lines 56
2. The general meet operation 58
3. Meet in three dimensions 60
4. Properties of meet 62
Chapter 7. Relative orientation 68
1. The two sides of a line 68
2. Relative position of arbitrary flats 69
3. The separation theorem 73
4. The coefficients of a hyperplane 75
Chapter 8. Projective maps 76
1. Formal definition 77
2. Examples 79
3. Properties of projective maps 81
4. The matrix of a map 82
Chapter 9. General two-sided spaces 86
1. Formal definition 86
2. Subspaces 87
Chapter 10. Duality 92
1. Duomorphisms 92
2. The polar complement 94
3. Polar complements as duomorphisms 98
4. Relative polar complements 100
5. General duomorphisms 101
6. The power of duality 102
Chapter 11. Generalized projective maps 104
1. Projective functions 104
2. Computer representation 110
Chapter 12. Projective frames 116
1. Nature of projective frames 116
2. Classification of frames 119
3. Standard frames 122
4. Coordinates relative to a frame 129
Chapter 13. Cross ratio 132
1. Cross ratio in unoriented geometry 132
2. Cross ratio in the oriented framework 135
Chapter 14. Convexity 140
1. Convexity in classical projective space 140
2. Convexity in oriented projective spaces 141
3. Properties of convex sets 143
4. The half-space property 147
5. The convex hull 150
6. Convexity and duality 152
Chapter 15. Affine geometry 160
1. The Cartesian connection 160
2. Two-sided affine spaces 162
Chapter 16. Vector algebra 176
1. Two-sided vector spaces 176
2. Translations 177
3. Vector algebra 177
4. The two-sided real line 180
5. Linear maps 180
Chapter 17. Euclidean geometry on the two-sided plane 182
1. Perpendicularity 182
2. Two-sided Euclidean spaces 186
3. Euclidean maps 187
4. Length and distance 192
5. Angular measure and congruence 196
6. Non-Euclidean geometries 198
Chapter 18. Representing flats by simplices 200
1. The simplex representation 200
2. The dual simplex representation 202
3. The reduced simplex representation 204
Chapter 19. Plücker coordinates 206
2. The canonical embedding 211
3. Plucker coefficients 212
4. Storage effíciency 213
5. The Grassmann manifolds 213
Chapter 20. Formulas for Plücker coordinates 216
1. Algebraic formulas 216
2. Formulas for computers 221
3. Projective maps in Plücker coordinates 226
4. Directions and parallelism 230
References 232
List of symbols 234
Index 236

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-6519-6 / 1483265196
• ISBN-13: 978-1-4832-6519-3 / 9781483265193