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.

Contenu

Chapter 0. Introduction
Chapter 1. Projective Geometry
1.1. The Classic Projective Plane
1.2. Advantages of Projective Geometry
1.3. Drawbacks of Classical Projective Geometry
1.4. Oriented Projective Geometry
1.5. Related Work
Chapter 2. Oriented Projective Spaces
2.1. Models of Two-Sided Space
2.2. Central Projection
Chapter 3. Flats
3.1. Definition
3.2. Points
3.3. Lines
3.4. Planes
3.5. Three-Spaces
3.6. Ranks
3.7. Incidence and Independence
Chapter 4. Simplices and Orientation
4.1. Simplices
4.2. Simplex Equivalence
4.3. Point Location Relative to a Simplex
4.4. The Vector Space Model
Chapter 5. The Join Operation
5.1. The Join of Two Points
5.2. The Join of a Point and a Line
5.3. The Join of Two Arbitrary Flats
5.4. Properties of Join
5.5. Null Objects
5.6. Complementary Flats
Chapter 6. The Meet Operation
6.1. The Meeting Point of Two Lines
6.2. The General Meet Operation
6.3. Meet in Three Dimensions
6.4. Properties of Meet
Chapter 7. Relative Orientation
7.1. The Two Sides of a Line
7.2. Relative Position of Arbitrary Flats
7.3. The Separation Theorem
7.4. The Coefficients of a Hyperplane
Chapter 8. Projective Maps
8.1. Formal Definition
8.2. Examples
8.3. Properties of Projective Maps
8.4. The Matrix of a Map
Chapter 9. General Two-Sided Spaces
9.1. Formal Definition
9.2. Subspaces
Chapter 10. Duality
10.1. Duomorphisms
10.2. The Polar Complement
10.3. Polar Complements as Duomorphisms
10.4. Relative Polar Complements
10.5. General Duomorphisms
10.6. The Power of Duality
Chapter 11. Generalized Projective Maps
11.1. Projective Functions
11.2. Computer Representation
Chapter 12. Projective Frames
12.1. Nature of Projective Frames
12.2. Classification of Frames
12.3. Standard Frames
12.4. Coordinates Relative to a Frame
Chapter 13. Cross Ratio
13.1. Cross Ratio in Unoriented Geometry
13.2. Cross Ratio in the Oriented Framework
Chapter 14. Convexity
14.1. Convexity in Classical Projective Space
14.2. Convexity in Oriented Projective Spaces
14.3. Properties of Convex Sets
14.4. The Half-Space Property
14.5. The Convex Hull
14.6. Convexity and Duality
Chapter 15. Affine Geometry
15.1. The Cartesian Connection
15.2. Two-Sided Affine Spaces
Chapter 16. Vector Algebra
16.1. Two-Sided Vector Spaces
16.2. Translations
16.3. Vector Algebra
16.4. The Two-Sided Real Line
16.5. Linear Maps
Chapter 17. Euclidean Geometry on the Two-Sided Plane
17.1. Perpendicularity
17.2. Two-Sided Euclidean Spaces
17.3. Euclidean Maps
17.4. Length and Distance
17.5. Angular Measure and Congruence
17.6. Non-Euclidean Geometries
Chapter 18. Representing Flats by Simplices
18.1. The Simplex Representation
18.2. The Dual Simplex Representation
18.3. The Reduced Simplex Representation
Chapter 19. Plücker Coordinates
19.2. The Canonical Embedding
19.3. Plücker Coefficients
19.4. Storage Efficiency
19.5. The Grassmann Manifolds
Chapter 20. Formulas for Plücker Coordinates
20.1. Algebraic Formulas
20.2. Formulas for Computers
20.3. Projective Maps in Plücker Coordinates
20.4. Directions and Parallelism
References
List of Symbols
Index