# Oriented Projective Geometry

## 1st Edition

### A Framework for Geometric Computations

**Authors:**Jorge Stolfi

**eBook ISBN:**9781483265193

**Imprint:**Academic Press

**Published Date:**28th July 1991

**Page Count:**246

## Description

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.

## Table of Contents

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

## Details

- No. of pages:
- 246

- Language:
- English

- Copyright:
- © Academic Press 1991

- Published:
- 28th July 1991

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483265193