Oriented Projective Geometry

A Framework for Geometric Computations

Jorge Stolfi

• Publisher: Academic Press
• Year: 2014
• Print ISBN: 9780126720259
• eBook ISBN: 9781483265193
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

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

• Front Cover
• Oriented Projective Geometry: A Framework for Geometric Computations
• Copyright Page
• Table of Contents
• Chapter 0. Introduction
• Chapter 1. Projective geometry
• Chapter 2. Oriented projective spaces
• Chapter 3. Flats
• Chapter 4. Simplices and orientation
• Chapter 5. The join operation
• Chapter 6. The meet operation
• Chapter 7. Relative orientation
• Chapter 8. Projective maps
• Chapter 9. General two-sided spaces
• Chapter 10. Duality
• Chapter 11. Generalized projective maps
• Chapter 12. Projective frames
• Chapter 13. Cross ratio
• Chapter 14. Convexity
• Chapter 15. Affine geometry
• Chapter 16. Vector algebra
• Chapter 17. Euclidean geometry on the two-sided plane
• Chapter 18. Representing flats by simplices
• Chapter 19. Plücker coordinates
• Chapter 20. Formulas for Plücker coordinates
• References
• List of symbols
• Index