Oriented Projective Geometry

Oriented Projective Geometry

A Framework for Geometric Computations

1st Edition - July 28, 1991

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  • Author: Jorge Stolfi
  • eBook ISBN: 9781483265193

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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


    List of Symbols


Product details

  • No. of pages: 246
  • Language: English
  • Copyright: © Academic Press 1991
  • Published: July 28, 1991
  • Imprint: Academic Press
  • eBook ISBN: 9781483265193

About the Author

Jorge Stolfi

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