Oriented Projective Geometry

A Framework for Geometric Computations

1st Edition - July 28, 1991
Author: Jorge Stolfi
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 6 5 1 9 - 3

Oriented Projective Geometry: A Framework for Geometric Computations proposes that oriented projective geometry is a better framework for geometric computations than classical… Read more

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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.

Chapter 0. IntroductionChapter 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 WorkChapter 2. Oriented Projective Spaces 2.1. Models of Two-Sided Space 2.2. Central ProjectionChapter 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 IndependenceChapter 4. Simplices and Orientation 4.1. Simplices 4.2. Simplex Equivalence 4.3. Point Location Relative to a Simplex 4.4. The Vector Space ModelChapter 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 FlatsChapter 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 MeetChapter 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 HyperplaneChapter 8. Projective Maps 8.1. Formal Definition 8.2. Examples 8.3. Properties of Projective Maps 8.4. The Matrix of a MapChapter 9. General Two-Sided Spaces 9.1. Formal Definition 9.2. SubspacesChapter 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 DualityChapter 11. Generalized Projective Maps 11.1. Projective Functions 11.2. Computer RepresentationChapter 12. Projective Frames 12.1. Nature of Projective Frames 12.2. Classification of Frames 12.3. Standard Frames 12.4. Coordinates Relative to a FrameChapter 13. Cross Ratio 13.1. Cross Ratio in Unoriented Geometry 13.2. Cross Ratio in the Oriented FrameworkChapter 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 DualityChapter 15. Affine Geometry 15.1. The Cartesian Connection 15.2. Two-Sided Affine SpacesChapter 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 MapsChapter 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 GeometriesChapter 18. Representing Flats by Simplices 18.1. The Simplex Representation 18.2. The Dual Simplex Representation 18.3. The Reduced Simplex RepresentationChapter 19. Plücker Coordinates 19.2. The Canonical Embedding 19.3. Plücker Coefficients 19.4. Storage Efficiency 19.5. The Grassmann ManifoldsChapter 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 ParallelismReferencesList of SymbolsIndex