Oriented Projective Geometry

Oriented Projective Geometry : A Framework for Geometric Computations

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Intended for computer graphics researchers and programmers, and mathematicians working in computational geometry. This book describes oriented projective geometry, a geometric model that combines the elegance and efficiency of classical projective geometry with the consistent handling of oriented lines and planes, signed angles, line segments, convex sets, and many other fundamental geometric computing concepts that classical theory does not support. The aim of this book is to assemble a consistent, practical and effective set of tools for computational geometry that can be used by graphics programmers in their everyday work. In keeping with this goal, formal derivations are kept to a minimum, and many definitions and theorems are illustrated with explicit examples in one, two, and three dimensions.
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Product details

  • Hardback | 248 pages
  • 152 x 230 x 22mm | 559.99g
  • Academic Press Inc
  • San Diego, United States
  • English
  • references, index
  • 0126720258
  • 9780126720259

Table of contents

Part 1 Projective geometry: the classic projective plane; advantages of projective geometry; drawbacks of classical projective geometry; oriented projective geometry; related work. Part 2 Oriented projective spaces: models of two-sided space; central projection. Part 3 Flats: definition; points; lines; planes; three-spaces; ranks; incidence and dependence. Part 4 Simplices and orientation: simplices; simplex equivalence; point location relative to a simplex; the vector space model. Part 5 The join operation: the join of two points; the join of a point and a line; the join of two arbitrary flats; properties of join; null objects; complementary flats. Part 6 The meeting operation: the meeting point of two lines; the general meet operation; meet in three dimensions; properties of meet. Part 7 Relative orientation: the two sides of a line; relative position of arbitrary flats; the separation theorem; the coefficients of a hyperplane. Part 8 Projective maps: formal definition; examples; properties of projective maps; the matrix of a map. Part 9 General: two-sided spaces - formal definition; subspaces. Part 10 Duality: duomorphisms; the polar complement; polar complements as duomorphisms; relative polar complements; general duomorphisms; the power of duality. Part 11 Generalized projective maps: projective functions; computer representation. Part 12 Projective frames: nature of projective frames; classification of frames; standard frames; coordinates relative to a frame. Part 13 Cross ratio: cross ratio in unoriented geometry; cross ratio in the oriented framework. Part 14 Convexity: convexity in classical projective space; convexity in oriented projective spaces; properties of convex sets; the half-space property; the convex hull; convexity and duality. Part 15 Affine geomerty: the Cartesian connection; two-sided affine spaces. Part 16 Vector albegra: two-sided vector spaces; translations; vector algebra; the two-sided real line; linear maps. Part 17 Euclidean geometry on the two-sided plane: perpendicularity; two-sided Euclidean spaces; Euclidean maps; length and distance; angular measure and congruence; non-Euclidean geometries. Part 18 Representing flats by simplices: the simplex representation; the dual simplex representation; the reduced simplex representation. Part 19 Plucker coordinates: the canonical embedding; Plucker coefficients; storage efficiency; the Grassmann manifolds. Part 20 Formulas for Plucker coordinates: algebraic formulas; formulas for computers; projective maps in Plucker coordinates; directions and parallelism.
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