Oriented projective geometry: a framework for geometric computations

Abstract

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.