E-book: Oriented Projective Geometry: A Framework for Geometric Computations

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.