Geometry & Topology 2024, 31: Projective Geometry
An alternative approach to inhomogeneous linear geometry is to map it onto projective space. Associate by equivalence x ~ y, if x and y are proportional or span the same line through 0 in ℝᵐ where m = n + 1. The projective space is
If V is an (n+1)-dimensional vector space over ℝ, then P(V) is the set of lines of V through 0. A point p in P(V) is an equivalence class of vectors in V, or a line ℝv through 0. The only structures in P(V) are derived from V. All statements or calculations for P(V) reduces to linear algebra in V, and the equivalence relation on points of V. With points p, q in P(V), the constructions αp + βq is meaningless as a point in P(V), as the equivalence class depends on the choice of p and q in their equivalence classes. It rather defines a set of vectors in a 2D subspace of V. It forms the line pq. If U is a vector subspace of V, the P(U) is the subset (U \ 0)/~ P(V) of lines through 0 in U. A set in P(V) M and the union of lines N in M with the vector subspace of V spanned by N and the "span" / "linear span" of M is <M> = P(U). This is the smallest projective linear subspace containing M. Linear dependency is taken from the dimension of the equivalence classes spanned by point vectors. For projective linear spaces E, F
A projective frame of reference is a set of n + 2 points, so that any n + 1 are linearly independent, spanning Pⁿ. There is a one-to-one correspondence between projective transformations and frames of reference. There is a unique projective linear transformation of P¹ taking any 3 distinct points P, Q, R in P¹ to any other 3.
An outside observer w.r.t. two hyperplanes of projective space can map perspectivity by mapping points from one hyperplane onto the points of intersection of the projective line with the other. The cross ratio of 4 points on a line is invariant under perspectivities. A hyperplane in n-dimensional projective space corresponds to some n-dimensional subspace in n+1 dimensional vector space. Said projective space modulo the hyperplane can be identified with n-dimensional affine space and the hyperplane with sets of parallel lines within it.
2 triangles in projective space with dimension of at least 2 are in perspective from some point O. The sides of the triangles meet in 3 colinear points (Desargues).
Two lines L, L' with a triple of points each, which are not on the intersection of said lines, then the 3 points QR' ∩ Q'R = A, PR' ∩ P'R = B, PQ' ∩ P'Q = C are collinear (Pappus). An axiomatic projective geometry is a two sets of points and lines, and an incidence relation: Incidence(Π) ⊂ Points(Π) ⨉ Lines(Π). An element of Incidence(Π) (P, L) is a line L passing through point P. Every line has at least 3 points, and every point has at least 3 lines through it. Any 2 distinct points have a unique line. and any 2 distinct lines meet in a unique point. These axioms are dual.