Polar space

In mathematics, in the field of geometry, a polar space of rank n (n ≥ 3), or projective index n − 1, consists of a set P, conventionally called the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms:

• Every subspace is isomorphic to a projective space P^d(K) with −1 ≤ d ≤ (n − 1) and K a division ring. (That is, it is a Desarguesian projective geometry.) For each subspace the corresponding d is called its dimension.
• The intersection of two subspaces is always a subspace.
• For each subspace A of dimension n − 1 and each point p not in A, there is a unique subspace B of dimension n − 1 containing p and such that A ∩ B is (n − 2)-dimensional. The points in A ∩ B are exactly the points of A that are in a common subspace of dimension 1 with p.
• There are at least two disjoint subspaces of dimension n − 1.

It is possible to define and study a slightly bigger class of objects using only the relationship between points and lines: a polar space is a partial linear space (P,L), so that for each point p ∈ P and each line l ∈ L, the set of points of l collinear to p is either a singleton or the whole l.

Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.