Polar space

Summary

A polar space is a geometry of points and lines with one rule. Each point is collinear with exactly one point or with all points of any line not through it. The main examples are the points and lines lying on a non-degenerate quadric. Others come from the polarities of symplectic and Hermitian forms, using the absolute points and totally isotropic lines. Polar spaces are the geometries of the classical groups other than the linear groups, and the non-degenerate ones of rank $2$ are the generalized quadrangles.