---
tag: Z7T0V2VAWJ
kind: concept
name: Polar space
msc: ["51A50", "51E24"]
sources:
  - url: https://en.wikipedia.org/wiki/Polar_space
same_as:
  wikidata: Q3058226
  wikipedia: 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|Q3Q4FF23VG]].
Others come from the [[polarities|727TZQJ4ZA]] 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|GQJCPFSYVK]].

## Explanations

### L1 human

Take the quadric surface $x_0 x_1 + x_2 x_3 = 0$ in projective space, which is covered by two families of straight lines.
Its points and the lines lying on it form a polar space.
Pick a line on the surface and a point of the surface off that line.
Then the point is joined by a line of the surface to exactly one point of the line.
In other polar spaces a point can also be joined to all points of a line, and this one-or-all rule turns out to characterize such geometries.