---
tag: 727TZQJ4ZA
kind: concept
name: Polarity
msc: ["51A10", "51A50"]
sources:
  - url: https://en.wikipedia.org/wiki/Duality_(projective_geometry)
---

## Summary

A polarity of a projective space is a correlation of order $2$.
It sends points to hyperplanes and hyperplanes to points, it reverses inclusion, and applying it twice gives the identity.
Polarities are the geometric form of non-degenerate reflexive forms, such as symmetric, alternating and Hermitian forms.
They underlie [[duality in projective geometry|DCMP1A0138]], the theory of pole and polar for conics, and [[polar spaces|Z7T0V2VAWJ]].

## Explanations

### L1 human

For a circle in the plane, each point outside it has a polar line, the line through the two points where the tangents from it touch the circle.
Each line has a pole, and going from point to line and back returns the starting point.
This exchange of points and lines that preserves incidence is a polarity.
In linear algebra, taking the orthogonal complement with respect to a dot product is the same kind of operation.