[[Abstract projective space]]
# Projective correlation

A **correlation** of a projective space $\mathcal{S}$ is a [[Collineätion]] $\alpha : \mathcal{S} \to \op{\mathcal{S}}$ into the [[Dual projective space]]. #m/def/geo

## Further terminology

- A correlation which is an [[involution]] is a **polarity**.
- The image of a point $P$ under a polarity $\alpha$ is called the **polar** (hyperplane) $P^\alpha$,
  while the image of a hyperplane $\mathcal{H}$ is called the **pole** (point) $\mathcal{H}^\alpha$.
- If $P \I Q^\alpha$ then $P$ and $Q$ are **conjugate**, if $P \I P^\alpha$ it is **self-conjugate**.

## Examples

- [[Orthogonal complement polarity]]

#
---
#state/develop | #lang/en | #SemBr