---
tag: 4695YA7F87
kind: definition
name: Collineation
msc: [51A10, 51N15]
defines: V7PRPN24CV
uses: [N32Z68F0D3, 0AFNNTRTJE]
sources:
  - url: https://en.wikipedia.org/wiki/Collineation
---

## Definition

Let $S$ and $S'$ be [[projective geometries|N32Z68F0D3]], with point sets $P$ and $P'$.
A *collineation* from $S$ to $S'$ is a bijection $f \colon P \to P'$ with the following property.
For all points $a, b, c$ of $P$, the points $a, b, c$ are [[collinear|0AFNNTRTJE]] exactly when $f(a), f(b), f(c)$ are collinear.
A collineation from $S$ to itself is a collineation of $S$.

## Remarks

Equivalently, $f$ is a bijection on points such that the image of every line is a line and every line of $S'$ is such an image.
The same definition applies to affine spaces and to other incidence structures whose lines are sets of points.

Example: every [[projective transformation|P1MKTF7MWQ]] $[v] \mapsto [Tv]$ of $\mathbb{P}(V)$ is a collineation.
Check: the line through $[u]$ and $[w]$ is $\mathbb{P}(W)$ for the plane $W$ spanned by $u, w$.
The map sends it onto $\mathbb{P}(T W)$, again a line, and the inverse map comes from $T^{-1}$.

Non-example: in the Fano plane $\mathbb{P}^2(\F_2)$, whose lines are the sets $\set{a, b, a + b}$ of nonzero vectors of $\F_2^3$, swap $e_1$ and $e_2$ and fix all other points.
The line $\set{e_1, e_3, e_1 + e_3}$ goes to $\set{e_2, e_3, e_1 + e_3}$, which is not a line, because $e_2 + e_3 \ne e_1 + e_3$.

The collineations of $S$ form a group under composition, the collineation group of $S$.