---
tag: V7PRPN24CV
kind: concept
name: Collineation
msc: [51A10, 51N15]
sources:
  - url: https://en.wikipedia.org/wiki/Collineation
same_as:
  wikidata: Q390741
  wikipedia: Collineation
---

## Summary

A collineation is a bijection between the points of two geometries that maps lines onto lines.
It is the notion of isomorphism for [[projective geometries|N32Z68F0D3]] and other incidence structures.
Every [[projective transformation|P1MKTF7MWQ]] of a projective space of dimension at least $2$ is a collineation.
The [[fundamental theorem of projective geometry|GKV4RVXBRE]] says that all collineations come from semilinear maps.
The collineations of a geometry form its collineation group, its group of symmetries.

## Explanations

### L1 human

A collineation moves the points of a geometry around so that points on a common line stay on a common line.
Any invertible linear change of coordinates does this.
Over the complex numbers, so does replacing every coordinate by its complex conjugate.
Swapping just two points of the plane and fixing all others does not: some line through one of them gets broken.