---
tag: JZ5J6REAMD
kind: concept
name: Translation plane
msc: ["51A40", "51E15"]
specializes: [DANFNN8C4Y]
sources:
  - url: https://en.wikipedia.org/wiki/Translation_plane
same_as:
  wikidata: Q382515
  wikipedia: Translation plane
---

## Summary

A translation plane is a [[projective plane|DANFNN8C4Y]] with a special line $\ell$.
The collineations that act like translations with respect to $\ell$ can move any point off $\ell$ to any other.
Removing $\ell$ leaves an affine plane with a full group of translations, as in the real plane.
Translation planes are exactly the planes coordinatized by [[quasifields|JQ22QH86DE]], and they form the largest well-understood family of non-Desarguesian planes.

## Explanations

### L1 human

In the ordinary plane one can slide every figure by a fixed vector, and these slides carry any point to any other point.
A translation plane is a projective plane that keeps this property for one chosen line, which plays the role of the line at infinity.
Each slide fixes every point at infinity and moves every line to a parallel line.
The real projective plane is a translation plane, and so is every finite plane $PG(2, q)$.
There are also translation planes that are not of this form, such as the Hall plane of order $9$.