---
tag: 8NCGCJ2YBH
kind: definition
name: Benz plane
msc: [51B05, 51B10]
defines: BG5A4CB6NB
uses: [DED008EJVH, F0JQ1TGKB0, VB2D03PD9Y]
sources:
  - url: https://en.wikipedia.org/wiki/Benz_plane
---

## Definition

A *Benz plane* is an incidence structure of points and circles that is a [[Möbius plane|DED008EJVH]], a [[Laguerre plane|F0JQ1TGKB0]] or a [[Minkowski plane|VB2D03PD9Y]].

## Remarks

The three types differ in their parallelism of points.
A Möbius plane has no parallel points, a Laguerre plane has one parallelism, and a Minkowski plane has two.
In each type, three pairwise non-parallel points lie on exactly one circle, and a touching axiom holds.

Example: the real plane with a point $\infty$ added, with circles and lines through $\infty$ as circles, is a Möbius plane, hence a Benz plane.

Non-example: a projective plane, with its lines as circles, is not a Benz plane.
Three non-collinear points lie on no line, so the axiom on three points fails for each of the three types.

The classical Benz planes are the geometries of plane sections of a sphere, a cylinder and a hyperboloid of one sheet.