---
tag: BG5A4CB6NB
kind: concept
name: Benz plane
msc: [51B05, 51B10]
sources:
  - url: https://en.wikipedia.org/wiki/Benz_plane
same_as:
  wikidata: Q3210599
  wikipedia: Benz plane
---

## Summary

Benz plane is the common name for three kinds of circle geometry: Möbius planes, Laguerre planes and Minkowski planes.
Each kind is an incidence structure of points and circles in which three suitable points lie on exactly one circle and circles can touch.
The classical examples are the plane sections of a sphere, of a circular cylinder and of a hyperboloid of one sheet.
The name honours Walter Benz, who developed their common theory.

## Explanations

### L1 human

Cut a sphere, a cylinder or a hyperboloid with planes and call the curves you get circles.
In each case three points that are not in some special position lie on exactly one circle.
The three resulting geometries share many features, and the word Benz plane names all three at once.
They differ in their parallel points: none on a sphere, one family of lines on a cylinder, two families on a hyperboloid.