---
tag: F0JQ1TGKB0
kind: concept
name: Laguerre plane
msc: [51B15]
specializes: [BG5A4CB6NB]
sources:
  - url: https://en.wikipedia.org/wiki/Laguerre_plane
same_as:
  wikidata: Q15825031
  wikipedia: Laguerre plane
---

## Summary

A Laguerre plane is an incidence structure of points and circles with a parallelism of points.
It is modelled on the plane sections of a circular cylinder.
Three pairwise non-parallel points lie on exactly one circle, and each circle meets each parallel class in exactly one point.
It is one of the three kinds of [[Benz plane|BG5A4CB6NB]].

## Explanations

### L1 human

Take a round cylinder and cut it with planes that are not parallel to its axis; the cuts are ellipses, called circles here.
Two points on the same vertical line of the cylinder are called parallel, and no circle passes through both.
Through three points with no two parallel there is exactly one such circle.
A flat picture of the same geometry uses parabolas $y = ax^2 + bx + c$ in the plane, with vertical lines as parallel classes.