---
tag: VB2D03PD9Y
kind: concept
name: Minkowski plane
msc: [51B20]
specializes: [BG5A4CB6NB]
sources:
  - url: https://en.wikipedia.org/wiki/Minkowski_plane
same_as:
  wikidata: Q4381561
  wikipedia: Minkowski plane
---

## Summary

A Minkowski plane is an incidence structure of points and circles with two parallelisms, modelled on the plane sections of a hyperboloid of one sheet.
Each hyperboloid of one sheet carries two families of lines, and these give the two parallelisms.
It is one of the three kinds of [[Benz plane|BG5A4CB6NB]], next to Möbius and Laguerre planes.

## Explanations

### L1 human

A hyperboloid of one sheet, the shape of a cooling tower, is covered by two families of straight lines.
Points on the same line of the first family are called $(+)$-parallel, and points on the same line of the second family $(-)$-parallel.
The circles are the plane sections that are not made of lines.
Through three points, no two of them parallel, there is exactly one circle.
An equivalent picture: the points are pairs $(x, y)$ of points of the projective line, and the circles are the graphs of the projective maps $x \mapsto (ax + b)/(cx + d)$.