---
tag: DED008EJVH
kind: concept
name: Möbius plane
msc: [51B10]
specializes: [BG5A4CB6NB]
sources:
  - url: https://en.wikipedia.org/wiki/M%C3%B6bius_plane
same_as:
  wikidata: Q1392693
  wikipedia: Möbius plane
---

## Summary

A Möbius plane is an incidence structure of points and circles that behaves like the circles on a sphere.
Any three points lie on exactly one circle, and circles can touch.
The classical example is the plane with one point at infinity, where circles and lines through infinity are the circles.
Möbius planes are one of the three kinds of [[Benz plane|BG5A4CB6NB]], and they give the axiomatic setting of [[inversive geometry|HFBW3SXQVM]].

## Explanations

### L1 human

On a sphere, any three points lie on exactly one circle.
Two circles can meet in two points, in no point, or touch in one point.
A Möbius plane keeps exactly these facts as axioms and forgets the rest.
Stereographic projection turns the sphere into the plane plus one point at infinity, and circles through that point become straight lines.