---
tag: 831H5X5JS4
kind: definition
name: Möbius plane
msc: [51B10]
defines: DED008EJVH
uses: [FGKNAFNHA3, 060W5CBZP6, 0J78KBPFZB, HNTE5FPS14]
sources:
  - url: https://en.wikipedia.org/wiki/M%C3%B6bius_plane
---

## Definition

Let $(\mathcal{P}, \mathcal{Z}, \in)$ be an [[incidence structure|FGKNAFNHA3]] whose elements of $\mathcal{P}$ are called points.
The elements of $\mathcal{Z}$ are subsets of $\mathcal{P}$, called circles.
The structure is a *Möbius plane* if:

1. any three distinct points lie on exactly one circle;
2. for every circle $z$, every point $P \in z$ and every point $Q \notin z$, there is exactly one circle $z'$ with $P, Q \in z'$ and $z \cap z' = \set{P}$;
3. every circle has at least three points, there is at least one circle, and there are four points not on a common circle.

## Remarks

Clause 2 is the touching axiom: $z'$ touches $z$ at $P$.

Example: the classical real Möbius plane has as points $\R^2 \cup \set{\infty}$, the [[Riemann sphere|0J78KBPFZB]].
Its circles are the Euclidean [[circles|060W5CBZP6]] and the sets $g \cup \set{\infty}$ for lines $g$.
Three distinct points lie on exactly one such circle: on a line with $\infty$ if they are collinear or one is $\infty$, and on the circumcircle otherwise.
Given $z$, $P \in z$ and $Q \notin z$, the circles through $P$ tangent to $z$ at $P$ form a pencil, and exactly one of them passes through $Q$.

Non-example: the Euclidean plane with only its circles as blocks fails clause 1, since three collinear points lie on no circle.

The plane sections of an [[ovoid|HNTE5FPS14]] in a three-dimensional projective space form a Möbius plane, called ovoidal; the classical plane comes from a sphere.
A finite Möbius plane has an order $n$: every circle has $n + 1$ points and there are $n^2 + 1$ points.