# Inversive plane

This term is related to: incidence geometry

View other terms related to incidence geometry | View facts related to incidence geometry

Template:Incidence structure property

## Contents

## Definition

### Symbol-free definition

An incidence structure is termed an **inversive plane** if it satisfes the following conditions (here the blocks are termed *circles*):

- Given three distinct points, there is a unique circle through all of them
- Given two points and a circle passing through only one of them, there is a unique circle passing through both points that intersects the earlier circle at only that one point
- There are at least four points, there exists a point-circle pair for which the point is not incident on the circle, and every circle has at least one point

### Definition in terms of affine planes

An **inversive plane** is an incidence structure such that the internal structure at any point in it, is an affine plane.

Alternative terms are **Mobius plane** and **conformal plane**.

## Circles in the inversive plane

### Intersection of circles

Given any two circles in the inversive plane, they can intersect in 0, 1, or 2 points.

- If the circles intersect in 0 points, they are termed
*disjoint* - If the circles intersect in 1 point, they are termed
*tangent* - If the circles intersect in 2 points, they are termed
*intersecting*

### Circles through a set of points

A set of points is termed *concyclic* or *concircular* if there is a circle passing through all the points in the set. Note that this becomes significant only when there are more than three points involved. Further, if there are two concircular sets and their intersection has cardinality at least 3, their union is also concircular.

### Related notions

## References

### Books

*Finite Geometries* by Peter Dembowski, 'Chapter 6. Inversive planes