---
tag: 8WBZ6Q949M
kind: definition
name: Inversive geometry
msc: [51B10, 51M05]
defines: HFBW3SXQVM
uses: [YXST7NGMTC, 06BRDF3F0E, 0J78KBPFZB, 0SG74WBFA2, 060W5CBZP6, 9QW65PGEPW]
sources:
  - url: https://en.wikipedia.org/wiki/Inversive_geometry
---

## Definition

Let $\hat{\C} = \C \cup \set{\infty}$ be the extended plane, the [[Riemann sphere|0J78KBPFZB]].
A *generalized circle* of $\hat{\C}$ is either a Euclidean [[circle|060W5CBZP6]] or a set $g \cup \set{\infty}$ with $g$ a line.
Extend each [[inversion in a circle|YXST7NGMTC]] to $\hat{\C}$ by swapping its center with $\infty$, and each [[reflection|06BRDF3F0E]] in a line by fixing $\infty$.
The *inversive group* is the group of bijections of $\hat{\C}$ generated by these maps.
*Inversive geometry* is the study of the properties of subsets of $\hat{\C}$ that are invariant under the inversive group.

## Remarks

In the language of the [[Erlangen program|9QW65PGEPW]], inversive geometry is the geometry of $\hat{\C}$ with the inversive group as its group of transformations.
The elements of the inversive group that preserve orientation are the [[Möbius transformations|0SG74WBFA2]] $z \mapsto (az + b)/(cz + d)$ with $ad - bc \ne 0$.
The others are the maps $z \mapsto (a\bar{z} + b)/(c\bar{z} + d)$ with $ad - bc \ne 0$.
Every element maps generalized circles to generalized circles and preserves the size of angles between curves.

Example: the property "these four points lie on a common generalized circle" is inversive, since the inversive group maps generalized circles to generalized circles.

Non-example: the property "this circle has radius $1$" is not inversive; the inversion $z \mapsto 4 / \bar{z}$ maps the circle $\abs{z} = 1$ to the circle $\abs{z} = 4$.

The same construction in $\R^n \cup \set{\infty}$, with inversions in spheres and reflections in hyperplanes, gives inversive geometry in dimension $n$.