---
tag: HFBW3SXQVM
kind: concept
name: Inversive geometry
msc: [51B10, 51M05]
sources:
  - url: https://en.wikipedia.org/wiki/Inversive_geometry
same_as:
  wikidata: Q9267097
  wikipedia: Inversive geometry
---

## Summary

Inversive geometry studies the plane with one point at infinity added.
It keeps the properties of figures that are preserved by inversions in circles and reflections in lines.
In it, lines count as circles through the point at infinity, and inversions map circles to circles.
Its transformations are the Möbius transformations and their composites with complex conjugation.
So it is the geometry of the Riemann sphere in the sense of the Erlangen program.

## Explanations

### L1 human

Inversive geometry treats lines and circles alike: a line is a circle that passes through a point at infinity.
Its basic moves are reflections in lines and inversions in circles, which turn circles into circles and keep angles between curves.
Lengths and straightness are not kept, so problems about tangent circles can often be simplified by an inversion that turns some circles into lines.
For example, two circles touching at a point $P$ become two parallel lines after an inversion centered at $P$.