---
tag: YXST7NGMTC
kind: concept
name: Inversion in a circle
msc: [51M05, 51B10]
sources:
  - url: https://en.wikipedia.org/wiki/Inversive_geometry
    locator: Inversion in a circle
---

## Summary

Inversion in a circle with center $O$ and radius $r$ sends a point $P \ne O$ to a point $P'$ on the ray from $O$ through $P$.
The point $P'$ is chosen so that $\abs{OP} \cdot \abs{OP'} = r^2$.
It swaps the inside and the outside of the circle and fixes the circle pointwise.
It maps circles and lines to circles and lines, and it preserves angles.
This makes it a basic tool of [[inversive geometry|HFBW3SXQVM]] and of geometric constructions.

## Explanations

### L1 human

Inversion is a mirror for circles instead of lines.
A point close to the center is sent far away, a point far away is sent close to the center, and points on the circle stay put.
For the unit circle centered at the origin, the point $(2, 0)$ goes to $(1/2, 0)$, since $2 \cdot 1/2 = 1$.
Doing the inversion twice brings every point back, just as reflecting twice in a mirror does.