---
tag: G03V3D097M
kind: definition
name: Inversion in a circle
msc: [51M05, 51B10]
defines: YXST7NGMTC
uses: [0XW7V9ZZ05, 060W5CBZP6, GP56NBN9RY, 0J78KBPFZB]
sources:
  - url: https://en.wikipedia.org/wiki/Inversive_geometry
    locator: Inversion in a circle
---

## Definition

Let $O$ be a point of the [[Euclidean plane|0XW7V9ZZ05]] and let $r > 0$.
The *inversion in the [[circle|060W5CBZP6]]* with center $O$ and radius $r$ sends each point $P \ne O$ to the unique point $P'$ such that:

1. $P'$ lies on the ray from $O$ through $P$;
2. $\abs{OP} \cdot \abs{OP'} = r^2$.

## Remarks

In coordinates with $O$ at the origin, the inversion is $x \mapsto r^2 x / \norm{x}^2$.
In complex notation with center $0$, it is $z \mapsto r^2 / \bar{z}$.
Adding a point $\infty$ and setting $O \mapsto \infty$ and $\infty \mapsto O$ extends it to a bijection of the [[Riemann sphere|0J78KBPFZB]].
It is an [[involution|GP56NBN9RY]]: applying it twice gives the identity.
The same formula defines inversion in a sphere of $\R^n$.

Example: for the circle with center $(0, 0)$ and radius $1$, the point $P = (2, 0)$ goes to $P' = (1/2, 0)$.
Check: $P'$ lies on the ray from the origin through $P$, and $\abs{OP} \cdot \abs{OP'} = 2 \cdot \frac{1}{2} = 1$.

Non-example: the map $x \mapsto -r^2 x / \norm{x}^2$ satisfies clause 2 but sends $P$ to the opposite ray, so clause 1 fails.