---
tag: 0SXDHQM2MH
kind: statement
role: theorem
name: Möbius transformations map circles and lines to circles and lines
aliases: [mobius-maps-preserve-generalized-circles]
msc: [30C35]
uses: [0B346DDAM5, 0CDMXGPSES]
sources:
  - url: https://en.wikipedia.org/wiki/M%C3%B6bius_transformation
    locator: Preservation of angles and generalized circles
---

## Statement

A generalized circle in the [[Riemann sphere|0CDMXGPSES]] $\widehat{\mathbb{C}}$ is either a circle in $\mathbb{C}$, or a set $L \cup \{\infty\}$ where $L$ is a line in $\mathbb{C}$.
Let $f$ be a [[Möbius transformation|0B346DDAM5]].
Then for each generalized circle $C$, the image $f(C)$ is a generalized circle.

## Remarks

A circle can go to a line: the map $z \mapsto \frac{1}{z}$ sends the circle $|z - 1| = 1$, which contains $0$, to the line $\operatorname{Re} w = \frac{1}{2}$ together with $\infty$.
The image of a generalized circle is a line together with $\infty$ exactly when the generalized circle contains the point that $f$ sends to $\infty$.
The proof checks the result for $z \mapsto z + b$, $z \mapsto az$ and $z \mapsto \frac{1}{z}$, because each Möbius transformation is a composition of such maps.

## Explanations

### L1 human

For a Möbius map, a line is only a circle through the point $\infty$.
So the theorem says that circles go to circles, if we count lines as circles.
For example, $z \mapsto 2z + 3$ sends each circle to a circle with twice the radius.