---
tag: 0B346DDAM5
kind: definition
name: Möbius transformation
msc: [30C35]
defines: 0SG74WBFA2
uses: [0CDMXGPSES]
sources:
  - url: https://en.wikipedia.org/wiki/M%C3%B6bius_transformation
    locator: Definition
  - url: https://en.wikipedia.org/wiki/M%C3%B6bius_transformation
    locator: Formula for the inverse transformation
---

## Definition

Let $a, b, c, d$ be complex numbers with $ad - bc \neq 0$.
The Möbius transformation with these coefficients is the map $f : \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ of the [[Riemann sphere|0CDMXGPSES]] to itself given by
$$f(z) = \frac{az + b}{cz + d}$$
for each $z \in \mathbb{C}$ with $cz + d \neq 0$. The map also has the following values.
If $c \neq 0$, then $f(-d/c) = \infty$ and $f(\infty) = a/c$.
If $c = 0$, then $f(\infty) = \infty$.

## Remarks

Each Möbius transformation is a bijection of $\widehat{\mathbb{C}}$, and its inverse is the Möbius transformation $w \mapsto \frac{dw - b}{-cw + a}$.
The Möbius transformations form a [[group|G5S1M9]] under composition, the Möbius group.
The coefficients $(\lambda a, \lambda b, \lambda c, \lambda d)$ with $\lambda \neq 0$ give the same map.
If $ad - bc = 0$, the formula gives a constant or is undefined, so it is not a Möbius transformation.

## Explanations

### L1 human

A Möbius transformation is a quotient of two linear functions of $z$.
Examples are $z \mapsto z + 1$, $z \mapsto 2z$ and $z \mapsto \frac{1}{z}$, and each Möbius transformation is a composition of maps of these simple types.
The point $\infty$ is necessary, because the denominator can be zero.