---
tag: 0SG74WBFA2
kind: concept
name: Möbius transformation
aliases: [mobius-transformation, linear-fractional-transformation, fractional-linear-transformation]
msc: [30C35]
sources:
  - url: https://en.wikipedia.org/wiki/M%C3%B6bius_transformation
    locator: lead section
same_as:
  wikidata: Q595742
  wikipedia: Möbius transformation
---

## Summary

A Möbius transformation is a map $z \mapsto \frac{az + b}{cz + d}$ of the Riemann sphere to itself, with $ad - bc \neq 0$, see the [[definition|0B346DDAM5]].
These maps are bijective and holomorphic, they preserve angles, and they [[map circles and lines to circles and lines|0SXDHQM2MH]].