# Möbius transformation

A *Möbius transformation ^{}* is a bijection

^{}on the extended complex plane $\u2102\cup \{\mathrm{\infty}\}$ given by

$$f(z)=\{\begin{array}{cc}\frac{a}{c}\hfill & \text{if}z=\mathrm{\infty}\hfill \\ \mathrm{\infty}\hfill & \text{if}z=-\frac{d}{c}\hfill \\ \frac{az+b}{cz+d}\hfill & \text{otherwise}\hfill \end{array}$$ |

where $a,b,c,d\in \u2102$ and $ad-bc\ne 0$

It can be shown that the inverse^{}, and composition^{} of two Möbius transformations are similarly defined, and so the Möbius transformations form a group under composition.

The geometric interpretation^{} of the Möbius group is that it is the group of automorphisms^{} of the Riemann sphere.

Any Möbius map can be composed from the elementary transformations - dilations^{}, translations and inversions^{}. If we define a line to be a circle passing through $\mathrm{\infty}$ then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.

Title | Möbius transformation |

Canonical name | MobiusTransformation |

Date of creation | 2013-03-22 12:23:19 |

Last modified on | 2013-03-22 12:23:19 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 21 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 30D99 |

Synonym | fractional linear transformation |

Synonym | linear fractional transformation |

Related topic | ProofOfConformalMobiusCircleMapTheorem |

Related topic | AutomorphismsOfUnitDisk |

Related topic | UnitDiskUpperHalfPlaneConformalEquivalenceTheorem |

Related topic | InversionOfPlane |