# Möbius transformation cross-ratio preservation theorem

A Möbius transformation^{} $f:z\mapsto w$ preserves the cross-ratios, i.e.

$$\frac{({w}_{1}-{w}_{2})({w}_{3}-{w}_{4})}{({w}_{1}-{w}_{4})({w}_{3}-{w}_{2})}=\frac{({z}_{1}-{z}_{2})({z}_{3}-{z}_{4})}{({z}_{1}-{z}_{4})({z}_{3}-{z}_{2})}$$ |

Conversely, given two quadruplets which have the same cross-ratio^{}, there
exists a Möbius transformation which maps one quadruplet to the other.

A consequence of this result is that the cross-ratio of $(a,b,c,d)$ is the value at $a$ of the Möbius transformation that takes $b$, $c$, $d$, to $1$, $0$, $\mathrm{\infty}$ respectively.

Title | Möbius transformation cross-ratio preservation theorem |
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Canonical name | MobiusTransformationCrossratioPreservationTheorem |

Date of creation | 2013-03-22 13:35:50 |

Last modified on | 2013-03-22 13:35:50 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 30E20 |

Related topic | CrossRatio |