# proof of converse of Möbius transformation cross-ratio preservation theorem

Suppose that $a,b,c,d$ are distinct. Consider the transform $\mu $ defined as

$$\mu (z)=\frac{(b-d)(c-d)}{(c-b)(z-d)}-\frac{b-d}{c-b}.$$ |

Simple calculation reveals that $\mu (b)=1$, $\mu (c)=0$, and $\mu (d)=\mathrm{\infty}$.
Furthermore, $\mu (a)$ equals the cross-ratio^{} of $a,b,c,d$.

Suppose we have two tetrads with a common cross-ratio $\lambda $. Then, as above, we may
construct a transform ${\mu}_{1}$ which maps the first tetrad to $(\lambda ,1,0,\mathrm{\infty})$ and a
transform ${\mu}_{2}$ which maps the first tetrad to $(\lambda ,1,0,\mathrm{\infty})$. Then ${\mu}_{2}^{-1}\circ {\mu}_{1}$ maps the former tetrad to the latter and, by the group property, it is also a
Möbius transformation^{}.

Title | proof of converse^{} of Möbius transformation cross-ratio preservation theorem |
---|---|

Canonical name | ProofOfConverseOfMobiusTransformationCrossratioPreservationTheorem |

Date of creation | 2013-03-22 17:01:51 |

Last modified on | 2013-03-22 17:01:51 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Proof |

Classification | msc 30E20 |