# 莫比乌斯变换

${\displaystyle f(z)={\frac {az+b}{cz+d}}}$

## 简介

${\displaystyle {\widehat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.}$

${\displaystyle {\mbox{Aut}}({\widehat {\mathbb {C} }})\,}$.

## 定义

${\displaystyle f(z)={\frac {az+b}{cz+d}}}$

${\displaystyle f(-d/c)=\infty }$${\displaystyle f(\infty )=a/c;}$

${\displaystyle f(\infty )=\infty .}$[1]

## 分解与基本性质

${\displaystyle f(z)={\frac {az+b}{cz+d}}}$

1. ${\displaystyle f_{1}(z)=z+d/c\!}$ （按d/c平移变换）；
2. ${\displaystyle f_{2}(z)=1/z\!}$ （关于单位圆反演变换然后关于实数轴做镜面反射）；
3. ${\displaystyle f_{3}(z)=(-(ad-bc)/c^{2})\cdot z\!}$ （做关于原点位似变换然后做旋转）；
4. ${\displaystyle f_{4}(z)=z+a/c\!}$（按a/c平移变换）。

${\displaystyle f_{4}\circ f_{3}\circ f_{2}\circ f_{1}(z)=f(z)={\frac {az+b}{cz+d}}.\!}$

${\displaystyle g_{i}=f_{i}^{(-1)}}$

${\displaystyle f^{(-1)}(z)=g_{1}\circ g_{2}\circ g_{3}\circ g_{4}(z)={\frac {dz-b}{-cz+a}}}$[3]:51

### 复比不变性

${\displaystyle {\frac {(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{2}-z_{3})(z_{1}-z_{4})}}={\frac {(w_{1}-w_{3})(w_{2}-w_{4})}{(w_{2}-w_{3})(w_{1}-w_{4})}}.}$

${\displaystyle z_{1},z_{2},z_{3},z_{4}}$ 中有一个或多个是无穷大时，复比就定义为相应逼近的极限。比如说当四个复数是 ${\displaystyle z_{1},z_{2},z_{3},\infty }$ 时，复比就是：

${\displaystyle {\frac {(z_{1}-z_{3})}{(z_{2}-z_{3})}}.}$

### 确定莫比乌斯变换

${\displaystyle f(z)={\frac {(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}}}$

## 矩阵表示

${\displaystyle g_{A}:\;z\;\mapsto {\frac {a_{1}z+a_{2}}{a_{3}z+a_{4}}}}$

${\displaystyle {\mathcal {GL}}_{2}(\mathbb {C} )\longrightarrow {\mathcal {M}}({\widehat {\mathbb {C} }})}$
${\displaystyle \varphi :\,A\quad \mapsto \quad g_{A}}$

${\displaystyle \varphi (AB)=\varphi (A)\circ \varphi (B)}$

${\displaystyle {\mathcal {M}}({\hat {\mathbb {C} }})\cong {\mathcal {SL}}_{2}(\mathbb {C} ){\bigg /}\left\{\mathbf {Id} _{2},-\mathbf {Id} _{2}\right\}=\mathbb {P} {\mathcal {SL}}_{2}(\mathbb {C} )}$

## 参考来源

1. ^ （英文）Mobius Transformation and the Extended Complex Plane 互联网档案馆存档，存档日期2015-04-17.，牛津大学数学系讲义
2. （英文）Gábor Tóth. Finite Möbius groups, minimal immersions of spheres, and moduli. Springer; 1 edition. 2001. ISBN 978-0387953236.
3. （英文）Knopp, Konrad, Elements of the Theory of Functions, New York: Dover, 1952, ISBN 978-0-486-60154-0