在几何学里, 莫比乌斯变换是一类从黎曼球面映射到自身的函数。用扩展复平面上的复数表示的话，其形式为：

其中 z, a, b, c, d 为满足 ad − bc ≠ 0的（扩展）复数。

莫比乌斯变换也可以被分解为以下几个变换：把平面射影到球面上，把球体进行旋转、位移等任何变换，然后把它射影回平面上。 莫比乌斯变换是以数学家奥古斯特·费迪南德·莫比乌斯的名字命名的，它也被叫做单应变换（homographic transformation）或分式线性变换（linear fractional transformation）。

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form

of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.