In mathematics, a **Möbius transformation**, named in honor of August Ferdinand Möbius, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞)

The general formula is given by

- the point is mapped to w = ∞
- the point z = ∞ is mapped to

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 - called the Möbius group.

The geometric interpretation of the Möbius group is that it is the group of automorphisms of the Riemann sphere. The bilinear transform is a special case of a Möbius transformation.

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 infinity, then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.

The **Möbius transformation cross-ratio preservation theorem** states that the
cross-ratio

Let be two Möbius transformations:

If these transformations are carried out in succession, first then to obtain , the result can be readily seen to be another Möbius transformation which appears as the product of the two matricies

The inverse of a Möbius transformation can be derived as

Any Möbius transformation will have two fixed points , invariant under transformation by . Either or both of these fixed points may be the point at infinity: this will happen when . If this is the case, then the transformation will be an affine transformation (some combination of rotation, dialation, and translation). If both points are at infinity, then the transformation is a translation .

The fixed points can be derived as the two roots of the quadratic equation

A Möbius transformation is uniqely defined by its two fixed points and by its characteristic constant .

The characteristic constant can be expressed in terms of its logarithm:

is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about and clockwise about . If is zero (or a multiple of ), then the transformation is said to beIf a transformation has fixed points , and expansion and rotation factors and , then will have .hyperbolic.

The inverse pole is directly opposite the pole relative to the point midway between the two fixed points:

A java applet allowing you to specify a transformation via its fixed points and so on may be found here.

*This page contains material from this article and this article at PlanetMath, used under the GFDL by permission.*