M�bius transformation

Möbius transformations should not be confused with the Möbius transform.

Table of contents

1 Geometry
2 Equations
3 Composition
4 Inversion
5 Fixed points, characteristic constant
6 Poles of the transformation
7 Specifying a transformation by three points
8 References
9 External link

Geometry

In mathematics, a M�bius 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

almost everywhere where a, b, c, d are any complex numbers satisfying ad - bc ≠ 0. There are two special cases not covered by the formula above:

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

We can have M�bius transformations over the real numbers, as well as for the complex numbers. In both cases, we need to augment the domain with a point at infinity.

It can be shown that the inverse and composition of two M�bius transformations are similarly defined, and so the M�bius transformations form a group under composition - called the M�bius group.

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

Any M�bius 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 M�bius transformation maps circles to circles, by looking at each elementary transformation.

The M�bius transformation cross-ratio preservation theorem states that the cross-ratio

is invariant under a M�bius transformation that maps from z to w.

Equations

The transformation

can be usefully expressed as a matrix

In this form, the matrix may be multiplied by any scalar λ and still represent the same transformation. This means that a M�bius transformation therefore has six real degrees of freedom.

Composition

Let be two M�bius transformations:

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

Inversion

The inverse of a M�bius transformation can be derived as

and so

Fixed points, characteristic constant

Any M�bius 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

Let us discuss the case where the fixed points are finite and the transformation does not perform an involution.

A M�bius transformation is uniqely defined by its two fixed points and by its characteristic constant .

All transformations with the same characteristic constant are similar. Every transformation is similar to some particular linear transformation having one fixed point at infinity and another at 0. A translation is similar to the identity transform, having .

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

When expressed in this way, becomes an expansion factor. It indicates how repulsive the fixed point is, and how attractive is. If , then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be elliptical.

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 be hyperbolic.

If a transformation has fixed points , and expansion and rotation factors and , then will have .

Poles of the transformation

The point

is called the pole of ; it is that point which is transformed to the point at infinity under .

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

A transform can therefore be specified with two fixed points and the pole .

This allows us to derive a formula for conversion between and given :

Which reduces down to

If anyone can work out how to do this without the square rot, that would be extremely cool.

Specifying a transformation by three points

Any set of three points

uniquely defines a transformation . To calculate this out, it is handy to make use of a transformation that is able to map three points onto (0,0), (1, 0) and the point at infinity.

One can get rid of the infinities by multiplying out by and as previously noted.

The matrix to map onto then becomes

You can multiply this out, if you want, but if you are writing code then it's easier to use temporary variables for the middle terms.

References

Not to be confused with: