Möbius transformations should not be confused with the Möbius transform.

Geometry

In mathematics, a Möbius transformation, also called a homographic function, 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 ∞.) It is named in honor of August Ferdinand Möbius.

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
• the point is mapped to

We can have Möbius 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 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

is invariant under a Möbius 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 Möbius transformation therefore has six real degrees of freedom.

Composition

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

Inversion

The inverse of a Möbius transformation can be derived as

and so

Fixed points, characteristic constant

Any Möbius transformation will have two fixed points γ₁,γ₂, invariant under transformation by . Either or both of these fixed points may be the point at infinity: this will happen when c = 0. 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 a = λ,b = λΔ,c = 0,d = λ.

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 Möbius transformation is uniqely defined by its two fixed points γ₁,γ₂ and by its characteristic constant k.

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 k = 1.

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

When expressed in this way, ρ becomes an expansion factor. It indicates how repulsive the fixed point γ₁ is, and how attractive γ₂ is. If ρ = 1, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be elliptical.

α is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about γ₁ and clockwise about γ₂. If α is zero (or a multiple of 2π), then the transformation is said to be hyperbolic.

If a transformation has fixed points γ₁,γ₂, and expansion and rotation factors ρ and α, then will have γ₁' = γ₁,γ₂' = γ₂,ρ' = nρ,α' = nα.

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 that point to which the point at infinity is transformed. The point midway between the two poles is always the same as the point midway between the two fixed points:

A transform can therefore be specified with two fixed points γ₁,γ₂ and the pole .

This allows us to derive a formula for conversion between k and given γ₁,γ₂:

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 z₂ - z₁ and Z₂ - Z₁ as previously noted.

The matrix to map z_1,2,3 onto Z_1,2,3 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:

• Möbius transform
• Möbius function

See also

• Projective geometry

External link

A java applet allowing you to specify a transformation via its fixed points and so on may be found here (http://www.users.bigpond.com/pmurray/Java/MoebApplet.html).

This page contains material from this article (http://planetmath.org/encyclopedia/MobiusTransformation.html) and this article (http://planetmath.org/?method=src&from=objects&id=4222&op=getobj) at PlanetMath, used under the GFDL by permission.