mbius transformation

Mbius transformations should not be confused with the Mbius transform.

Geometry

In mathematics, a Mbius 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 Mbius. The general formula is given by

w = \frac{a z + b}{c z + d}

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 $z = -d/c\,$ is mapped to $w=\infty$
the point $z=\infty$ is mapped to $w = a/c\,$

We can have Mbius 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 Mbius transformations are similarly defined, and so the Mbius transformations form a group under composition - called the Mbius group. The geometric interpretation of the Mbius group is that it is the group of automorphisms of the Riemann sphere. The bilinear transform is a special case of a Mbius transformation. Any Mbius 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 Mbius transformation maps circles to circles, by looking at each elementary transformation. The Mbius transformation cross-ratio preservation theorem states that the cross-ratio

\frac{(w_1-w_2)(w_3-w_4)}{(w_1-w_4)(w_3-w_2)} = \frac{(z_1-z_2)(z_3-z_4)}{(z_1-z_4)(z_3-z_2)} is invariant under a Mbius transformation that maps from z to w.

sample pictures

In the images below, the red and blue dots are the fixed points of each trasformation. We then gave a grid of green crosses, and a line to the magenta dot which is where the cross is moved to under the transformation.

Identity transformation

Hyperbolic transformations

In these transformations, the log of the characteristic constant has a zero imaginary component. These transformations tend to move all points away from one fixed point towards the other fixed point. If one of the fixed points is at infinty, the this is equivalent to doing an affine expansion or contraction around a point.

Elliptical transformations

In these transformations, the log of the characteristic constant has a zero real component. These transformations tend to move all points in circles around the two fixed points . If one of the fixed points is at infinty, the this is equivalent to doing an affine rotation around a point.

Loxodromic transformations

In these transformations, the log of the characteristic constant has non-zero real and imaginary components. These transformations tend to move all points in S-shaped paths from one fixed point to the other.

Equations

The transformation

w = \frac{a z + b}{c z + d}

can be usefully expressed as a matrix

\mathfrak{H} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

In this form, the matrix may be multiplied by any scalar λ and still represent the same transformation. This means that a Mbius transformation on C therefore has six real degrees of freedom.The matrix view of a Mobius transformation corresponds to a projectivity on the projective line over C.

Composition

Let

\mathfrak{H}_1, \mathfrak{H}_2

be two Mbius transformations:

   \mathfrak{H}_1 = \begin{pmatrix} a_1 & b_1 \\ c_1 & d_1 \end{pmatrix}, \;\;   \mathfrak{H}_2 = \begin{pmatrix} a_2 & b_2 \\ c_2 & d_2 \end{pmatrix}

If these transformations are carried out in succession, first

z_1 = \mathfrak{H}_1(z)

then

Z = \mathfrak{H}_2(z_1)

to obtain

Z = \mathfrak{H}_2 \mathfrak{H}_1(z)

, the result can be readily seen to be another Mbius transformation

\mathfrak{H}_3

which appears as the product of the two matrices

\mathfrak{H}_1, \mathfrak{H}_2

\mathfrak{H}_3 = \mathfrak{H}_2 \mathfrak{H}_1 =

\begin{pmatrix} a_2 a_1 + b_2 c_1 & a_2 b_1 + b_2 d_1 \\ c_2 a_1 + d_2 c_1 & c_2 b_1 + d_2 d_1 \end{pmatrix} Thus, Mbius transformations form a group.

Inversion

The inverse of a Mbius transformation

\mathfrak{H}

can be derived as

z = \mathfrak{H}^{-1}(Z) = \frac{d Z - b}{-c Z + a}

and so

\mathfrak{H}^{-1} = \begin{pmatrix} \;\;d & -b \\ -c & \;\;a \end{pmatrix}

Fixed points, characteristic constant

Any Mbius transformation

\mathfrak{H}

will have two fixed points

\gamma_1, \gamma_2

, invariant under transformation by

\mathfrak{H}

. 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, dilation, and translation). If both points are at infinity, then the transformation is a translation

a = \lambda, b=\lambda\Delta, c = 0, d=\lambda

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

c \gamma^2 - (a - d) \gamma - b = 0 \;

\gamma = \frac{(a - d) \pm \sqrt{(a - d)^2 + 4 c b}}{2 c}

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

\mathfrak{H}

transformation is uniquely defined by its two fixed points

\gamma_1, \gamma_2

and by its characteristic constant

k

\mathfrak{H} =

\begin{pmatrix}

   k \gamma_2 - \gamma_1 & (1 - k) \gamma_1     \gamma_2 \\   k          - 1        &         \gamma_2 - k \gamma_1

\end{pmatrix} 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:

\rho e^{\alpha i} = k \;

When expressed in this way,

\rho

becomes an expansion factor. It indicates how repulsive the fixed point

\gamma_1

is, and how attractive

\gamma_2

is. If

\rho = 1

, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be elliptical.

\alpha

is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about

\gamma_1

and clockwise about

\gamma_2

. If

\alpha

is zero (or a multiple of

2 \pi

), then the transformation is said to be hyperbolic.

Iterating a trasformation

If a transformation

\mathfrak{H}

has fixed points

\gamma_1, \gamma_2

, and expansion and rotation factors

\rho

and

\alpha

, then

\mathfrak{H}' = \mathfrak{H}^n

will have

\gamma_1' = \gamma_1, \gamma_2' = \gamma_2, \rho' = n\rho, \alpha' = n\alpha

. This can be used to continuously iterate a transformation. These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants

Poles of the transformation

The point

z_\infty = - \frac{d}{c}

is called the pole of

\mathfrak{H}

; it is that point which is transformed to the point at infinity under

\mathfrak{H}

. The inverse pole

Z_\infty = \frac{a}{c}

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:

\gamma_1 + \gamma_2 = z_\infty + Z_\infty

These four points are the vertexes of a parallelogram which is sometimes called the characteristic parallelogram of the transformation. A transform

\mathfrak{H}

can be specified with two fixed points

\gamma_1, \gamma_2

and the pole

z_\infty

\mathfrak{H} =

\begin{pmatrix}

   Z_\infty & - \gamma_1 \gamma_2 \\   1        & - z_\infty

\end{pmatrix}, \;\;

   Z_\infty = \gamma_1 + \gamma_2 - z_\infty

This allows us to derive a formula for conversion between

k

and

z_\infty

given

\gamma_1, \gamma_2

z_\infty = \frac{k \gamma_1 - \gamma_2}{1 - k}

k

= \frac{\gamma_2 - z_\infty}{\gamma_1 - z_\infty} = \frac{Z_\infty - \gamma_1}{Z_\infty - \gamma_2} = \frac {a - c \gamma_1}{a - c \gamma_2} Which reduces down to

k = \frac{(a + d) + \sqrt {(a - d)^2 + 4 b c}}{(a + d) - \sqrt {(a - d)^2 + 4 b c}}

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

   Z_1 = \mathfrak{H}(z_1), \;\;   Z_2 = \mathfrak{H}(z_2), \;\;   Z_3 = \mathfrak{H}(z_3)

uniquely defines a transformation

\mathfrak{H}

. 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.

\mathfrak{H}_1 = \begin{pmatrix}

\frac{z_2 - z_3}{z_2 - z_1} & -z_1 \frac{z_2 - z_3}{z_2 - z_1} \\ 1 & -z_3 \end{pmatrix}, \;\; \mathfrak{H}_2 = \begin{pmatrix} \frac{Z_2 - Z_3}{Z_2 - Z_1} & -Z_1 \frac{Z_2 - Z_3}{Z_2 - Z_1} \\ 1 & -Z_3 \end{pmatrix} One can get rid of the infinities by multiplying out by

z_2 - z_1

and

Z_2 - Z_1

as previously noted.

\mathfrak{H}_1 = \begin{pmatrix}

z_2 - z_3 & z_1 z_3 - z_1 z_2 \\ z_2 - z_1 & z_1 z_3 - z_3 z_2 \end{pmatrix} , \;\; \mathfrak{H}_2 = \begin{pmatrix} Z_2 - Z_3 & Z_1 Z_3 - Z_1 Z_2 \\ Z_2 - Z_1 & Z_1 Z_3 - Z_3 Z_2 \end{pmatrix} The matrix