# Möbius Transformation

A complex rational function of the form

\[f(z) = \frac{a z + b}{c z + d}\]

# 1 Fixed Points

Following wikipedia, and being a bit sloppy (the chance of a random transformation having a fixed point at infinity is slim):

\(f(\gamma) = \gamma\) implies

\[\gamma_{1,2} = \frac{(a - d) \pm \sqrt{(a-d)^2 + 4bc}}{2c}\]

(assuming \(c \neq 0\)).

Then \[k = f'(\gamma_1) = \frac{a - c \gamma_1}{a - c \gamma_2}\] is the dilation associated with the transformation.

I use the convention that \(\gamma_1\) is the attractive fixed point, with \(|k| \le 1\); otherwise reciprocate \(k\) and swap \(\gamma_1\) with \(\gamma_2\).

Converting the other way,

\[\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} \gamma_1 - k \gamma_2 & (k - 1) \gamma_1 \gamma_2 \\ 1 - k & k \gamma_1 - \gamma_2 \end{pmatrix}\]

# 2 Interpolation

In my 2015 blog post Interpolating Moebius transformations I tried to interpolate between different transformations by using matrix diagonalisation.

An alternative method that seems generally better (at least for iterated function system fractals where fixed points are related directly to appearance) is to find the fixed points and dilation factor as above for each keyframe, and interpolate those, converting back to coefficients in matrix form at each between frame.