A Family of Möbius Transformations

A Möbius transformation is a function $f:{\mathbb C} \to {\mathbb C}$ of the form $$f(z) = \frac{az+b}{cz+d}$$ where the complex constants $a$, $b$, $c$, and $d$ satisfy the condition $ad-bc\neq0$.

A Möbius transformation is completely defined by its action on just three points. In addition, one can show that they always map circles and lines to circles and lines. These facts allow us to define a particular class of Möbius transformations that are real valued, i.e. $f:{\mathbb C} \to {\mathbb R}$. They are defined by the cross ratio formula $$f(z) = [z,z_1,z_2,z_3] \equiv \frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}.$$ They have the property that $f(z_1) = 0$, $f(z_2)=1$, and $f(z_3)=\infty$ (properly defined).

The animation to the left is showing the action of a particular family of Möbius transformations that map the unit circle to the unit circle. Consider $$M_1(z) = [z,1,-1,i]$$ and $$M_2(z,\theta) = [z,1,-1,e^{i\theta}].$$ Then, $M(z,\theta) = M_2^{-1}(M_1(z),\theta)$ maps the unit circle to the unit circle for each $\theta$.

For each fixed $\theta$, applying this transformation to each pixel in the original image gives the periodic sequence of images seen to the left. The unit circle is shown along with the special values $1$, $-1$, $i$ and $e^{i\theta}$ (the red dots). A characterizing feature of this sequence is that pixels on the unit circle stay on the unit circle throughout the entire sequence.

This work was inspired by Exercise 3.16 of A First Course in Complex Analysis, Version 1.53 by Matthias Beck, Gerald Marchesi, Dennis Pixton & Lucas Sabalka.

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.