All automorphisms of the complex unit disk $\Delta=\{z\in\mathbb{C}:|z|<1\}$ to itself, can be written in the form $f_{a}(z)=e^{{i\theta}}\frac{z-a}{1-\overline{a}z}$ where $a\in\Delta$ and $\theta\in S^{1}$.

This map sends $a$ to $0$, $1/\overline{a}$ to $\infty$ and the unit circle to the unit circle.