I'm studying the space of relations with given size in the group of isometries of the Poincaré disk. The unitary disk in the complex plane with a suitable metric is the Poincaré disk, and on this model the (orientation-preserving) isometries are given by the Möbius transformations on the disk:

$$f(z)=e^{i\theta}\frac{z+a}{\bar az + 1}, |a|<1$$

Equivalently, we can write these transformations as

$$f(z) = \frac{cz+a}{\bar az +\bar c}, |a|<|c|$$

I'm trying to find a classification of these isometries in elliptic, parabolic and hyperbolic, which the respectively parameters.

For instance, is easy to see that $f(z)=\frac{z+a}{\bar az + 1}$ is a hyperbolic transformation since it fixes $\pm\frac{a}{|a|}$ and the parameter can be calculated knowing that $-a\mapsto0$.

I had some tries but it still far from complete. Any reference or tips?