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?

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?

I'm studing 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?

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?