Reading through a book of Farb & Magalit (p. 271/in pdf 288 shortly before Prop 10.3) I stumbled upon (after some cleaning) the following equation for hyperbolic metric:
$dist(z, Az)=cosh^{-1}(trace(A)/2)$
for some non-elliptic $A \in PSL_2(\mathbb R)$ and $A(\cdot)$ the typical möbius transformation. I have some problems proving that equation (mod ignoring some constants). My first steps first regarding the case $z=i$ (this should suffice for non-elliptic transformations?) and using this equation were quite fruitless. I would be happy for some help/hint/link were this equation comes from.
Edit: I found the mistake (if someone happens do struggle at the same point and visits this page): The $dist(z, Az)$ should actually be the translation length of the möbius transformation, so the minimum of all these possible $z \in \mathbb H$ or in the context where this question arose the length of the geodesic arc connecting the two corresponding sides of the fundamental area. This is exactly what this equation says for hyperbolic $A$ (s. wiki). Not sure, why $A$ couldn't be parabolic in this context, but will figure this out too (this is outside the question).