I'm trying to build a geometric datastructure in hyperbolic space. For that purpose, I'm using the Poincaré disk model.
The distance between two points can be calculated with the hyperbolic law of cosines, as mentioned here.
As mentioned in the title, I'm interested in the distance between a point and a line segment. Let's say I have points $a$, $b$, and $c$. What is the distance between $c$ and the shortest path between $a$ and $b$?
With appropriate transformations, one could put $a$ in the origin and have the connecting path be a segment of the diameter. Would this make things easier?