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?
I suggest you reflect the point in the hyperbolic line. This is just an inversion in a circle. Then you can compute the distance between the point and its image, and divide that by two.
Moving one of the points which span the line into the origin may help as well, but it's not immediately obvious. And quite an unsymmetric approach.
You might want to instead put the single point, $c$, in the origin. Then the line between $a$ and $b$ is a circle but the orthogonal line between $c$ and that is a diameter. So you'd have to find the point on the circle which is closest to the origin, which is a lot simpler. Then you are back to point-point distance.