Following this question, I'm looking for a coordinate transformation which leaves distances unchanged. Does such a transformation exist? The isometries for the poincaré disk looked promising, but only conserve angles, not distances.
Edit: I'm using the hyperbolic law of cosines to get the distances between two points in polar coordinates. By constructing a triangle with the origin and the two query points, the length of the third side, which is the desired distance, can be calculated.
I tried a circle inversion with a circle centered outside the disk.
Assuming I have point $a$ near the left border of $P$ and a point $b$ immediately to the left of the arc segment $c$ marked in red. After mirroring both points at $C$, they are closer than they were before. Do I have to adjust the metric or am I doing something wrong?
Ultimately I'm looking for a lower bound for the distances of a point to all points in a (convex) shape.