Is there anyone to help me tile on a Poincaré disc? In fact, I'm going to tile triangle tiles on a surface in hyperbolic geometry ; is there any algorithmic method to do so?
I've created such tessellations in my Ph.D. thesis. I'd start by identifying the corners of one central triangle, using the hypberolic law of cosines. Then I'd describe the reflections in the edges of said triangle. To make sure that I'd create each triangle of the tiling exactly once, I'd create a finite state automaton representing the Coxeter group of the tiling, and do a breadth first traversal of that. You'll probably need some criterion to stop the iteration once you're close enough to the rim of the Poincaré model. This is obviously just a rough outline. For details feel free to read my work, and contact me if you got any more specific problems.