The words that leave your mouth, do not create proper images in my head:
If you want to create a game taking in the continuous hyperbolic space, I recommend using the Minkowski hyperboloid model internally, and the Poincaré disk model display, just as HyperRogue does. As te shmup section shows, the tesselation might be useful too, to help with the precision of local computations. Use hyperpoint.cpp, or write your own.
