I am currently attempting to draw polygons in hyperbolic space with angles that are integer multiples of pi, i.e., $\pi / k$ for k an integer. I am particularly interested in determining the vertex positions of such polygons, especially in the context of the Upper Half Plane (UHP) or the Poincaré Disk model. While I have come across a generalization of the second hyperbolic law of cosines to polygons by Glushchenko et al. (2020), which deals with the analytical calculation of multi-beam interference of coherent light, I am curious if there are simpler or more standard approaches commonly used in the community for this purpose. Any guidance or references would be greatly appreciated.
** I know that there is a discussion for the case of hyperbolic triangles so I googled further ended up finding Glushchenko et al. (2020).