Most introductory textbooks on the modular group begin with an introduction of it as the group generated by the two Mobius transformations:
$$z'=z+1$$ $$z'=-\frac{1}{z}$$,
and immediately after that, they describe the canonical fundamental domain of the modular group as a certain hyperbolic triangle with angles $\pi/3,\pi/3,0$, which under the action of the modular group tiles up the whole hyperbolic plane. My question refers to the opposite process: given an arbitrary hyperbolic triangle with angles $\pi/n,\pi/m,\pi/l$ (where n,m,l are positive integers), how to construct the generators $T,S$ of the group of Mobius transformation such that this triangle is its fundamental domain?