I’m trying to investigate hyperbolic tessellations and how to construct them. After doing lots of reading I’ve found that given a hyperbolic triangle with angles $\frac{\pi}{m}, \frac{\pi}{n}, \frac{\pi}{l} $ you can create a tessellation. I understand the theory behind this but I’m struggling to find a general way to construct triangles with this property in the unit disc.
My method so far has been as follows:
Choose $m$ and $n$ so that $m+n<\frac{1}{2}$, assume the third angle is $\frac{\pi}{2} $.
Assume the vertices $v_1, v_2, v_3$ are such that $v_1$ is the origin, $v_2$ lies on the x axis and $v_3$ is somewhere in the top right quadrant of the disc, call $\alpha=\frac{\pi}{m}, \beta= \frac{\pi}{2}, \gamma= \frac{\pi}{n}$ the angles at each vertex and the lengths of the opposite sides to each vertex a,b and c.
The sides of the triangle are $\ell_1$: a line segment of the x axis, $\ell_2$: the line $y=xtan(\frac{\pi}{m})$, and $\ell_3$: a d line which meets $\ell_2$ at the angle $\gamma$ and $\ell_1$ at the angle $\beta$.
Using the hyperbolic law of cosines: $cos(\gamma)= -cos(\alpha)cos(\beta)+sin(\alpha)sin(\beta)cosh(c)$, we can find a formula for c which is the length distance of $\ell_1$, this gives the position of $v_2$.
Then create the d line passing through $v_2$, this gives $\ell_3$.
This method doesn’t seem to be working as it isn’t always creating triangles- and I’m entirely stumped how to proceed.
Any help would be much appreciated!