how can I construct an hyperbolic (8,3) tiling ( see https://en.wikipedia.org/wiki/Octagonal_tiling ) in the Poincare Disk model or Klein Disk model of hyperbolic geometry ?
or:
What are the hyperbolic lengths of all relevant distances (distance center - vertex, distance center midpoint of an edge , length of an edge ed)
If you know the three angles, you can get the lengths of the sides, using the Hyperbolic analog of the formulas for spherical triangles. You do have to be careful, though, I think that in the Law of Cosines, there may be a minus sign slipped in. Sorry to be so sketchy in this answer, I have a ton of other things hanging over me.
Notice that formulas here using radian, not degree.
If you want to calculate the tiling with schlafli symbol {8,3}, then p=8 and q=3.
To calculate edge length=e: solve this
the angle of a polygon = $2*asin(\frac{cos(\frac{\pi}{p})}{cosh(\frac{e}{2})})$
Since q=3, then that angle must be $\frac{2\pi}{3}$. You can solve the function $2*asin(\frac{cos(\frac{\pi}{p})}{cosh(\frac{e}{2})})=\frac{2\pi}{3}$ yourself, that's easy.
halfedgelength = $\frac{edgelength}{2}$
distance center - vertex = circumradius = $asinh(\frac{sinh(halfedgelength)}{sin(\frac{\pi}{p})})$
distance center - midpoint of an edge = inradius = $asinh(\frac{tanh(halfedgelength)}{tan(\frac{\pi}{p})})$
