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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)

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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.

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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 = $asinh(\frac{tanh(halfedgelength)}{tan(\frac{\pi}{p})})$

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