The trigonometry of hyperbolic tessellations

http://dx.doi.org/10.4153/CMB-1997-019-0
Canad. Math. Bull. 40(1997), 158-168

  Published:1997-06-01
 Printed: Jun 1997

For positive integers $p$ and $q$ with $(p-2)(q-2) > 4$ there is, in the hyperbolic plane, a group $[p,q]$ generated by reflections in the three sides of a triangle $ABC$ with angles $\pi /p$, $\pi/q$, $\pi/2$. Hyperbolic trigonometry shows that the side $AC$ has length $\psi$, where $\cosh \psi = c/s$, $c = \cos \pi/q$, $s = \sin\pi/p$. For a conformal drawing inside the unit circle with centre $A$, we may take the sides $AB$ and $AC$ to run straight along radii while $BC$ appears as an arc of a circle orthogonal to the unit circle. The circle containing this arc is found to have radius $1/\sinh \psi = s/z$, where $z = \sqrt{c^2-s^2}$, while its centre is at distance $1/\tanh \psi = c/z$ from $A$. In the hyperbolic triangle $ABC$, the altitude from $AB$ to the right-angled vertex $C$ is $\zeta$, where $\sinh\zeta = z$.