The trigonometry of hyperbolic tessellations

Abstract

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 π/p, π/q, π/2. Hyperbolic trigonometry shows that the side AC has length ψ, where cosh ψ = c/s, c = cos π/q, s = sin π/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 ψ = s / z, where z = √c2 - s2, while its centre is at distance 1 / tanh ψ = c/z from A. In the hyperbolic triangle ABC, the altitude from AB to the right-angled vertex C is ζ, where sinh ζ = z.