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The Trigonometry of Hyperbolic Tessellations

Published online by Cambridge University Press:  20 November 2018

H. S. M. Coxeter
H. S. M. Coxeter*
Affiliation:
Department of Mathematics University of Toronto Toronto, Ontario M5S 2G3
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Abstract

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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 radiiwhile 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 = , 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.

Keywords

51F1551N3052A55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 2 , 01 June 1997 , pp. 158 - 168
Copyright
Copyright © Canadian Mathematical Society 1997

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