The Trigonometry of Hyperbolic Tessellations

Coxeter, H. S. M.

Canadian mathematical bulletin, Tome 40 (1997) no. 2, pp. 158-168

Résumé

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.

Détail

DOI : 10.4153/CMB-1997-019-0

Mots-clés : 51F15, 51N30, 52A55

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