Abstract view
The trigonometry of hyperbolic tessellations


Published:19970601
Printed: Jun 1997
Abstract
For positive integers $p$ and $q$ with $(p2)(q2) >
4$ 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 $z
= \sqrt{c^2s^2}$, while its centre is at distance $1/\tanh
\psi = c/z$ from $A$. In the hyperbolic triangle $ABC$,
the altitude from $AB$ to the rightangled vertex $C$ is
$\zeta$, where $\sinh\zeta = z$.
MSC Classifications: 
51F15, 51N30, 52A55 show english descriptions
Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55] Geometry of classical groups [See also 20Gxx, 14L35] Spherical and hyperbolic convexity
51F15  Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55] 51N30  Geometry of classical groups [See also 20Gxx, 14L35] 52A55  Spherical and hyperbolic convexity
