Trig in hyperbolic geometry

I recently wrote posts about spherical analogs of the Pythagorean theorem, the law of cosines, and the law of sines. The corresponding formulas for hyperbolic space mostly just replace circular functions with hyperbolic functions, i.e. replace sine with hyperbolic sine and cosine with hyperbolic cosine.

Triangles on a sphere or on a hyperbolic space like a pseudosphere have two kinds of angles: the sides come together at an angle, but the sides themselves are angles. By convention, the former are denoted with upper case letters and the latter with lower case letters.

The translation rule from spherical to hyperbolic geometry is to change functions of sides from circular to hyperbolic, but to leave functions of intersection angles alone. Or in typographical terms, put an h on the end of functions of a lower case letter but not functions of a upper case letter.

You can find these formulas, for example, in [1].

Hyperbolic Pythagorean theorem

The Pythagorean theorem on a sphere

cos(c) = cos(a) cos(b).

becomes

cosh(c) = cosh(a) cosh(b)

in hyperbolic geometry. Here you simply change cos to cosh.

Hyperbolic law of sines

The law of sines on a sphere

sin(a) / sin(A) = sin(b) / sin(B) = sin(c) / sin(C).

becomes

sinh(a) / sin(A) = sinh(b) / sin(B) = sinh(c) / sin(C).

in hyperbolic geometry. Here sin becomes sinh, but only before a lower case letter, i.e. when applied to a side.

Hyperbolic law of cosines

The law of cosines on a sphere

cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C).

becomes

cosh(c) = cosh(a) cosh(b) - sinh(a) sinh(b) cos(C).

Note the negative sign, a small exception to our conversion rule.

We could rewrite the law of cosines on a sphere to be

cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C)

where stands for curvature, which equals 1 for a unit sphere. Then our theorem translation rule holds exactly:

cosh(c) = cosh(a) cosh(b) + sinh(a) sinh(b) cos(C)

in a hyperbolic space with curvature = -1.

Related post

See this post on the Unified Pythagorean Theorem for a version of the Pythagorean theorem that holds in spherical, plane, and hyperbolic geometry.

Maybe there are analogous unified laws of sines and cosines. This is left as an exercise for the reader.

[1] William P. Thurston. Three-Dimensional Geometry and Topology, Volume 1. Princeton University Press, 1997.

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