Given two distinct points A and B in the hyperbolic plane, there are always two horocycles that contain both A and B.
A horocyle is a curve of constant geodesic curvature κ=1.
In the Poincaré disk model of the hyperbolic plane, a horocycle is represented by a circle that is tangent to the boundary of the disk (aka the circle of infinitely distant “ideal points”).
In the hyperboloid model of the hyperbolic plane, a horocycle is the intersection of the hyperboloid –t^2 + x^2 + y^2 = –1 in (2+1)-dimensional Minkowski spacetime and a plane with a null (aka “lightlike”) normal vector.
What is the area of the region bounded by the two horocycles that pass through two given points? This should only depend on the distance between the points, δ. The computations I performed in both the Poincaré disk model and the hyperboloid model give the same answer:
A(δ) = 4 sinh(δ/2) – 8 atan(tanh(δ/4))
That these two calculations agreed makes me fairly confident that this formula is correct ... but frustratingly I can’t locate anything on the web that says anything about this area.
I can’t believe this simple result isn’t stated *anywhere* on the internet. Even if the formula I’m claiming is wrong, whatever the correct result is, someone like Nikolai Lobachevsky or Eugenio Beltrami surely figured it out in the 19th century, so it must be mentioned somewhere.
So I’m wondering if anyone is aware of a reliable reference on hyperbolic geometry that gives a formula for the area between the two horocycles that contain two given points.
