A strong triangle inequality in hyperbolic geometry

TL;DR

This paper investigates the probability that a strong triangle inequality holds in hyperbolic geometry, showing it occurs approximately 79% of the time under certain angle distributions.

Contribution

It introduces a probabilistic framework for analyzing triangle inequalities in hyperbolic geometry and provides theoretical methods to compute these probabilities with arbitrary precision.

Findings

The strong triangle inequality holds approximately 79% of the time.

Theoretical results enable precise probability calculations with bounded errors.

Abstract

For a triangle in the hyperbolic plane, let $\alpha,\beta,\gamma$ denote the angles opposite the sides $a,b,c$, respectively. Also, let $h$ be the height of the altitude to side $c$. Under the assumption that $\alpha,\beta, \gamma$ can be chosen uniformly in the interval $(0,\pi)$ and it is given that $\alpha+\beta+\gamma<\pi$, we show that the strong triangle inequality $a + b > c + h$ holds approximately 79\% of the time. To accomplish this, we prove a number of theoretical results to make sure that the probability can be computed to an arbitrary precision, and the error can be bounded.