Random Spherical Triangles

TL;DR

This paper investigates the probability distribution of the perimeter of a random spherical triangle, providing new insights and an exact value at a specific point, building on historical formulas for area density.

Contribution

The paper explores the existence of a closed-form expression for the perimeter density of random spherical triangles and determines its exact value at pi.

Findings

Exact perimeter density at pi determined

No known closed-form expression for perimeter density

Historical formulas for area density are referenced

Abstract

Let Delta be a random spherical triangle (meaning that vertices are independent and uniform on the unit sphere). A closed-form expression for the area density of Delta has been known since 1867; a complicated integral expression for the perimeter density was found in 1994. Does there exist a closed-form expression for the latter? We attempt to answer this question from several directions. An outcome of our work is the exact value of the perimeter density at the point pi.