Random Spherical Triangles
Steven R. Finch, Antonia J. Jones

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.
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Taxonomy
TopicsScientific Research and Discoveries
