A probabilistic proof of the spherical excess formula

TL;DR

This paper provides a probabilistic proof of Girard's angle excess formula for spherical triangles, connecting geometric properties with probabilistic projections of convex cones.

Contribution

It introduces a novel probabilistic approach to prove the spherical excess formula, linking convex cone projections with spherical geometry.

Findings

Probabilistic proof of Girard's angle excess formula

Characterization of convex cone projections in 3D

Insight into geometric probabilities in spherical geometry

Abstract

This note offers a probabilistic proof of Girard's angle excess formula for the area of a spherical triangle, based on the observation that an unbounded 3-dimensional convex cone, with single vertex at the origin, has only three kinds of 2-dimensional orthogonal projections: a 2-dimensional convex cone with one vertex, a 2-dimensional half-plane (this is an outcome of probability zero), and a 2-dimensional plane.