Correlation between Angle and Side

TL;DR

This paper investigates the expected value of the product of an angle and its opposite side in a random spherical triangle under various constraints, revealing connections to special mathematical constants.

Contribution

It derives explicit expected values for angle-side products in spherical triangles with specific constraints, highlighting the emergence of Apery's and Catalan's constants.

Findings

E(alpha*a)=3.05... when a side is pi/2

E(alpha*a)=2.87... when an angle is pi/2

E(alpha*a)=pi^2/2 - 2 without constraints

Abstract

Let alpha be an arbitrary angle in a random spherical triangle Delta and a be the side opposite alpha. (The sphere has radius 1; vertices of Delta are independent and uniform.) If some other side is constrained to be pi/2, then E(alpha*a)=3.05.... If instead some other angle is fixed at pi/2, then E(alpha*a)=2.87.... In our study of the latter scenario, both Apery's constant and Catalan's constant emerge. We also review Miles' 1971 proof that E(alpha*a)=pi^2/2-2 when no constraints are in place.