Correlation between Angle and Side
Probability
2022-03-22 v4 Classical Analysis and ODEs
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.
Cite
@article{arxiv.1012.0781,
title = {Correlation between Angle and Side},
author = {Steven R. Finch},
journal= {arXiv preprint arXiv:1012.0781},
year = {2022}
}
Comments
13 pages