Directional distributions and the half-angle principle

TL;DR

This paper explores the mathematical properties of angle halving in directional distributions, revealing its connections to key distributions on the circle and its extension to higher dimensions, with implications for stereographic projection.

Contribution

It introduces the concept of angle halving for directional distributions, linking it to the wrapped Cauchy and angular central Gaussian distributions, and discusses its extension to higher dimensions.

Findings

Angle halving identifies wrapped Cauchy with angular central Gaussian distributions on the circle.

It provides a geometric interpretation of stereographic projection in higher dimensions.

The operation's effect on distributions becomes more complex beyond the circle.

Abstract

Angle halving, or alternatively the reverse operation of angle doubling, is a useful tool when studying directional distributions. It is especially useful on the circle where, in particular, it yields an identification between the wrapped Cauchy distribution and the angular central Gaussian distributions, as well as a matching of their parameterizations. The operation of angle halving can be extended to higher dimensions, but its effect on distributions is more complicated than on the circle. In all dimensions angle halving provides a simple way to interpret stereographic projection from the sphere to Euclidean space.