Directional distributions and the half-angle principle

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.