A note on the geodesic normal distribution on the sphere

TL;DR

This paper introduces a new formulation of the geodesic normal distribution on the sphere that eliminates the need for tangent space projection, with density contours forming ellipses directly on the sphere.

Contribution

It proposes a tangent-space-independent definition of the geodesic normal distribution on the sphere, offering a novel geometric characterization.

Findings

Density contours are exactly ellipses on the sphere

The distribution is defined directly on the sphere without tangent space projection

Provides an alternative geometric characterization of the distribution

Abstract

This paper presents an alternative formulation of the geodesic normal distribution on the sphere, building on the work of Hauberg (2018). While the isotropic version of this distribution is naturally defined on the sphere, the anisotropic version requires projecting points from the hypersphere onto the tangent space. In contrast, our approach removes the dependence on the tangent space and defines the geodesic normal distribution directly on the sphere. Moreover, we demonstrate that the density contours of this distribution are exactly ellipses on the sphere, providing intriguing alternative characterizations for describing this locus of points.