# Approximation to uniform distribution in SO(3)

## TL;DR

This paper develops bounds and explicit formulas for energies on SO(3) using determinantal point processes, introduces an efficient sampling algorithm, and compares uniform point constructions on the rotation group.

## Contribution

It provides new bounds for Green and Riesz energies on SO(3), explicit Green function calculations, and a practical sampling algorithm for points in SO(3).

## Key findings

- Upper bounds for Green and Riesz energies established.
- Explicit Green function computed for SO(3).
- An effective sampling algorithm for SO(3) points is proposed.

## Abstract

Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO(3), with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on SO(3). The variance of rotation matrices sampled by the determinantal point process is estimated, and formulas for the L2 -norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interest. Also a simple but effective algorithm to sample points in SO(3) is given.