Generate Non-Parallel Axes

Generate n 'well-spaced' axes -- uniform sampling of 3D rotation axes

Generates n axes-of-rotation that are 'well-spaced', that is all axes are as far from being parallel as possible. The resulting axes minimize clustering. This is one way to generate a uniform sampling of 3D rotation axes.

INPUTS:
n, the number of axis to uniformly sample.

BACKGROUND:
This is a generalization of the Thomson problem (J.J. Thomson 1904). The Thomson problem determines the minimum energy configuration for n electrons confined to the surface of a sphere: http://en.wikipedia.org/wiki/Thomson_problem

To generate well-spaced axes, additional constraints are imposed by "coupling" each electron with an additional electron at its antipodal point, i.e. x and -x in 3-space.
The solution generates "maximally" separated directions in 3-space, where maximally means that the directions are as far from being parallel as possible.

CREDITS:
Based on code from Hao Peng and Yongyang Yu and their unpublished “Optimization on the surface of a hyper sphere”, https://www.cs.purdue.edu/homes/pengh/reports/590OP.pdf