Generate Non-Parallel Axes

Generate n 'well-spaced' axes -- uniform sampling of 3D rotation axes
Updated 10 Dec 2013

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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.

n, the number of axis to uniformly sample.

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:

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.

Based on code from Hao Peng and Yongyang Yu and their unpublished “Optimization on the surface of a hyper sphere”,

Cite As

Aaron T. Becker's Robot Swarm Lab (2024). Generate Non-Parallel Axes (, MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R2013a
Compatible with any release
Platform Compatibility
Windows macOS Linux

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Version Published Release Notes

On advice from Anton Semechko, vectorized the code for a 10x speedup and explained the difference between uniform_axes and points_on_a_sphere sampling.