Generate Non-Parallel Axes

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

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更新 2013/12/10

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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.

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Based on code from Hao Peng and Yongyang Yu and their unpublished “Optimization on the surface of a hyper sphere”, https://www.cs.purdue.edu/homes/pengh/reports/590OP.pdf

引用

Aaron T. Becker's Robot Swarm Lab (2023). Generate Non-Parallel Axes (https://www.mathworks.com/matlabcentral/fileexchange/44515-generate-non-parallel-axes), MATLAB Central File Exchange. 取得済み .

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作成: R2013a
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1.1.0.0

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.

1.0.0.0