Points on $\operatorname{SO}(3)$ with low logarithmic energy

Carlos Beltrán; Federico Carrasco; Damir Ferizović; Pedro R. López-Gómez

Points on $\operatorname{SO}(3)$ with low logarithmic energy

Classical Analysis and ODEs 2025-07-21 v2

Authors: Carlos Beltrán , Federico Carrasco , Damir Ferizović , Pedro R. López-Gómez

Abstract

We describe several randomized collections of $3\times 3$ rotation matrices and analyze their associated logarithmic energy. The best one (i.e. the one attaining the lowest expected logarithmic energy) is constructed by choosing $r$ spherical points, which come from the zeros of a randomly chosen degree $r$ polynomial, and considering at each of these points a set of $s$ evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of $n=rs$ rotation matrices.

Keywords

logarithmic energy SO(3) rotation matrices spherical points random polynomial zeros energy minimization random matrix collections upper bound

Cite

@article{arxiv.2506.13388,
  title  = {Points on $\operatorname{SO}(3)$ with low logarithmic energy},
  author = {Carlos Beltrán and Federico Carrasco and Damir Ferizović and Pedro R. López-Gómez},
  journal= {arXiv preprint arXiv:2506.13388},
  year   = {2025}
}

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32 pages

Points on $\operatorname{SO}(3)$ with low logarithmic energy

Abstract

Keywords

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