Distributing points uniformly on the unit sphere under a mirror reflection symmetry constraint

Uniformly distributed point sets on the unit sphere with and without symmetry constraints have been found useful in many scientific and engineering applications. Here, a novel variant of the Thomson problem is proposed and formulated as an unconstrained optimization problem. While the goal of the Thomson problem is to find the minimum energy configuration of $N$ electrons constrained on the surface of the unit sphere, this novel variant imposes a new symmetry constraint---mirror reflection symmetry with the $x$-$y$ plane as the plane of symmetry. Qualitative features of the two-dimensional projection of the optimal configurations are briefly mentioned and compared to the ground-state configurations of the two dimensional system of charged particles laterally confined by a parabolic potential well.