Distributing Points on the Spheres

We study four different methods for distributing points on the sphere and numerically analyze their relative merits with respect to certain metrics. The analysis of the merits of the algorithms is based on the calculation of the metrics V₀ and V₁ . Given points z₁, ... , z_N on the sphere S² of unit radius we set

V_s = V_s(z₁, . . . , z_N) =

where s>0 is a positive real number and |z_j - z_k| denotes the distance in R³ of z_j and z_k . The metric V₀ = V₀ (z₁, . . . , z_N) is more appropriately defined for points on the sphere S²_½ of unit diameter and is defined as

V₀ =

Here also  |z_j - z_k| refers to the distance of z_j and z_k in R³ .

For each s≥0, the configuration of the points achieving the minimum of V_s, in some sense, forms "evenly" distributed points on the sphere. For example, an equilibrium sate of N electronic charges on a sphere produces minimum Coulomb potential, V₁ . We have investigated some practical methods by which one can obtain an approximate optimal configuration of points without any explicit practice of minimization methods. Refer to the paper for more details about these methods.