The distribution of points on the sphere and corresponding cubature formulae

In applications, for instance in optics and astrophysics, there is a need for high-accuracy integration formulae for functions on the sphere. To construct better formulae than previously used, almost equidistantly spaced nodes on the sphere and weights belonging to these nodes are required. This problem is closely related to an optimal dispersion problem on the sphere and to the theories of spherical designs and multivariate Gauss quadrature formulae. We propose a two-stage algorithm to compute optimal point locations on the unit sphere and an appropriate algorithm to calculate the corresponding weights of the cubature formulae. Points as well as weights are computed to high accuracy. These algorithms can be extended to other integration problems. Numerical examples show that the constructed formulae yield impressively small integration errors of up to 10-12.