Quadrature on a spherical surface

Abstract: Approximately calculating integrals over spherical surfaces in $\mathbb{R}^3$ can be done by simple extensions of one dimensional quadrature rules. This, however, does not make use of the symmetry or structure of the integration domain and potentially better schemes can be devised by directly using the integration surface in $\mathbb{R}^3$. We investigate several quadrature schemes for integration over a spherical surface in $\mathbb{R}^3$, such as Lebedev quadratures and spherical designs, and numerically test their performance on a set of test functions.