Constructing fully symmetric cubature formulae for the sphere

Abstract

We construct symmetric cubature formulae of degrees in the 13-39 range for the surface measure on the unit sphere. We exploit a recently published correspondence between cubature formulae on the sphere and on the triangle. Specifically, a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function ( u 1 u 2 u 3 ) − 1 / 2 (u_{1}u_{2}u_{3})^{-1/2} , where u 1 u_{1} , u 2 u_{2} , and u 3 u_{3} are homogeneous coordinates.