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 cor
respondence between cubature formulae on the sphere and on the triangle. Sp
ecifically, a fully symmetric cubature formula for the surface measure on t
he unit sphere corresponds to a symmetric cubature formula for the triangle
with weight function (u(1)u(2)u(3))(-1/2), where u(1), u(2), and u(3) are
homogeneous coordinates.