Parametric Families of Minimal Positive Cubature Formulas of Degrees 4 and 6 on the 2-Sphere

We study minimal positive cubature formulas on the two-dimensional sphere of degrees four and six, with ten and eighteen nodes, respectively. We prove the regularity of Popov’s symmetric formulas and establish that their local families of deformations are two-dimensional, up to rotations. We obtain algebraic relations between the weights and pairwise inner products of the nodes. For degree four, we construct a one-parameter family with half-turn symmetry connecting the formulas of Popov and Heo–Xu. We also give a self-contained proof that eighteen nodes are minimal for degree six.