Lebedev quadrature

Remarks

Proof idea: for any $f$ let $\bar f = \frac{1}{48} \sum_{g \in \mathcal{O}_h} f \circ g$. Both $Q$ and the integral are invariant under $\mathcal{O}_h$, so $Q(f) = Q(\bar f)$ and $\int f \, d\sigma = \int \bar f \, d\sigma$, and $\bar f$ is an invariant polynomial of the same degree bound. This reduction is an instance of a theorem of Sobolev on invariant cubature. It cuts the number of equations from about $(p+1)^2$, the dimension of the spherical harmonics of degree at most $p$, to the number of invariant ones. Example: the $6$-point rule with nodes $\pm e_1, \pm e_2, \pm e_3$ and weights $\frac{4\pi}{6}$ is invariant. On $S^2$ the invariant polynomials of degree at most $3$ are constants, because odd ones change sign under $x \mapsto -x$ and the only invariant quadratic is $x_1^2 + x_2^2 + x_3^2 = 1$. Since the weights add up to $4\pi = \sigma(S^2)$, the rule is exact up to degree $3$; it fails for $x_1^4$, where it gives $\frac{4\pi}{3}$ instead of $\frac{4\pi}{5}$.

Going deeper

For a student

Vyacheslav Lebedev computed such rules with many points and high degree, and they are standard in quantum chemistry, where integrals over directions appear in density functional theory. Compared with product rules in spherical coordinates, they use fewer points for the same degree and do not crowd points near the poles. The underlying principle, reducing exactness to the invariant functions of a symmetry group, is used for cubature on many other domains.