Hybrid Spherical Designs

Spherical t -designs are finite point sets on the unit sphere that enable exact integration of polynomials of degree at most t via equal-weight quadrature. This concept has recently been extended to spherical t -design curves by the use of normalized path integrals. However, explicit examples of such curves are rare. We construct new spherical t -design curves for small t based on edge-transitive convex polytopes. We then introduce hybrid t -designs that combine points and curves for exact polynomial integration of higher degree. Our constructions use the edges and vertices of dual pairs of convex polytopes and polynomial invariants of their symmetry group.