Hybrid spherical designs

TL;DR

This paper introduces hybrid spherical designs combining points and curves for exact polynomial integration on the sphere, using convex polytopes and symmetry groups, including a notable design for t=19.

Contribution

It constructs new spherical t-design curves from convex polytopes and introduces hybrid designs that integrate points and curves for higher degrees.

Findings

Constructed spherical t-design curves for small t.

Developed hybrid t-designs combining points and curves.

Achieved a hybrid t-design for t=19.

Abstract

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 the edges of a distinct subclass of convex polytopes. We then introduce hybrid $t$-designs that combine points and curves for exact polynomial integration of higher degree. Our constructions are based on the vertices and edges of dual pairs of convex polytopes and polynomial invariants of their symmetry group. A notable result is a hybrid $t$-design for $t=19$.