Efficient Spherical Designs with Good Geometric Properties

Abstract: Spherical $t$-designs on $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ provide $N$ nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most $t$. This paper considers the generation of efficient, where $N$ is comparable to $(1+t)^d/d$, spherical $t$-designs with good geometric properties as measured by their mesh ratio, the ratio of the covering radius to the packing radius. Results for $\mathbb{S}^{2}$ include computed spherical $t$-designs for $t = 1,...,180$ and symmetric (antipodal) $t$-designs for degrees up to $325$, all with low mesh ratios. These point sets provide excellent points for numerical integration on the sphere. The methods can also be used to computationally explore spherical $t$-designs for $d = 3$ and higher.