Efficient Spherical Designs with Good Geometric Properties

Spherical t-designs on 𝕊 d ⊂ ℝ d + 1 $$\mathbb {S}^{d}\subset \mathbb {R}^{d+1}$$ 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 𝕊 2 $$\mathbb {S}^{2}$$ 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.