Point sets on the sphere ${\mathbb{S}}^2$ with small spherical cap discrepancy

Abstract

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order $N^{-1/2}$. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given.

References in corpus (1)

• Quasi-Monte Carlo rules for numerical integration over the unit sphere ${\mathbb{S}}^2$

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