Point Sets on the Sphere \mathbb{S}^{2} with Small Spherical Cap Discrepancy

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.