Discrete & Computational Geometry

, Volume 48, Issue 4, pp 990–1024 | Cite as

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

Article

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.

Keywords

Discrepancy Isotropic discrepancy Lambert map Level curve Level set Numerical integration Quasi-Monte Carlo Spherical cap discrepancy 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute of Mathematics AGraz University of TechnologyGrazAustria
  2. 2.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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