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arXiv · 1109.3265

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

Christoph Aistleitner·Johann Brauchart·Josef Dick
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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.

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Christoph Aistleitner, Johann Brauchart, Josef Dick. 2011-09-15. Point sets on the sphere $\mathbb{S}^2$ with small spherical cap discrepancy. https://doi.org/10.1007/s00454-012-9451-3

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