Quasi-equal area subdivision algorithm for uniform points on a sphere with application to any geographical data distribution

Abstract

This paper describes a quasi-equal area subdivision algorithm based on equal area spherical subdivision to obtain approximated solutions to the problem of uniform distribution of points on a 2-dimensional sphere, better known as Smale's seventh problem. The algorithm provides quasi-equal area triangles, starting by splitting the Platonic solids into subsequent spherical triangles of identical areas. The main feature of the proposed algorithm is that the final adjacent triangles share common vertices that can be merged. It applies reshaping to the final triangles in order to remove obtuse triangles. The proposed algorithm is fast and efficient to generate a large number of points. Consequently, they are suitable for various applications requiring a large number of distributed points. The proposed algorithm is then applied to two geographical data distributions that are modeled by quasi-uniform distribution of weighted points.