An introduction to computational geometry for curves and surfaces
An introduction to computational geometry for curves and surfaces
Energy functionals, numerical integration and asymptotic equidistribution on the sphere
Journal of Complexity
Cubature over the sphere S2 in Sobolev spaces of arbitrary order
Journal of Approximation Theory
An icosahedron-based method for even binning of globally distributed remote sensing data
Computers & Geosciences
Uniform spherical grids via equal area projection from the cube to the sphere
Journal of Computational and Applied Mathematics
Discrepancy, separation and Riesz energy of finite point sets on the unit sphere
Advances in Computational Mathematics
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We construct a new area preserving map from the unit sphere to the regular octahedron, both centered at the origin. Its inverse map allows the construction of uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the faces of the octahedron. We prove that our new grids are diameter bounded and then, for the resulting configurations of points we calculate some Riesz s-energies and we compare them with the optimal ones. For some configurations, we also calculate the point energies and we list the minimum and maximum values, concluding that these values are very close. Finally, we show how we can map a hemisphere of the Earth onto a square, using our new area preserving projection. The simplicity and the symmetry of our formulas lead to fast computations.