Area preserving maps and volume preserving maps between a class of polyhedrons and a sphere

Abstract

For a class of polyhedrons denoted ${\mathbb{K}}_n(r,\epsilon )$, we construct a bijective continuous area preserving map from ${\mathbb{K}}_n(r,\epsilon )$ to the sphere ${\mathbb{S}}^2(r)$, together with its inverse. Then we investigate for which polyhedrons ${\mathbb{K}}_n(r^{\prime },\epsilon )$ the area preserving map can be used for constructing a bijective continuous volume preserving map from ${\bar{\mathbb{K}}}_n(r^{\prime },\epsilon )$ to the ball $\overline{{\mathbb{S}}^2}(r)$. These maps can be further used in constructing uniform and refinable grids on the sphere and on the ball, starting from uniform and refinable grids of the polyhedrons ${\mathbb{K}}_n(r,\epsilon )$ and ${\bar{\mathbb{K}}}_n(r^{\prime },\epsilon )$, respectively. In particular, we show that HEALPix grids can be obtained by mappings polyhedrons ${\mathbb{K}}_n(r,\epsilon )$ onto the sphere.

25 pages, 6 figures