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

TL;DR

This paper constructs area and volume preserving maps between certain polyhedrons and spheres or balls, enabling the creation of uniform grids like HEALPix on spherical surfaces.

Contribution

It introduces explicit bijective continuous area and volume preserving maps between specific polyhedrons and spheres or balls, facilitating grid generation.

Findings

Constructed bijective continuous area-preserving maps for a class of polyhedrons.

Identified conditions for extending these maps to volume-preserving maps.

Demonstrated how to generate uniform grids, including HEALPix, via these mappings.

Abstract

For a class of polyhedrons denoted $\mathbb K_n(r,\varepsilon)$, we construct a bijective continuous area preserving map from $\mathbb K_n(r,\varepsilon)$ to the sphere $\mathbb S^{2}(r)$, together with its inverse. Then we investigate for which polyhedrons $\mathbb K_n(r',\varepsilon)$ the area preserving map can be used for constructing a bijective continuous volume preserving map from $\bar{\mathbb K}_n(r',\varepsilon)$ to the ball $\bar{\mathbb S^{2}}(r)$. 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 $\mathbb K_n(r,\varepsilon)$ and $\bar{\mathbb K}_{n}(r',\varepsilon)$, respectively. In particular, we show that HEALPix grids can be obtained by mappings polyhedrons $\mathbb K_n(r,\varepsilon)$ onto the sphere.