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

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.