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.
Categories: math.MG