An Area Preserving Projection from the Regular Octahedron to the Sphere

Abstract

In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere $${\mathbb{S}^2}$$ , both centered at the origin. The construction scheme consists of two steps. First, each face F i of the octahedron is mapped to a curved planar triangle $${\mathcal{T}_i}$$ of the same area. Afterwards, each $${\mathcal{T}_i}$$ is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of $${\mathbb{S}^2}$$ . The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.