Uniform spherical grids via equal area projection from the cube to the sphere

We construct an area preserving map from the cube to the unit sphere S, both centered at the origin. More precisely, each face Fi of the cube is first projected to a curved square Si of the same area, and then each Si is projected onto the sphere by inverse Lambert azimuthal equal area projection, with respect to the points situated at the intersection of the coordinate axes with S. This map is then used to construct uniform and refinable grids on a sphere, starting from any grid on a square.