Comparing Area and Shape Distortion on Polyhedral-Based Recursive Partitions of the Sphere

Regular grid sampling structures in the plane are a common spatial framework for many studies. Constructing grids with desirable properties such as equality of area and shape is more difficult on a sphere. We studied the distortion characteristics of recursive partitions of the surface of the globe starting with the octahedron and icosahedron polyhedral models. We used five different methods for mapping from the polyhedral model to the surface of the sphere: the Gnomonic projection, Fuller's Dymaxion projection, Snyder's equal area polyhedral projection, direct spherical subdivision, and a recursive polyhedral projection. We increased partition density using both a 4-fold and a 9-fold ratio at each level of recursive subdivision by subdividing to the 8th level with the 4-fold density ratio (65 536 cells per polyhedral face) and to the fifth level with the 9-fold density ratio (59 049 cells per polyhedral face). We measured the area and perimeter of each cell at each level of recursion for each method on e...