Averaged ratio between complementary profiles for evaluating shape distortions of map projections and spherical hierarchical tessellations

We describe a metric named averaged ratio between complementary profiles to represent the distortion of map projections, and the shape regularity of spherical cells derived from map projections or non-map-projection methods. The properties and statistical characteristics of our metric are investigated. Our metric (1) is a variable of numerical equivalence to both scale component and angular deformation component of Tissot indicatrix, and avoids the invalidation when using Tissot indicatrix and derived differential calculus for evaluating non-map-projection based tessellations where mathematical formulae do not exist (e.g., direct spherical subdivisions), (2) exhibits simplicity (neither differential nor integral calculus) and uniformity in the form of calculations, (3) requires low computational cost, while maintaining high correlation with the results of differential calculus, (4) is a quasi-invariant under rotations, and (5) reflects the distortions of map projections, distortion of spherical cells, and the associated distortions of texels. As an indicator of quantitative evaluation, we investigated typical spherical tessellation methods, some variants of tessellation methods, and map projections. The tessellation methods we evaluated are based on map projections or direct spherical subdivisions. The evaluation involves commonly used Platonic polyhedrons, Catalan polyhedrons, etc. Quantitative analyses based on our metric of shape regularity and an essential metric of area uniformity implied that (1) Uniform Spherical Grids and its variant show good qualities in both area uniformity and shape regularity, and (2) Crusta, Unicube map, and a variant of Unicube map exhibit fairly acceptable degrees of area uniformity and shape regularity.