Distortion Analysis of Approximate Equal-area Grids Based on Octahedron

In this article, a method for constructing an approximate equal area grid based on the octahedron and Snyder projection is introduced. An octahedron that has equal surface area to a sphere is built; each face of this octahedron is considered as initial subdivision surface. Then, each initial surface is subdivided into hierarchical triangles using a quaternary triangular subdivision scheme, which are projected onto the surface of sphere using Snyder projection. The arc projecting polyhedron triangles onto the spherical surface is modified by using great circle line instead of Snyder projection arc. A new approximate equal area global discrete grid system is constructed. Based on the analysis of difference between the great circle arc and Snyder projection arc, the values of area, edge length, and angle of different subdivision level grids are calculated according to corresponding spherical calculation equation. Based the calculation results, different levels of approximately equidistant grid areas, lengths, angles, and spatial distributions of the deformation are analyzed. Results indicate that with increasing osubdivision levels, 1) the difference in grid areas is very little; area errors of 99.8% of the grids are between -10% and 10%. The girds with heavy area distortion are near the lines between the middle point and three vertexes of octahedron surface, when subdivision level is equal to 10; 2) the ratio increments of the maximum and minimum values of grid areas and edge length show a trend toward convergence, converging to 1.73 and 3.03 respectively.