As a coordinate system for the Earth's surface based upon latitude and lon- gitude is refined, the areas of facets at the poles become small. The ratio of the area of the smallest facet to the largest one approaches zero. A coordinate system developed by Geoffrey Dutton that produces triangular facets of roughly uniform size over the entire Earth's surface is described. The ratio of the area of the smallest facet to the largest one approaches 6/11. A method is described to refine that coordinate system within a specified polygon. A data structure to use the coordinate system is described. Using that coordinate system as a horizontal basis, a three-dimensional coordinate system consisting of prisms, each of which has a base in the surface mesh, is developed. Methods to compute intersections of rays, along which the radiative-transfer equation is integrated, with faces of prisms of the mesh, and therefrom to compute interpolation coefficients necessary to compute a Jacobian matrix, are described.

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