latitudes. However, geodesic grids have a nearly homogeneous distribution of points over the sphere.

In mathematics, a geodesic is the equivalent of a straight line, but on a curved surface. On a spherical surface, such as that of Earth, a geodesic is the shortest path between two points, specifically a segment of a great circle. A spherical geodesic grid is defined by spherical, equilateral triangles whose edges are geodesics. One way of defining this grid is to begin with an icosahedron, the geometric solid shown in Fig. 3.10a with 20 triangular faces (major triangles), 12 vertices, and 30 edges, where the vertices touch the surface of a sphere. The vertices may then be connected by geodesics on the sphere, producing spherical triangles. A grid may be created by dividing the major triangles into smaller ones (grid triangles) using a variety of approaches. For example, bisecting each edge of the icosahedron and connecting the bisection points produces four new equilateral triangles for each original one (Fig. 3.10b). The vertices of these new triangles can then be projected onto the sphere along a radial from the center (Fig. 3.10c), and then connected by geodesics to again produce spherical triangles (Fig. 3.10d). Even though the distances between adjacent points look uniform, they are not exactly so. A hint at the asymmetries from one part of the surface to the next can be seen in the fact that the “new” vertex facing the viewer in the upper-center (Fig. 3.10d) is surrounded by six adjoining triangles, while the “original” icosahedron vertex to its right is surrounded by only five. Williamson (1968) and Sadourny et al. (1968) describe another approach for dividing the major triangles into grid triangles, where the inequality in the distance between points is less than that resulting from the method just described. Figure 3.11 shows an example of the distribution of grid points over the sphere.

Some applications of spherical geodesic grids employ the above triangular cells, while others use a related grid with hexagonal cells. To obtain the latter, Voronoi cells are constructed based on the triangular grid, where such cells consist of the set of all points that are closer to a particular vertex than to any other vertex. For the twelve original vertices in the icosahedral grid (e.g., in Fig. 3.11), the Voronoi cells are pentagons. For all the rest, they are hexagons. Figure 3.12 illustrates the geometric relationship between the triangular

In the generation of a spherical geodesic grid, the major triangles of the icosahedron (a) are subdivided, where (b) shows one approach. The vertices of the new triangles are projected (c) onto the sphere that is coincident with the vertices of the icosahedron. Geodesic lines are then drawn between the new vertices to generate spherical grid triangles (d).