It was recalled that the spherical image of the boundary of a microstructural feature was a map of the directions of its surface normals on the unit orientation sphere. For closed surfaces, it was a topological invariant. In the case of curved-faced polyhedra, such as those which were found in grain structures or other tessellations of 3-dimensional space, the corners, edges and faces all contributed to the spherical image. The geometries of the spherical images of corners, edges and faces were described here. These geometrical concepts were applied to individual cells, to the cell set, and to the triple lines and quadruple points in the corresponding tessellation. Equations were derived which related the contributions of these spherical images to the number of faces on a single cell and to the average number of faces per grain for the structure. The application of these relationships to grain structures that were governed by local capillary equilibrium demonstrated that it was unlikely that the average number of faces per grain would deviate significantly from 13.4.

R.T.DeHoff: Acta Metallurgica et Materialia, 1994, 42[8], 2633-43