ABSTRACT
This chapter outlines the three classic methods of defining points between an initial set provided by spherical Polyhedra. So called the Alternate (of Ford), Triacon, and Skew, these methods form the backbone of subdivision. Spherical polyhedra, particularly the icosahedron, provide a convenient starting point for subdividing spheres because they evenly distribute an initial set of points on the sphere. When points are connected by arcs or chords with points immediately around them, a triangular grid results. High frequency grids can contain harmonics or sub-grids of lower frequency grids. Selective removal of chords produces diamond and hex-pent grids. Each Class layout is described and trigonometric and matrix techniques are defined for calculating grid vertices and, in Class III, projecting them to the surface of the sphere. Class I Alternate grids develop a principal triangle that is related to the face of the reference Polyhedron. Class II Triacon grids benefit from a standard right spherical triangle work area called a Schwarz triangle. Class III Skew grids offer a general projection method from selected points on a two-dimensional BC grid. Numeric examples show how all three Classes develop spherical grids and how rotation, reflection, and translation of these small-area grids can cover the entire sphere without overlaps or gaps.
