Higher Dimensions

Abstract

A polyhedron is a 3D element with planar faces. It is simple when all vertices are of order three. Coordinates for simple polyhedra were developed in Chap. 7 of Part I. The GADJ algorithm generalizes to polyhedra. If a polyhedron is bounded by n planes, a unique surface of maximal order n − 4 on which the denominator vanishes may be computed from the divisor group (excluding the vertices) of the boundary planes. The denominator is found easily by the GADJ algorithm. Let vertices j and j + 1 have the common edge (j, j + 1). The planes intersecting on (j, j + 1) are $\mathrm{P}_{\mathrm{j},\mathrm{j}+1}^{+}$ and $\mathrm{P}_{\mathrm{j},\mathrm{j}+1}^{-}$. The third plane at vertex j is P_j and at j + 1 is P_j+1. The other boundary planes common to vertices j and j + 1 are F_j, j+1 (Fig. 12.1).