V.10 - UNDERSTANDING SIMPLOIDS

Abstract

Publisher Summary This chapter presents algorithms for splitting a simploid into a collection of simplices and splitting a simplex into a pair of simploids on opposite sides of a plane. A simploid is a polytope isomorphic to a product of simplices, that is, the vertices of a product of simplices can be moved around a bit, and the result is still a simploid if all the faces remain flat, and the incidence of vertices, edges, and faces does not change. In three dimensions, there are three kinds of simploids: the (3)-simploids, the (2, 1)-simploids, and the (1,1,1) simploids. Although a simploid is not necessarily a product, the structure is the same. Thus, algorithms for manipulating products of simplices, which often work by operating separately on the separate factors, are easily converted to algorithms for working on general simploids. A fundamental operation in constructive solid geometry is computing the intersection of two objects. When the objects are polytopes, the problem can be broken down further into many applications of a simplex/half-space intersection algorithm. In an application that works primarily with simplices, the simploid is a helpful intermediate step; the simploid can be decomposed into simplices immediately. On the other hand, an application that manipulates simploids can cut a simploid by a plane by decomposing the simploid into simplices and then cutting the simplices into simploids.