On the intrinsic geometry of polyhedra: Convex polygon coordinates
There is a very extensive literature dealing with convex polytopes from the standpoints of combinatorics and numerical analysis. By contrast, the current paper adopts an alternative viewpoint that regards a polytope as an autonomous space in its own right, with its own intrinsic geometry. Our attention is focused on the complete set of all the coordinate systems that serve to locate a point of the polytope; divorced, for example, from the smoothness issues that are of concern for applications in numerical analysis. We use the efficient and appropriate algebraic language of barycentric algebras to elicit the convex structure of the set of polytope coordinate systems. Specializing to convex polygons, we examine the chordal coordinate systems that are determined by the triangulations of the polygon. An algorithm to compute the coordinates of a point within such a system is presented. The algorithm relies on a coalgebra structure that transports a probability distribution on one side of a triangle to distributions on each of the remaining two sides. The Catalan number enumeration of the polygon triangulations (well-known within combinatorics) is then obtained in a natural geometric fashion from the parsing trees of the coalgebra structure of our algorithm.