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On the intrinsic geometry of polyhedra: Convex polygon coordinates

Anna B. Romanowska, Jonathan D. H. Smith, Anna Zamojska-Dzienio

Abstract

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.

On the intrinsic geometry of polyhedra: Convex polygon coordinates

Abstract

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.
Paper Structure (34 sections, 12 theorems, 59 equations, 1 figure, 4 tables)

This paper contains 34 sections, 12 theorems, 59 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Notational conventions
  4. Barycentric algebras
  5. The basic definition
  6. Subalgebras, homomorphisms, and products
  7. Free algebras and simplices
  8. Construction of the free barycentric algebra
  9. Probability distributions and weighted averages
  10. Kernel functions
  11. Partitions of unity and linear precision
  12. Interpolation
  13. Coordinate systems for polytopes
  14. Function spaces
  15. Barycentric coordinates in the function space
  16. ...and 19 more sections

Key Result

Lemma 2.11

If the barycentric property E:brycmbkn holds, then the partition of unity property E:PartoOne follows.

Figures (1)

  • Figure 1: Oriented, non-crossing chordal decompositions of hexagons.

Theorems & Definitions (43)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 33 more