Günter Rote, Francisco Santos, and Ileana Streinu:

Expansive motions and the polytope of pointed pseudo-triangulations

In: Discrete and Computational Geometry—The Goodman–Pollack Festschrift. Editors: Boris Aronov, Saugata Basu, János Pach, and Micha Sharir, Algorithms and Combinatorics, vol. 25, Springer Verlag, Berlin 2003, pp. 699–736. doi:10.1007/978-3-642-55566-4_33, arXiv:math/0206027 [math.CO].

Abstract

We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of interior pseudo-triangulation edges.

For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an n-gon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1-dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement.

Our methods produce as a by-product a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by Streinu (2000).

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Last update: October 13, 2003.