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].


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).

  PostScript file (gzipped)   pdf file
other papers about this subject
Last update: October 13, 2003.