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Expansive Motions and the Polytope of Pointed Pseudo-Triangulations | | Guenter Rote
; Francisco Santos
; Ileana Streinu
; | | Date: |
4 Jun 2002 | | Journal: | In "Discrete and Computational Geometry -- The Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds), Algorithms and Combinatorics 25, Springer Verlag, Berlin, June 2003, pp. 699-736. | | Subject: | Combinatorics; Metric Geometry MSC-class: 52C25 (primary) 52C35, 68U05, 52B55 (secondary) | math.CO math.MG | | 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). | | Source: | arXiv, math.CO/0206027 | | Services: | Forum | Review | PDF | Favorites | |
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