Expansive Motions and the Polytope of Pointed Pseudo-Triangulations

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).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially to the "Further remarks" in Section 5) + changed to final book format. This version is to appear in "Discrete and Computational Geometry -- The Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds), series "Algorithms and Combinatorics", Springer Verlag, Berli

Trees with Convex Faces and Optimal Angles

We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons. Among all such drawings we seek the one maximizing the angular resolution of the drawing. We find linear time algorithms for solving this problem, both for plane trees and for trees without a fixed embedding. In any such drawing, the edge lengths may be set independently of the angles, without crossing; we describe multiple strategies for setting these lengths.Comment: 12 pages, 10 figures. To appear at 14th Int. Symp. Graph Drawing, 200

The polytope of non-crossing graphs on a planar point set

For any finite set \A of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set \A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on \A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension $2n_i +n -3$ where $n_i$ is the number of points of \A in the interior of \conv(\A). The vertices of this polytope are all the pseudo-triangulations of \A, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has been reshape

Optimization of star research algorithm for esmo star tracker

This paper explains in detail the design and the development of a software research star algorithm, embedded on a star tracker, by the ISAE/SUPAERO team. This research algorithm is inspired by musical techniques. This work will be carried out as part of the ESMO (European Student Moon Orbiter) project by different teams of students and professors from ISAE/SUPAERO (Institut Supe ́rieur de l’Ae ́ronautique et de l’Espace). Till today, the system engineering studies have been completed and the work that will be presented will concern the algorithmic and the embedded software development. The physical architecture of the sensor relies on APS 750 developed by the CIMI laboratory of ISAE/SUPAERO. First, a star research algorithm based on the image acquired in lost-in-space mode (one of the star tracker opera- tional modes) will be presented; it is inspired by techniques of musical recognition with the help of the correlation of digital signature (hash) with those stored in databases. The musical recognition principle is based on finger- printing, i.e. the extraction of points of interest in the studied signal. In the musical context, the signal spectrogram is used to identify these points. Applying this technique in image processing domain requires an equivalent tool to spectrogram. Those points of interest create a hash and are used to efficiently search within the database pre- viously sorted in order to be compared. The main goals of this research algorithm are to minimise the number of steps in the computations in order to deliver information at a higher frequency and to increase the computation robustness against the different possible disturbances