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Reconstructing Point Set Order Types from Radial Orderings
"... Abstract. We consider the problem of reconstructing the combinatorial structure of a set of n points in the plane given partial information on the relative position of the points. This partial information consists of the radial ordering, for each of the n points, of the n − 1 other points around it. ..."
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Abstract. We consider the problem of reconstructing the combinatorial structure of a set of n points in the plane given partial information on the relative position of the points. This partial information consists of the radial ordering, for each of the n points, of the n − 1 other points around it. We show that this information is sufficient to reconstruct the chirotope, or labeled order type, of the point set, provided its convex hull has size at least four. Otherwise, we show that there can be as many as n − 1 distinct chirotopes that are compatible with the partial information, and this bound is tight. Our proofs yield polynomialtime reconstruction algorithms. These results provide additional theoretical insights on previously studied problems related to robot navigation and visibilitybased reconstruction. 1
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ProjectTeam VEGAS Effective Geometric Algorithms for Visibility and Surfaces
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Laboratoire Lorrain de Recherche en Informatique et ses Applications  2004 Activity Report
, 2004
"... The research, both theoretical and applied, is centered around five major themes: • highperformance computing, networks and visualization, • teleoperations and intelligent assistants, • language engineering, document engineering, scientific and technical information engineering, • sofware and compu ..."
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The research, both theoretical and applied, is centered around five major themes: • highperformance computing, networks and visualization, • teleoperations and intelligent assistants, • language engineering, document engineering, scientific and technical information engineering, • sofware and computer system quality and safety, • bioinformatics and applications in genomics. Research activity is performed by 25 research teams including 15 INRIA project teams, with the assistance of 8 research support services. In 2003, three new research teams have been created: Design on objectoriented langages and systems, Madynes on management of dynamic networks and services, and Algorille on algorithms for the Grid. The research teams Cassis et Modbio became INRIA project teams. In autumn 2003, 4 INRIA and 3 CNRS permanent researchers, 5 assistant professors joined the research teams. More than half of them are coming from outside Lorraine. Detailled activities and results are described in this annual report. Without attempting any exhaustivity, some of them mentioned below illustrate applicative potential and scientific relevance
On the number of radial orderings of planar point sets
, 2012
"... Given a set S of n points in the plane, a radial ordering of S with respect to a point p (not in S) is a clockwise circular ordering of the elements in S by angle around p. If S is twocolored, a colored radial ordering is a radial ordering of S in which only the colors of the points are considered. ..."
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Given a set S of n points in the plane, a radial ordering of S with respect to a point p (not in S) is a clockwise circular ordering of the elements in S by angle around p. If S is twocolored, a colored radial ordering is a radial ordering of S in which only the colors of the points are considered. In this paper, we obtain bounds on the number of distinct noncolored and colored radial orderings of S. We assume a strong general position on S, not three points are collinear and not three lines—each passing through a pair of points in S—intersect in a point of R2 \ S. In the colored case, S is a set of 2n points partitioned into n red and n blue points, and n is even. We prove that: the number of distinct radial orderings of S is at most O(n4) and at least Ω(n3); the number of colored radial orderings of S is at most O(n4) and at least Ω(n); there exist sets of points with Θ(n4) colored radial orderings and sets of points with only O(n2) colored radial orderings.