7,243 research outputs found

    Triangulating the Real Projective Plane

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    We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P2 if this necessary condition holds. As far as we know, this is the first computational result on the real projective plane

    The Complexity of Finding Small Triangulations of Convex 3-Polytopes

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    The problem of finding a triangulation of a convex three-dimensional polytope with few tetrahedra is proved to be NP-hard. We discuss other related complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof appeared at the proceedings of SODA 200

    Memory-Constrained Algorithms for Simple Polygons

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    A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of O(logn)O(\log n) bits, where nn is the size of the input. We assume that a simple nn-gon is given by the ordered sequence of its vertices. We show that we can find a triangulation of a plane straight-line graph in O(n2)O(n^2) time. We also consider preprocessing a simple polygon for shortest path queries when the space constraint is relaxed to allow ss words of working space. After a preprocessing of O(n2)O(n^2) time, we are able to solve shortest path queries between any two points inside the polygon in O(n2/s)O(n^2/s) time.Comment: Preprint appeared in EuroCG 201

    Triangulating stable laminations

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    We study the asymptotic behavior of random simply generated noncrossing planar trees in the space of compact subsets of the unit disk, equipped with the Hausdorff distance. Their distributional limits are obtained by triangulating at random the faces of stable laminations, which are random compact subsets of the unit disk made of non-intersecting chords coded by stable L\'evy processes. We also study other ways to "fill-in" the faces of stable laminations, which leads us to introduce the iteration of laminations and of trees.Comment: 34 pages, 5 figure

    TRIANGULAR MATRIX REPRESENTATIONS OF SKEW MONOID RINGS

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    Let R be a ring and S a u.p.-monoid. Assume that there is a monoid homomorphism &#945; : S &#8594; Aut (R). Suppose that &#945; is weakly&#12288;rigid and lR(Ra) is pure as a left ideal of R for every element a &#8712; R. Then the skew monoid ring R*S induced by &#945; has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is R*S.</p

    3D Reconstruction with Low Resolution, Small Baseline and High Radial Distortion Stereo Images

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    In this paper we analyze and compare approaches for 3D reconstruction from low-resolution (250x250), high radial distortion stereo images, which are acquired with small baseline (approximately 1mm). These images are acquired with the system NanEye Stereo manufactured by CMOSIS/AWAIBA. These stereo cameras have also small apertures, which means that high levels of illumination are required. The goal was to develop an approach yielding accurate reconstructions, with a low computational cost, i.e., avoiding non-linear numerical optimization algorithms. In particular we focused on the analysis and comparison of radial distortion models. To perform the analysis and comparison, we defined a baseline method based on available software and methods, such as the Bouguet toolbox [2] or the Computer Vision Toolbox from Matlab. The approaches tested were based on the use of the polynomial model of radial distortion, and on the application of the division model. The issue of the center of distortion was also addressed within the framework of the application of the division model. We concluded that the division model with a single radial distortion parameter has limitations
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