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Avtomat. i Telemekh., 2012, Issue 3, Pages 79–90 (Mi at3779)  

This article is cited in 7 scientific papers (total in 7 papers)

Applications of Mathematical Programming

Approximating sets on a plane with optimal sets of circles

P. D. Lebedev, A. V. Ushakov

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

Abstract: We study optimal networks on a plane. We generalize the Chebyshev center of a set on the case of several points. We propose numerical and analytic methods for finding a placement of a fixed number of points that minimizes the Hausdorff deviation of a given set from these points. We develop and experiment with software for computing a network of two or three points for the case of flat figures. We show examples of modeling optimal coverings of polyhedra by sets of one, two, or three circles. Based on these networks, we propose an approximation of flat, in general nonconvex, sets by collections of circles.

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English version:
Automation and Remote Control, 2012, 73:3, 485–493

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Presented by the member of Editorial Board: . . 

Received: 06.06.2011

Citation: P. D. Lebedev, A. V. Ushakov, “Approximating sets on a plane with optimal sets of circles”, Avtomat. i Telemekh., 2012, no. 3, 79–90; Autom. Remote Control, 73:3 (2012), 485–493

Citation in format AMSBIB
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\by P.~D.~Lebedev, A.~V.~Ushakov
\paper Approximating sets on a~plane with optimal sets of circles
\jour Avtomat. i Telemekh.
\yr 2012
\issue 3
\pages 79--90
\mathnet{http://mi.mathnet.ru/at3779}
\transl
\jour Autom. Remote Control
\yr 2012
\vol 73
\issue 3
\pages 485--493
\crossref{https://doi.org/10.1134/S0005117912030071}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. P. D. Lebedev, A. A. Uspenskii, V. N. Ushakov, “Algoritmy nailuchshei approksimatsii ploskikh mnozhestv ob'edineniyami krugov”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 4, 88–99  mathnet
    2. P. D. Lebedev, D. S. Bukharov, “Approksimatsiya mnogougolnikov nailuchshimi naborami krugov”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 6:3 (2013), 72–87  mathnet
    3. V. N. Ushakov, A. S. Lakhtin, P. D. Lebedev, “Optimization of the Hausdorff distance between sets in Euclidean space”, Proc. Steklov Inst. Math. (Suppl.), 291, suppl. 1 (2015), 222–238  mathnet  crossref  mathscinet  isi  elib
    4. V. N. Ushakov, P. D. Lebedev, “Algorithms for the construction of an optimal cover for sets in three-dimensional Euclidean space”, Proc. Steklov Inst. Math. (Suppl.), 293, suppl. 1 (2016), 225–237  mathnet  crossref  mathscinet  isi  elib
    5. V. N. Ushakov, P. D. Lebedev, “Algoritmy optimalnogo pokrytiya mnozhestv na ploskosti $\mathbb{R}^2$”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:2 (2016), 258–270  mathnet  crossref  mathscinet  elib
    6. A. L. Kazakov, P. D. Lebedev, “Algorithms for constructing optimal $n$-networks in metric spaces”, Autom. Remote Control, 78:7 (2017), 1290–1301  mathnet  crossref  mathscinet  isi  elib
    7. A. R. Alimov, I. G. Tsar'kov, “Chebyshev centres, Jung constants, and their applications”, Russian Math. Surveys, 74:5 (2019), 775–849  mathnet  crossref  crossref  mathscinet  adsnasa  isi
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