Geodesic
Algorithms of optimal covering of 2D sets with dynamical metrics
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 60 (2022), pp. 58-72

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The paper deals with the problem of constructing the thinnest covering for a convex set by a set of similar elements. As a distance between two points, we use the shortest time it takes to achieve one point from another, and the boundary of each covering circle is an isochron. Such problems arise in applications, particularly in sonar and underwater surveillance systems. To solve the problems of covering with such circles and balls, we previously proposed algorithms based both on variational principles and geometric methods. The purpose of this article is to construct coverings when the characteristics of the medium change over time. We propose a computational algorithm based on the theory of wave fronts and prove the statement about its properties. Illustrative calculations are performed.
Keywords: optimal covering, wave front, dynamical metric, Chebyshev center. 
@article{IIMI_2022_60_a3,
     author = {P. D. Lebedev and A. A. Lempert and A. L. Kazakov},
     title = {Algorithms of optimal covering of {2D} sets with dynamical metrics},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {58--72},
     publisher = {mathdoc},
     volume = {60},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2022_60_a3/}
}
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P. D. Lebedev; A. A. Lempert; A. L. Kazakov. Algorithms of optimal covering of 2D sets with dynamical metrics. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 60 (2022), pp. 58-72. http://geodesic.mathdoc.fr/item/IIMI_2022_60_a3/