Geodesic
Algorithms of optimal packing construction in ellipse
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 10 (2017) no. 3, pp. 67-79

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It is often necessary to realize an approximation of sets with the union of congruent elements in the theory of control. One way for this approximation is a packing of the union of disks with equal radii into a planar figure. Two versions of an optimal packing problem are considered in the present paper: the number of elements is fixed and to maximize their radii is required in one, the radius is fixed and to maximize the number of elements is required in another one. Iterative methods imitating their centers repulsing from each other and from the boarder are applied in the first version. Constructions of the Chebyshev center, orthogonal projections and points repulsing are used for them. Packing with a hexagonal pattern (closed to optimal) is considered in the second version. Software complex for packing into eclipses with different ratio of axes is developed.
Keywords: packing; Hausdorff distance; maximization; Chebyshev center; direction derivative. 
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     author = {V. N. Ushakov and P. D. Lebedev and N. G. Lavrov},
     title = {Algorithms of optimal packing construction in ellipse},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2017_10_3_a5/}
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V. N. Ushakov; P. D. Lebedev; N. G. Lavrov. Algorithms of optimal packing construction in ellipse. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 10 (2017) no. 3, pp. 67-79. http://geodesic.mathdoc.fr/item/VYURU_2017_10_3_a5/