Covering a rectangle with six and seven circles

Variable-Size Rectangle Covering

In wireless communication networks, optimal use of the directional antenna is very important. The directional antenna coverage (DAC) problem is to cover all clients with the smallest number of directional antennas. In this paper, we consider the variable-size rectangle covering (VSRC) problem, which is a transformation of the DAC problem. There are n points above the base line y = 0 of the two-...

Packing non-identical circles within a rectangle with open length

Packing non-identical circles inside a rectangle witnesses a wide range of industrial applications. However, the non-convex constraints in this problem make it intractable using exact analytical approaches. Even via heuristic methods, the solution time for industrial-scale instances sometimes is too long to be acceptable. This article aims to challenge the existing methods for the benchmark ins...

A Note on Rectangle Covering with Congruent Disks

In this note we prove that, if Sn is the greatest area of a rectangle which can be covered with n unit disks, then 2 ≤ Sn/n < 3 √ 3/2, and these are the best constants; moreover, for ∆(n) := (3 √ 3/2)n − Sn, we have 0.727384 < lim inf ∆(n)/ √ n < 2.121321 and 0.727384 < lim sup∆(n)/ √ n < 4.165064. The problem of covering sets in the plane with figures of prescribed shape has been extensively s...

The Six Circles Theorem revisited

Partial Covering of a Circle by Equal Circles. Part II: The Case of 5 Circles

How must n equal circles of given radius r be placed so that they cover as great a part of the area of the unit circle as possible? In this Part II of a two-part paper, a conjectured solution of this problem for n = 5 is given for r varying from the maximum packing radius to the minimum covering radius. Results are obtained by applying a mechanical model described in Part I. A generalized tense...