Circle Covering With a Margin

Abstract

The problem of the thinnest (i.e., minimum density) covering of the plane with equal circles, solved by Kershner in 1939, can be interpreted as searching for the most economical distribution of transmission towers over a very large plane region, all towers having equal circular range, and collectively to provide reception at each point of the region. The problem of the thinnest 2-fold circle covering can be interpreted similarly, with the stronger requirement that the region should remain covered even if one of the towers stops functioning. Here we consider the intermediate variation, in which the region's coverage is to be maintained even if one of the towers experiences partial loss of power resulting in a certain decrease of its range radius. Specifically, we say that a covering of the plane with unit circular disks has margin µ (where 0 ≤ µ ≤ 1) if any one arbitrarily chosen disk can be replaced with a concentric disk of radius r = 1 - µ and the plane still remains covered. This concept provides a continuous transition from simple covering (margin 0) to 2-fold covering (margin 1). In this paper we determine the minimum density $\vartheta _L^{1 + \mu }$ among all lattice circle coverings that have margin µ, and the lattice λµ that produces the minimum density. Somewhat surprisingly, the optimal lattice λµ behaves discontinuously and is non-unique at a pair of values of µ. As µ varies from 0 to 1, at one point λµ changes suddenly from an equilateral triangular lattice into a square lattice and then, at another point, into a certain rectangular lattice.