Covering the Plane with Equal Sectors

Abstract

In the regular covers, a region in the plane is split into the equal regular polygons (tiles), and all the tiles are covered equally with some geometric figures. In this paper, a tile is an equilateral triangle. We proposed and analyzed the regular covers with equal sectors in which the number of sectors per unit area is minimal. The problem of minimizing the number of sectors per unit area is closely related to the problem of the least dense coverage, but does not coincide with it completely. We found the optimal number of sectors covering one tile in the case when every sector is involved in covering only one tile, and the vertices of the sectors which cover one tile are located in one point.The results can be used in different applications, for example, for design of the cost-effective sensor networks with equal directed sensors when the coverage area of the sensor is a sector.