Covering a plane with ellipses

Abstract

This paper is devoted to the construction of regular min-density plane coverings with ellipses of one, two and three types. This problem is relevant, for example, to power-efficient surface sensing by autonomous above-grade sensors. A similar problem, for which discs are used to cover a planar region, has been well studied. On the one hand, the use of ellipses generalizes a mathematical problem; on the other hand, it is necessary to solve these types of problems in real applications of wireless sensor networks. This paper both extends some previous results and offers new regular covers that use a small number of ellipses to cover each regular polygon; these covers are characterized by having minimal known density in their classes and give the new upper bounds for densities in these classes as well.

Covering the Plane with Equal Sectors

“…To evaluate the sign of the difference f( ) -g( ) it is sufficient to study the function is smaller. So, comparing three values, we find the minimum value of the objective function (6), and the optimal number of sectors covering one tile. Statement 2 is proved too.…”

“…An optimal cover with disks in the class COV3(4, 2) is proposed in [22]. In [6], several new regular covers with ellipses of one, two or three types are introduced. As a result, the optimal covers and upper bounds for the minimal density in the different classes are found.…”

“…The cover of a plane region A is such set of figures C, that each point in A belongs to at least one figure in C. It is believed that the sensor's energy consumption is proportional to the covered area [2,3,5,6,18,24].…”

Sensor Networks and Optimal Regular Covering of the Plane With Equal Sectors

“…Then the main goal in WSN is lifetime maximization [5,8,9,10,15,17,20,23]. If only sensing energy consumption is taken into account, then the problem can be reduced to the construction of the least density covers [3,5,7,16,21,23].…”

“…A cover of a plane region is fully specified if one knows the figures included in the cover, the location of each figure and its orientation. For example, if the figures in a cover are ellipses, then it is sufficient to know the semi-axes, the location of the centre and the slope of the axis of each ellipse [7].…”