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

Abstract

We suppose that in the sensor network, the sensing areas of the sensors are equal sectors, and consider the problem of regular covering of the plane with minimal number of identical sensors per unit area. In the regular cover, the plane is split into the equal regular polygons -"tiles" (equilateral triangles, squares or regular hexagons), and all the tiles are covered equally. We solved the problem for the special case when every sensor covers one tile, and all the sensors covering one tile are placed in one point.

Covering the Plane with Equal Sectors

“…is negative, hence the derivative is also negative, and the function f( ) takes a minimum value when 9 ) ( k (here and below we denote by r the integer part of real r, and by r the smallest integer not less than r). This means that for k /9 is advisable to cover one triangle with 9 sectors. Then 9 sin 2) If /9 k /3, then it is easy to see that the minimum of the objective function )…”

“…Papers [1,13] analyzed a problem of maximizing the number of objects covered by a given number of sensors with sensing area the sector. Several of our publications are devoted to the topic [7][8][9].…”

“…Given the cost of sensors, then naturally the problem arises of finding the plane cover with equal sectors in which the minimum amount of sectors are used to cover the unit area. Moreover, due to technical reasons the parameters of the sector (angle and radius) often cannot take arbitrary independent values [7][8][9]21]. If we take into account the cost of installation and communication of the sensors, it should be preferred to place several sensors (the vertices of the sectors) in the same site (point).…”