Minimal number of sensors for 3D coverage

Abstract

This paper presents some theoretical results on the smaller number Nk(a, b, c) of sensors to achieve k coverage for the 3D rectangular area [0, a] × [0, b] × [0, c]. The first properties outline some theoretical results for the numbers Nk(a, b, c), including symmetry, subadditivity, and monotony on each variable. We use then these results to establish some lower and upper bounds for Nk(a, b, c). The main contribution proposes a result concerning the minimal density of sensors to achieve k-coverage.