Applications in smart environments can make use of a user positioning service for various purposes such as activity recognition, context awareness and indoor navigation. We propose a novel solution to the user positioning problem based on inplane detection, which employs infrared light emitters and sensors placed horizontally on fixed positions along the circumference. The presented solution outputs all convex areas that are likely to contain the cross section of a user. The performance is evaluated in simulations of various indoor environments and for various numbers of users present using centroid offset, precision and ratio of equivalent diameters metrics. The simulation results show that the proposed solution determines the individual positions of multiple users in a room with very high performance.

We consider the problem of determining the minimum number Nd of unit disks that is required to block all rays emanating from a point P in the two-dimensional space, where each disk has at least a distance d to point P and to any other disk. We study the asymptotic behavior of Nd, as d tends to infinity. By deriving upper bounds and lower bounds, we prove that pi2/16 ≤ lim_{d -> infinity} N_d/d2 ≤ 18/pi2, where the upper bound is based on establishing an interesting link between unit disks positioned on a regular triangular grid and Farey sequences from number theory. By positioning point P as well as the centers of the disks on the grid points of such a triangular grid, we create hexagonal rings of disks around P. We prove that we need exactly d-1 of these hexagons to block all rays emanating from P.