Optimal Cutting of a Polygon by Lasers

Indoor localization is a challenging and important problem. Although GPS technology is very effective for outdoor localization, it cannot be used in the indoor settings, since GPS depends on line of sight to satellites. Hence, other techniques are being developed for indoor environment. Use of lasers is one option that is being investigated to perform localization and target tracking. When multiple lasers are deployed, their beams induce an arrangement, cutting the region into cells, allowing one to track the movement of a target from one cell to a neighboring cell crossing the path of a laser beam. The accuracy of the localization depends on the sizes of the cells. Thus, it is desirable to cut the polygon into cells with small diameter (the maximum Euclidean distance between two points in a cell) or small area. A related problem is known as the “cake cutting problem” in which cuts of a polygonal “cake” are made with an infinite straight line “knife”; in contrast, laser cuts are chords of a polygon and do not pass through the polygon boundary. Another related problem is that of covering a simple polygon with a minimum number of convex polygons; this problem is known to be NP-hard [3].