Probing convex polygons with a wedge

Minimizing the number of probes is one of the main challenges in reconstructing geometric objects with probing devices. In this paper, we investigate the problem of using an ω-wedge probing tool to determine the exact shape and orientation of a convex polygon. An ω-wedge consists of two rays emanating from a point called the apex of the wedge and the two rays forming an angle ω. A valid ω-probe of a convex polygon O contains O within the ω-wedge and its outcome consists of the coordinates of the apex, the orientation of both rays and the coordinates of the closest (to the apex) points of contact between O and each of the rays. We present algorithms minimizing the number of probes and prove their optimality. In particular, we show how to reconstruct a convex n-gon (with all internal angles of size larger than ω) using 2n−2 ω-probes; if ω = π/2, the reconstruction uses 2n − 3 ω-probes. We show that both results are optimal. Let NB be the number of vertices of O whose internal angle is at most ω, (we show that 0 ≤ NB ≤ 3). We determine the shape and orientation of a general convex n-gon with NB = 1 (respectively NB = 2, NB = 3) using 2n− 1 (respectively 2n + 3, 2n + 5) ω-probes. We prove optimality for the first case. Assuming the algorithm knows the value of NB in advance, the reconstruction of O with NB = 2 or NB = 3 can be achieved with 2n + 2 probes,which is optimal.