Probing Convex Polygons with a Wedge

Abstract

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 $\omega$-wedge probing tool to determine the exact shape and orientation of a convex polygon. An $\omega$-wedge consists of two rays emanating from a point called the apex of the wedge and the two rays forming an angle $\omega$. To probe with an $\omega$-wedge, we set the direction that the apex of the probe has to follow, the line $\overrightarrow L$, and the initial orientation of the two rays. A valid $\omega$-probe of a convex polygon $O$ contains $O$ within the $\omega$-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 $\omega$) using $2n-2$ $\omega$-probes; if $\omega = \pi/2$, the reconstruction uses $2n-3$ $\omega$-probes. We show that both results are optimal. Let $N_B$ be the number of vertices of $O$ whose internal angle is at most $\omega$, (we show that $0 \leq N_B \leq 3$). We determine the shape and orientation of a general convex $n$-gon with $N_B=1$ (respectively $N_B=2$, $N_B=3$) using $2n-1$ (respectively $2n+3$, $2n+5$) $\omega$-probes. We prove optimality for the first case. Assuming the algorithm knows the value of $N_B$ in advance, the reconstruction of $O$ with $N_B=2$ or $N_B=3$ can be achieved with $2n+2$ probes,- which is optimal.