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 $ω$-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 $ω$. To probe with an $ω$-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 $ω$-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 $N_B$ be the number of vertices of $O$ whose internal angle is at most $ω$, (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 $2n+3$, $2n+5$) $ω$-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.