Reconstructing a convex polygon from its $ω$-cloud

Abstract

An $ω$-wedge is the closed set of points contained between two rays that are emanating from a single point (the apex), and are separated by an angle $ω< π$. Given a convex polygon $P$, we place the $ω$-wedge such that $P$ is inside the wedge and both rays are tangent to $P$. The set of apex positions of all such placements of the $ω$-wedge is called the $ω$-cloud of $P$. We investigate reconstructing a polygon $P$ from its $ω$-cloud. Previous work on reconstructing $P$ from probes with the $ω$-wedge required knowledge of the points of tangency between $P$ and the two rays of the $ω$-wedge in addition to the location of the apex. Here we consider the setting where the maximal $ω$-cloud alone is given. We give two conditions under which it uniquely defines $P$: (i) when $ω< π$ is fixed/given, or (ii) when what is known is that $ω< π/2$. We show that if neither of these two conditions hold, then $P$ may not be unique. We show that, when the uniqueness conditions hold, the polygon $P$ can be reconstructed in $O(n)$ time with $O(1)$ working space in addition to the input, where $n$ is the number of arcs in the input $ω$-cloud.