Reconstructing a Convex Polygon from Its $\omega$-cloud

Abstract

An $\omega$-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 $\omega < \pi$. Given a convex polygon P, we place the $\omega$-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 $\omega$-wedge is called the $\omega$-cloud of P.

We investigate reconstructing a polygon P from its $\omega$-cloud. Previous work on reconstructing P from probes with the $\omega$-wedge required knowledge of the points of tangency between P and the two rays of the $\omega$-wedge in addition to the location of the apex. Here we consider the setting where the maximal $\omega$-cloud alone is given. We give two conditions under which it uniquely defines P: (i) when $\omega < \pi$ is fixed/given, or (ii) when what is known is that $\omega < \pi /2$. 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 $\omega$-cloud.

E. A. was partially supported by F.R.S.-FNRS, and by SNF Early PostDoc Mobility project P2TIP2-168563. P. B., J. C., and S. V. were partially supported by NSERC.