Reconstructing a convex polygon from its ω -cloud
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.
|Series||Lecture Notes in Computer Science|
Arseneva, E. (Elena), Bose, P, De Carufel, J.-L. (Jean-Lou), & Verdonschot, S. (Sander). (2019). Reconstructing a convex polygon from its ω -cloud. In Lecture Notes in Computer Science. doi:10.1007/978-3-030-19955-5_3