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.
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Notes
- 1.
In [4] probing with an \(\omega \)-wedge is defined for a wider class of convex objects. Since here we focus on the objects being convex polygons, we restrict our definition accordingly.
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Arseneva, E., Bose, P., De Carufel, JL., Verdonschot, S. (2019). Reconstructing a Convex Polygon from Its \(\omega \)-cloud. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_3
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