Visibility Extension via Reflection
release_hli6wyxlgjhehbzx75pzminecy
by
Arash Vaezi, Bodhayan Roy, Mohammad Ghodsi
2020
Abstract
This paper studies a variant of the Art-gallery problem in which "walls" can
be replaced by reflecting-edges, which allows the guard to see further
and thereby see a larger portion of the gallery. The art-gallery is a simple
closed polygon P, a guard is a point p in P, and a guard sees another
point q in P if the segment pq is contained in the interior of
P. The visibility region of p is the set of points q in P that are
visible from p. If we let an edge of the polygon allow reflections, then the
visibility region should be changed accordingly. We study visibility with
specular and diffuse reflections. Moreover, the number of times a ray can be
reflected can be taken as a parameter. For vertex guarding polygons with
k diffuse reflections, we establish an upper bound on the optimum solution.
For this problem, we generalize the O(logn)-approximation ratio algorithm of
the Art Gallery Problem. For a bounded k, the generalization gives a
polynomial-time algorithm with O(log n)-approximation ratio for both cases
diffuse and specular reflections. Furthermore, We show that several cases of
the generalized problem are NP-hard. We also illustrate that if P is a funnel
or a weak visibility polygon, then the problem becomes more straightforward and
can be solved in polynomial time.
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2011.03107v1
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