Visibility Extension via Reflection
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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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