We extend the concept of the polygon visible from a source point S in a simple polygon by considering visibility with two types of reflection, specular and diffuse. In specular reflection a light ray reflects from an edge of the polygon according to the rule: the angle of incidence equals the angle of reflection. In diffuse reflection a light ray reflects from an edge of the polygon in all inward directions. Several geometric and combinatorial properties of visibility polygons under these two types of reflection are described, when at most one reflection is permitted. We show that the visibility polygon Vs(S) under specular reflection may be nonsimple, while the visibility polygon Vd(S) under diffuse reflection is always simple. We present a (n 2 ) worst-case bound on the combinatorial complexity of both Vs(S) and Vd(S) and describe simple O(n 2 log 2 n) time algorithms for constructing the sets.

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We show that the region lit by a point light source inside a simple n-gon after at most k reflections off the boundary has combinatorial complexity O(n 2k ), for any k ≥ 1. A lower bound of ((n/k − (1)) 2k ) is also established which matches the upper bound for any fixed k. A simple near-optimal algorithm for computing the illuminated region is presented, which runs in O(n 2k log n) time and O(n 2k ) space for k > 1, and in O(n 2 log 2 n) time and O(n 2 ) space for k = 1.

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