Visibility with Multiple Reflections

We show that the region lit by a point light source inside a simple n-gon after at most k reeections oo 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 xed 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.