Region Intervisibility in Terrains

A polyhedral terrain is the graph of a continuous piecewise linear function defined over the triangles of a triangulation in the xy-plane. Two points on or above a terrain are visible to each other if the line-of-sight does not intersect the space below the terrain. In this paper, we look at three related visibility problems in terrains. Suppose we are given a terrain T with n triangles and two regions R1 and R2 on T , i.e., two simply connected subsets of at most m triangles. First, we present an algorithm that determines, for any constant > 0, within O(n m) time and storage whether or not R1 and R2 are completely intervisible. We also give an O(mn) time algorithm to determine whether every point in R1 sees at least one point in R2. Finally, we present an O(m n logn) time algorithm to determine whether there exists a pair of points p ∈ R1 and q ∈ R2, such that p and q see each other.