Visibility Maps of Segments and Triangles in 3D

Let T be a set of n triangles in three-dimensional space, let s be a line segment, and let t be a triangle, both disjoint from T . We consider the visibility map of s with respect to T , i.e., the portions of T that are visible from s. The visibility map of t is defined analogously. We look at two different notions of visibility: strong (complete) visibility, and weak (partial) visibility. The trivial Ω(n) lower bound for the combinatorial complexity of the strong visibility map of both s and t is almost tight: we prove an O(n log n) upper bound for both structures. Furthermore, we prove that the weak visibility map of s has; complexity Θ(n), and the weak visibility map of t has complexity Θ(n). If T is a polyhedral terrain, the complexity of the weak visibility map is Ω(n) and O(n), both for a segment and a triangle. We also present efficient algorithms to compute all discussed structures, i.e., in most cases computation is only an O(log n) factor worse than the worst-case complexity.