Triangle Guarding

We consider the problem of triangle guarding (4-guarding) a simple, 2D polygon Q. A polygon Q is 4-guarded if every point q of Q is contained in the convex hull of some three guards that can all see q. This rather odd condition approximates a desire, for example, to see all sides of q or to locate (via triangulation) q from at least two angularly well-separated views. If we restrict the guards to lie in Q and Q has a transparent boundary then the vertices of the convex hull of Q (CH (Q)) form the minimum set of 4-guards. We examine two other variations of the problem. In the first, the guards may be placed within a transparent border surrounding Q. For this variation, we describe a polynomial time algorithm, related to the classic algorithm for finding min-link separators, that finds a minimum set of 4-guards. In the second, we consider the case where Q’s boundary is opaque and show that, as in the traditional art gallery problem, finding a minimum set of vertex 4-guards is NP-hard. Transparent Fences Given simple polygons P and Q with Q contained in P , where |Q|+ |P | = n, what is the minimum number and placement of guards in P needed to 4-guard Q, assuming that both Q and P have transparent boundaries? Our algorithm for this problem is inspired by an algorithm for finding min-link separators between two convex polygons, one contained in the other [1]. The following basic lemmas motivate using this approach. Define legal regions to be regions of the outer polygon P , that are not contained in the interior of CH (Q) (Figure 1).