Guard placement for efficient point-in-polygon proofs

We consider the problem of placing a small number of angle guards inside a simple polygon P so as to provide efficient proofs that any given point is inside P. Each angle guard views an infinite wedge of the plane, and a point can prove membership in P if it is inside the wedges for a set of guards whose common intersection contains no points outside the polygon. This model leads to a broad class of new art gallery type problems, which we call "sculpture garden" problems and for which we provide upper and lower bounds. In particular, we show there is a polygon P such that a "natural" angle-guard vertex placement cannot fully distinguish between points on the inside and outside of P (even if we place a guard at every vertex of P), which implies that Steiner-point guards are sometimes necessary. More generally, we show that, for any polygon P, there is a set of n + 2(h - 1) angle guards that solve the sculpture garden problem for P, where h is the number of holes in ...

• Type: Conference
• Year: 2007
• Where: COMPGEOM
• Authors: David Eppstein, Michael T. Goodrich, Nodari Sitchinava
• Keywords: Angle Guards, COMPGEOM 2007, Discrete Geometry, Polygons, Sculpture Garden