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 ...

An algorithm is presented for computing geodesic furthest neighbors for all the vertices of a simple polygon, where geodesic denotes the fact that distance between two points of the polygon is defined as the length of an Euclidean shortest path connecting them within the polygon. The algorithm runs in O(n log n) time and uses O(n) space; n being the number of vertices of the polygon. As a corol...

Given a simple polygon P in the plane, we present new algorithms and data structures for computing the weak visibility polygon from any query line segment in P . We build a data structure in O(n) time and O(n) space that can compute the visibility polygon for any query line segment s in O(k log n) time, where k is the size of the visibility polygon of s and n is the number of vertices of P . Al...

The scattering support is an estimate of the support of a source or scatterer, based on a limited set of far field measurements. In this paper, we suppose that the far field is measured at all wavenumbers, but only at a few, say N, angles θi ∈ . From these measurements, we produce a -convex polygon (a convex polygon with normals in the θi directions). We show that this polygon must be contained...

An example is presented of a cubic Bézier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bézier curve. These examples complement known upper bounds on the number of subdivisions sufficient for a control polygon to be ambient isotopic to its...

In this paper we solve the following optimization problem: Given a simple polygon P , what is the maximum-area polygon that is axially symmetric and is contained by P? We propose an algorithm for solving this problem, analyze its complexity, and describe our implementation of it (for the case of a convex polygon). The algorithm is based on building and investigating a planar map, each cell of w...

Behavior of convex solution polygons to a general crystalline motion is investigated. A polygon is called admissible if the set of its normal angles equals that of the Wulff shape. We prove that if the initial polygon is not an admissible polygon, then all edges disappear simultaneously, or edge disappearing occurs at most finitely many instants and eventually a convex solution polygon becomes ...