A visibility relation can be viewed as a graph: the uncountable graph of a visibility relationship between points in a polygon P is called the point visibility graph (PVG) of P. In this paper we explore the use of perfect graphs to characterize tractable subprob-lems of visibility problems. Our main result is a characterization of which polygons are guaranteed to have weakly triangulated PVGs, ...

What is the minimal area that a convex lattice polygon with exactly n vertices can have? This question, and several similar ones, are considered in this survey.

The inaccessibility of a point p in a bounded domain D ⊂ R is the minimum of the lengths of segments through p with boundary at ∂D. The points of maximum inaccessibility ID are those where the inaccessibility achieves its maximum. We prove that for strictly convex domains, ID is either a point or a segment, and that for a planar polygon ID is in general a point. We study the case of a triangle,...

We describe an algorithm which finds the externally visible vertices of a polygon, and which generates a new geometric construction we call the convex ropes of each visible vertex. The convex ropes give the range of angles from which each vertex is visible, and give all the pairs of vertices which are reachable by a straight robot finger. All of the convex ropes can be found in expected time or...

This paper is devoted to similarity and symmetry measures for convex shapes whose deenition is based on Minkowski addition and the Brunn-Minkowski inequality. This means in particular that these measures are region-based, in contrast to most of the literature, where one considers contour-based measures. All measures considered in this paper are invariant under translations; furthermore, they ca...

We give the first polynomial-time algorithm for the problem of finding a minimumperimeter k-gon that encloses a given n-gon. Our algorithm is based on a simple structural result, that an optimal k-gon has at least one “flush” edge with the ngon. This allows one to reduce the problem to computing the shortest k-link path in a simple polygon. As a by-product we observe that the minimum-perimeter ...

Given a simple polygon in the plane, a ip is de ned as follows: consider the convex hull of the polygon. If there are no pockets do not perform a ip. If there are pockets then re ect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erd}os conjectured that every simple polygon will become convex after a nite number of ips. The result was rst prove...

The first purpose of this paper is to propose a formulation of general area-preserving motion of polygonal curves by using a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to admissible class used in the so-called crystalline curvature flow. Actually, if the initial curve is a convex polygon, then our polygonal flow is nothing but the crystall...

Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polyt...