In the Art Gallery problem a polygon is given and the goal is to place as few guards as possible so that the entire area of the polygon is covered. We address a closely related problem: how to place a fixed number of guards on the vertices or the edges of a simple polygon so that the total guarded area inside the polygon is maximized. Recall that an optimization problem is called APX-hard, if t...

The relative convex hull, or the minimum-perimeter polygon (MPP) of a simple polygon A, contained in a second polygon B, is a unique polygon in the set of nested polygons between A and B. The computation of the minimum-length polygon (MLP), as a special case for isothetic polygons A and B, is useful for various applications in image analysis and robotics. The paper discusses the first recursive...

In a convex drawing of a plane graph G, every facial cycle of G is drawn as a convex polygon. A polygon for the outer facial cycle is called an outer convex polygon. A necessary and sufficient condition for a plane graph G to have a convex drawing is known. However, it has not been known how many apices of an outer convex polygon are necessary for G to have a convex drawing. In this paper, we s...

In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudo-triangles and convex polygons. We call the resulting decomposition PTconvex. We are interested in minimum decompositions, i.e., in decomposing the input polygon into the least number of subpolygons. Allowing subpolyg...

A flipturn on a polygon consists of reversing the order of edges inside a pocket of the polygon, without changing lengths or slopes. Any polygon with n edges must be convexified after at most (n− 1)! flipturns. A recent paper showed that in fact it will be convex after at most n(n−3)/2 flipturns. We give here lower bounds. We construct a polygon such that if pockets are chosen in a bad way, at ...

A polygon is called k-guardable if it is possible to find a collection G of points in the interior of the edges of P such that every point in P is visible from at least k elements of G, and such that no edge of P contains more than one element of G. In this paper we prove that every simple polygon can be 1-guarded using at most n/2 guards, and that every polygon with 0 or 1 polygonal hole can...

The generation of random simple polygon and the pseudo-triangulation of a polygon are regarded as the proposed problems in computational geometry. The production of a random polygon is used in the context of the consideration of the accuracy of algorithms. In this paper, a new algorithm is presented to generate a simple spiral polygon on a set of random points S in the plane using convex hull l...

A classic problem in computer graphics is to decompose a simple polygon into a collection of triangles whose vertices are only those of the simple polygon. By definition, a simple polygon is an ordered sequence of n points, ~ V0 through ~ Vn−1. Consecutive vertices are connected by an edge 〈~ Vi, ~ Vi+1〉, 0 ≤ i ≤ n − 2, and an edge 〈~ Vn−1, ~ V0〉 connects the first and last points. Each vertex ...

In this paper we present a polynomial time algorithm for computing a Hausdorff core of a polygon with a single reflex vertex. A Hausdorff core of a polygon P is a convex polygon Q contained inside P which minimizes the Hausdorff distance between P andQ. Our algorithm essentially consists of rotating a line about the reflex vertex; this line defines a convex polygon by cutting P . To determine t...

Polygon approximation plays a vital role in abquitious applications like multimedia, geographic and object recognition. An extensive number of polygonal approximation techniques for digital planar curves have been proposed over the last decade, but there are no survey papers on recently proposed techniques. Polygon is a collection of edges and vertices. Objects are represented using edges and v...