We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π -guards that collectively monitor a simple polygon with n vertices? Our results are as follows: 1. Any simple polygon with n vertices can be monitored by at most n/2 general vertex π -guards. This bo...

Let Σ = { S1 , . . . , Sn } be a finite set of disjoint line segments in the plane. We conjecture that its visibility graph, Vis(Σ), is hamiltonian. In fact, we make the stronger conjecture that Vis(Σ) has a hamiltonian cycle whose embedded version is a simple polygon (i.e., its boundary edges are non-crossing visibility segments). We call such a simple polygon a spanning polygon of Σ. Existenc...

We present a diagram that captures containment information for scalable rotated and translated versions of a convex polygon. For a given polygon P and a contact point q in a point set S, the diagram parameterizes possible translations, rotations, and scales of the polygon in order to represent containment regions for each additional point v in S. We present geometric and combinatorial propertie...

Instead of using the polygon defined by adjacent vertices to a vertex (called the ball) or its kernel [1], we propose a modified polygon that is easy to compute, convex and an approximation of the kernel. We call this polygon the “quick kernel ball region.” This novel algorithm is presented in details. It is easy to implement and effective in constraining a vertex to remain within its feasible ...

Given a simple polygon P , we consider the problem of finding a convex polygon Q contained in P that minimizes H(P, Q), where H denotes the Hausdorff distance. We call such a polygon Q a Hausdorff core of P . We describe polynomial-time approximations for both the minimization and decision versions of the Hausdorff core problem, and we provide an argument supporting the hardness of the problem.

VQ = {(q, a) ∈ Z : gcd(q, a) = 1, max{|a|, |q|} ≤ Q}. Then the Jarńık polygon PQ is the unique (up to translation) convex polygon whose sides are precisely the vectors in VQ. In other words, PQ is the polygon whose vertices can be obtained by starting from an arbitrary point in R and adding the vectors in VQ one by one, traversing those vectors in a counterclockwise direction. For example, the ...

Let S be used to denote a nite set of planar geometric objects. Deene a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have som...

Low efficiency of interference calculation has become the bottleneck that restricts further development of the performance of evolutionary algorithm for the polygon layout. To solve the problem, in this paper, we propose an algorithm of calculating overlapping area between two irregular polygons. For this algorithm, at first, two irregular polygons are respectively decomposed into the minimum n...

We present a diagram that captures containment information for scalable translations of a convex polygon. For a given polygon P and contact point q in a set S, the diagram parameterizes possible translations and scales of the polygon in order to represent containment regions for each other point v in S. We present geometric and combinatorial properties for this diagram, and describe how it can ...

We continue the investigation of computational aspects of restricted-orientation convexity (O-convexity) in two dimensions. We introduce one notion of an O-halfplane, for a set O of orientations, and we investigate O-connected convexity. The O-connected convex hull of a nite set X can be computed in time O(jXj log jXj + jOj). The O-connected hull is a basis for determining the O-convex hull of ...