A convex hole (or empty convex polygon) of a point set P in the plane is a convex polygon with vertices in P , containing no points of P in its interior. Let R be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of n random points chosen independently and uniformly over R is Θ(logn/(log log n)), regardless of the shape of R.

Let us say that an $n$-sided polygon is semi-regular if it circumscriptible, and its angles are all equal but possibly one which then larger than the rest. Regular polygons, in particular, semi-regular. The main result of paper that, class convex polygons uniquely determined by just three geometric quantities: area, perimeter, a third quantity depending only on interior appears heat trace asymp...

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...

This paper presented study on convex drawing of planar graph. In graph theory, a planar graph is a graph that can be embedded in the plane. A planar graph is one that can be drawn on a plane in such a way that there are no “edge crossings,” i.e. edges intersect only at their common vertices. Convex polygon has all interior angles less than or equal to 180°. A graph is called a convex drawing if...

We present a new polygon decomposition problem, the anchored area partition problem, which has applications to a multiple-robot terrain-covering problem. This problem concerns dividing a given polygon P into n polygonal pieces, each of a speciied area and each containing a certain point (site) on its boundary. We rst present the algorithm for the case when P is convex and contains no holes. The...

Geometric intersection graphs are graphs determined by the intersections of certain geometric objects. We study the complexity of visualizing an arrangement of objects that induces a given intersection graph. We give a general framework for describing classes of geometric intersection graphs, using arbitrary finite base sets of rationally given convex polygons and rationally-constrained affine ...

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...

Poincaré’s classical theorem of fundamental polygons is a widely known, valuable tool that gives sufficient conditions for a (convex) hyperbolic polygon, equipped with so-called side-pairing transformations, to be a fundamental domain for a discrete subgroup of isometries. Poincaré first published the theorem in dimension two in 1882. In the past century, there have been several published proof...