In astronomy, the objective determination of large empty spaces or voids in the spatial distribution of galaxies is part of the characterization of the large scale structure of the universe. This paper proposes a new method to find voids that starting from local longest-edges in a Delaunay triangulation builds the largest possible empty or almost empty polygons around them. A polygon is conside...

In this paper we give a parallel algorithm for finding the nearest-neighbor vertex of each vertex of a convex polygon. Our algoritb..z::J. runs in O(log n) time using O(njlogn) processors, in the parallel computation model CREW PRA.lvr (ConcurrentRead, Exclusive-Write Parallel RAM). This implies that the all nearest-neighbors problem for a convex polygon can be solved in O(n/p+logn) time using ...

Let P~,..., Pk be pairwise non-intersecting simple polygons with a total of n vertices and s start vertices. A start vertex, in general, is a vertex both of which neighbors have larger x coordinate. We present an algorithm for triangulating P~,..., Pk in time O(n + s log s). s may be viewed as a measure of non-convexity. In particular, s is always bounded by the number of concave angles + 1, an...

In this paper, we consider the problem of computing the minimum area triangle that circumscribes a given n-sided convex polygon touching edge-to-edge. In other words, we compute the minimum area triangle that is the intersection of 3 half-planes out of n half-planes defined by a given convex polygon. Previously, O(n logn) time algorithms were known which are based on the technique for computing...

This paper presents a heuristic algorithm based on Continuous Grasp to finding the largest ellipse, with prescribed eccentricity, inscribed in a non-convex polygon and a case study of the efficiency of this approach. We describe Continuous Grasp, present a mathematical formulation of the global optimization problem and report the results of an experimental evaluation.

We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon, etc. In some cases enumeration gives new expression for sequences enumerating these structures.

The problem of finding the convex hull of a planar set of points P, that is, finding the smallest convex region enclosing P, arises frequently in computer graphics. For example, to fit P into a square or a circle, it is necessary and sufficient that H(P), the convex hull of P, fits; and since it is usually the case that H(P) has many fewer points than P has, it is a simpler object to manipulate...

Given a 2-coloring of the vertices of a regular n-gon P , how many parallel lines are needed to separate the vertices into monochromatic subsets? We prove that bn/2c is a tight upper bound, and also provide anO(n log n) time algorithm to determine the direction that gives the minimum number of lines. If the polygon is a non-regular convex polygon, then n − 3 lines may be necessary, while n − 2 ...

Given two points p, q and a sequence of n lines (n>1) in the plane, we want to find the shortest path of touring all the given lines that starts at p and ends at q. In this paper, we solve the problem by reducing it to the problem of finding the shortest path that tours all the segments in a convex polygon from p to q. We first introduce how to construct the convex polygon. Then, we propose the...

In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasi-polynomials of nonintegral convex polygons. Define a pseudo-integral polygon, or PIP, to be a convex rational polygon whose Ehrhart quasipolynomial is a polynomial. The numbers of lattice points on the interior and on the boundary of a PIP determin...