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In this paper we show an equivalence relationship between additively weighted Voronoi cells in R d , power diagrams in R d and convex hulls of spheres in R d. An immediate consequence of this equivalence relationship is a tight bound on the complexity of : (1) a single additively weighted Voronoi cell in dimension d; (2) the convex hull of a set of… (More)

In this paper we present a dynamic algorithm for the construction of the additively weighted Voronoi diagram of a set of weighted points on the plane. The novelty in our approach is that we use the dual of the additively weighted Voronoi diagram to represent it. This permits us to perform both insertions and deletions of sites easily. Given a set B of n… (More)

Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm… (More)

<i>It is well known that the Delaunay Triangulation is a spanner graph of its vertices. In this paper we show that any bounded aspect ratio triangulation in two and three dimensions is a spanner graph of its vertices as well. We extend the notion of spanner graphs to environments with obstacles and show that both the Constrained Delaunay Triangulation and… (More)

This paper presents a dynamic algorithm for the construction of the Euclidean Voronoi diagram of a set of convex objects in the plane. We consider first the Voronoi diagram of smooth convex objects forming pseudo-circles set. A pseudo-circles set is a set of bounded objects such that the boundaries of any two objects intersect at most twice. Our algorithm… (More)

One of the earliest and most well known problems in computational geometry is the so-called art gallery problem. The goal is to compute the minimum possible number guards placed on the vertices of a simple polygon in such a way that they cover the interior of the polygon. In this paper we consider the problem of guarding an art gallery which is modeled as a… (More)

We present a global iterative algorithm for constructing spatial ¥ § ¦-continuous interpolat-ing¨-splines, which preserve the shape of the polygonal line that interpolates the given points. Furthermore, the algorithm can handle data exhibiting two kinds of degeneracy, namely copla-nar quadruples and collinear triplets of points. The convergence of the… (More)

In this paper we present an efficient algorithm for the computation of the segment Voronoi diagram in two dimensions. Our algorithm can handle not only disjoint segments or segments that share endpoints, but also segments that may intersect at their interior. It is incremental and the expected cost of inserting n (possibly intersecting) sites (points or… (More)

We present a new methodology for agent modeling that is scalable and efficient. It is based on the integration of nonlinear dynamical systems and kinetic data structures. The method consists of three-layers, which together model 3D agent steering, crowds and flocks among moving and static obstacles. The first layer, the local layer employs nonlinear… (More)

We study the predicates involved in an efficient dynamic algorithm for computing the Apollonius diagram in the plane, also known as the additively weighted Voronoi diagram. We present a complete algorithmic analysis of these predicates, some of which are reduced to simpler and more easily computed primitives. This gives rise to an exact and efficient… (More)