This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.

We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two- and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.

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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...

The theme of this paper is a rather simple method that has proved very potent in the analysis of the expected performance of various randomized algorithms and data structures in computational geometry. The method can be described as "analyze a randomized algorithm as if it were running backwards in time, from output to input." We apply this type of analysis to a variety of algorithms, old and new, and obtain solutions with optimal or near optimal expected performance for a plethora of problems in computational geometry, such as computing Delaunay triangulations of convex polygons, computing convex hulls of point sets in the plane or in higher dimensions, sorting, intersecting line segments, linear programming with a fixed number of variables, and others. 1 Introduction The curious phenomenon that randomness can be used profitably in the solution of computational tasks has attracted a lot of attention from researchers in recent years. The approach has proved useful in such diverse area...

Abstract. This paper describes a linear-time algorithm for computing the intersection of two convex polyhedra in 3-space. Applications of this result to computing intersections, convex hulls, and Voronoi diagrams are also given. Key words, computational geometry, convex polyhedra AMS(MOS) subject classifications. 68Q25, 68H05 1. Introduction. Giventwo

Given two intersecting polyhedra P , Q and a direction d, find the smallest translation of Q along d that renders the interiors of P and Q disjoint. The same problem can also be posed without specifying the direction, in which case the minimum translation over all directions is sought. These are fundamental problems that arise in robotics and computer vision. We develop techniques for implicitly building and searching convolutions and apply them to derive efficient algorithms for these problems. 1 Introduction The computation of spatial relationships among geometric objects is a fundamental problem in such areas as robotics, computer-aided design, VLSI layout, and computer graphics. In a dynamic environment where objects are mobile, intersection or proximity among objects has obvious applications. Consider, for instance, the problem of collision detection in robot motion planning. The Euclidean distance is a commonly used measure in these areas. Numerous efficient algorithms are known...

We give an algorithm for triangulating n-vertex polygonal regions (with holes) so that no angle in the nal triangulation measures more than pi/2. The number of triangles in the triangulation is only O(n), improving a previous bound of O(n²), and the worst-case running time is O(n log² n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm.

Consider a set of n points in d-dimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has an upper bound of O(n d s (n)), where s (n) is the maximum length of a (n; s)-Davenport-Schinzel sequence [AgShSh 89, DaSc 65] and s is a constant depending on the motions of the point sites. Our results are a linear-factor improvement over the naive O(n d+2 ) upper bound on the number of topological events. In addition, we show that if only k points are moving (while leaving the other n \Gamma k points fixed), there is an upper bound of O(kn d\Gamma1 s (n) + (n \Gamma k)...

The Delaunay Tree is a hierarchical data structure that has been introduced in [6] and analyzed in [7,4]. For a given set of sites S in the plane and an order of insertion for these sites, the Delaunay Tree stores all the successive Delaunay triangulations. As proved before, the Delaunay Tree allows the insertion of a site in logarithmic expected time and linear expected space, when the insertion sequence is randomized.