Showing papers on "Delaunay triangulation published in 1983"

On the shape of a set of points in the plane

Abstract: A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " \alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets. It is shown that a-shapes are subgraphs of the closest point or furthest point Delaunay triangulation. Relying on this result an optimal O(n \log n) algorithm that constructs \alpha -shapes is developed.

Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams

Abstract: We discuss the following problem: given n points in the plane (the “sites”), and an arbitrary query point q, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites, and then locating the query point in one of its regions. We give two algorithms, one that constructs the Voronoi diagram in O(n lg n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, the Delaunay triangulation, and are simple enough to be of practical value. The simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem, and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is embeddings of graphs in two-dimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror-image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams.

Magnetic field computation using Delaunay triangulation and complementary finite element methods

Abstract: A two-dimensional finite element analysis package is described which automatically generates optimal finite element meshes for magnetic field problems. The system combines the concept of Delaunay triangulation with variational principles to provide a grid which adapts to the characteristics of the solution. In this procedure, two different approximate solutions to the magnetic field are derived, the difference between the two approximate solutions providing an element by element measure of the accuracy of the solution. By refining those elements having the largest errors and recomputing the solution iteractively, finite element meshes having a uniforrn error density are obtained. The system is menu oriented and utilizes multiple command and display windows to create and edit the object description interactively. Matrix solution is by means of a rapid pre-conditioned conjugate gradient algorithm, and a wide variety of post-processing operations are supported. Application of the mesh generator to a variety of problems in magnetic field design shows it to be one of the most powerful and easy to use systems yet devised.

Fast Triangulation of Simple Polygons

Abstract: We present a new algorithm for triangulating simple polygons that has four advantages over previous solutions [GJPT, Ch].

Segmenting Dot Patterns by Voronoi Diagram Concavity

Abstract: This correspondence defines a signed distance, called ``internal concavity,'' on paths of the Voronoi diagram of a dot pattern. An algorithm using internal concavity to segment dot patterns is described. The segmentation algorithm produces subsets of the Dirichlet tessellation (Delaunay triangulation) of the dot pattern.

Design of a recursive, shape controlling mesh generator

Abstract: A recursive, shape controlling triangulation method is described. The method is designed to produce a labelling which implies reduced fill in the solution of (finite element) equations assembled from such a triangulation and allows simple implementation of a nested disection algorithm for irregular domains. This approach saves a substantial amount of time usually spent on discovering a suitable relabelling of the triangulation. In addition, the matrix of the resulting system is then endowed with a recursive doubly bordered block diagonal form. This allows us to develop a recursive parallel bisection method for the solution of the system of equations.

Electromagnetic Modeling Techniques Using Finite Element Methods

Abstract: In designing and in evaluating finite element packages for industrial electromagnetics applications. the: following questions are typical: How easy is the package to use? How accurate is the solution? Can I get smooth flux plots? What about the electric and magnetic fields?