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 study several versions of the problem of generating triangular meshes for finite element methods. We show how to triangulate a planar point set or polygonally bounded domain with triangles of bounded aspect ratio; how to triangulate a planar point set with triangles having no obtuse angles; how to triangulate a point set in arbitrary dimension with simplices of bounded aspect ratio; and how to produce a linear-size Delaunay triangulation of a multi-dimensional point set by adding a linear number of extra points. All our triangulations have size (number of triangles) within a constant factor of optimal, and run in optimal time O(n log n+k) with input of size n and output of size k. No previous work on mesh generation simultaneously guarantees well-shaped elements and small total size. 1. Introduction Geometric partitioning problems ask for the decomposition of a geometric input into simpler objects. These problems are fundamental in many areas, such as solid modeling, computeraided ...

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.

We present a simple new algorithm for triangulating polygons and planar straightline graphs. It provides "shape" and "size" guarantees: All triangles have a bounded aspect ratio. The number of triangles is within a constant factor of optimal. Such "quality" triangulations are desirable as meshes for the nite element method, in which the running time generally increases with the number of triangles, and where the convergence and stability may be hurt by very skinny triangles. The technique we use - successive refinement of a Delaunay triangulation - extends a mesh generation technique of Chew by allowing triangles of varying sizes. Compared with previous quadtree-based algorithms for quality mesh generation, the Delaunay refinement approach is much simpler and generally produces meshes with fewer triangles. We also discuss an implementation of the algorithm and evaluate its performance on a variety of inputs.

Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles.

Let X be a complex of vertices and piecewise linear constraining facets embedded in E d . Say that a simplex is strongly Delaunay if its vertices are in X and there exists a sphere that passes through its vertices but passes through and encloses no other vertex. Then X has a d-dimensional constrained Delaunay triangulation if each k-dimensional constraining facet in X with k d \Gamma 2 is a union of strongly Delaunay k-simplices. This theorem is especially useful in E 3 for forming tetrahedralizations that respect specified planar facets. If the bounding segments of these facets are subdivided so that the subsegments are strongly Delaunay, then a constrained tetrahedralization exists. Hence, fewer vertices are needed than in the most common practice in the literature, wherein additional vertices are inserted in the relative interiors of facets to form a conforming (but unconstrained) Delaunay tetrahedralization. 1 Introduction Many applications can benefit from triangulations...

A plane geometric graph C in ! 2 conforms to another such graph G if each edge of G is the union of some edges of C. It is proved that for every G with n vertices and m edges, there is a completion of a Delaunay triangulation of O(m 2 n) points that conforms to G. The algorithm that constructs the points is also described. Keywords. Discrete and computational geometry, plane geometric graphs, Delaunay triangulations, point placement. Appear in: Discrete & Computational Geometry, 10 (2), 197--213 (1993) 1 Research of the first author is supported by the National Science Foundation under grant CCR-8921421 and under the Alan T. Waterman award, grant CCR-9118874. Any opinions, finding and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation. Work of the second author was conducted while he was on study leave at the University of Illinois. 2 Department of Computer Scienc...

We describe methods for triangulating polygonal regions of the plane so that no triangle has a large angle. Our main result is that a polygon with n sides can be triangulated with O(n 2 ) nonobtuse triangles. We also show that any triangulation (without Steiner points) of a simple polygon has a refinement with O(n 4 ) nonobtuse triangles. Finally we show that a triangulation whose dual is a path has a refinement with only O(n 2 ) nonobtuse triangles. Keywords: Computational geometry, mesh generation, triangulation, angle condition. 1. Introduction One of the classical motivations for problems in computational geometry has been automatic mesh generation for finite element methods. In particular, mesh generation has motivated a number of triangulation algorithms, such as finding a triangulation that minimizes the maximum angle. 1 A triangulation algorithm takes a geometric input, typically a point set or polygonal region, and produces an output that is a triangulation of ...

Nonmanifold geometric domains having small angles present special problems for triangular and tetrahedral mesh generators. Although small angles inherent in the input geometry cannot be removed, one would like to find a way to triangulate a domain without creating any new small angles. Unfortunately, this problem is not always soluble. I discuss how mesh generation algorithms based on Delaunay refinement can be modified to ensure that they always produce a mesh, and to ensure that poor quality triangles or tetrahedra appear only near small input angles. 1 Introduction The Delaunay refinement algorithms for triangular mesh generation introduced by Jim Ruppert [4] and Paul Chew [1] are almost entirely satisfying in theory and in practice. However, one unresolved problem has limited their applicability: they do not always mesh domains with small angles well---or at all---especially if these domains are nonmanifold. This problem is not just true of Delaunay refinement algorithms; it ste...

The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas of finite element methods and graph drawing are sketched.