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 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...

Abstract This paper settles the following two longstanding open problems: 1. What is the worst-case approximation ratio between the greedy and the minimum weight triangulation? 2. Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum? The answer to the first question is that the known \Omega (pn) lower bound is tight. The second question is answered in the affirmative by using a slight modification of an O(n log n) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constant-factor approximation of the minimum weight convex partition can be obtained within the same time bounds. 1 Introduction Let S be any set of n points in the plane. A triangulation of S is a maximal straight-line graph whose vertices are the points in S. Any triangulation of S partitions the convex hull of S into empty triangles. A triangulation that has received special attention is the minimum weight triangulation, in which the optimization criteria is to minimize the total edge length. This triangulation has some good properties [2] and is e.g. useful for numerical approximation of bivariate data [20].

The edge-insertion paradigm improves a triangulation of a finite point set in R² iteratively by adding a new edge, deleting intersecting old edges, and retriangulating the resulting two polygonal regions. After presenting an abstract view of the paradigm, this paper shows that it can be used to obtain polynomial time algorithms for several types of optimal triangulations.

Abstract. We show that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time O(n 2). The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modi cations the algorithm works for arbitrary normed metrics. Key words. Computational geometry,point sets, triangulations, two dimensions, minmax edge length, normed metrics AMS(MOS) subject classi cations. 68U05, 68Q25, 65D05 Appear in: SIAM Journal on Computing, 22 (3), 527{551, (1993)

this paper we present a simple, practical algorithm that computes the greedy triangulation in expected time O(n log n) and space O(n) for points uniformly distributed over any convex shape. A variant of this algorithm should also be fast for many other distributions. We first describe a surprisingly simple method for testing the compatibility of a candidate edge with edges in a partially constructed greedy triangulation. The new edge is tentatively added to the embedding of the partial GT and at most four constant time tests are done involving edges lying clockwise and counterclockwise from the candidate edge at each vertex. Even though there can be O(n) edges adjacent to one of the endpoints, we are able to show that if we can determine where in angular order the new edge falls among a subset of at most 10 of those edges then we can perform the compatibility test and if necessary update the triangulation. Our method therefore provides a \Theta(1) time edge test that requires only \Theta(1) time to update the structure, \Theta(n) time for initialization, and \Theta(n) space. This compares favorably with the previous method of Gilbert [10], which requires \Theta(log n) time for an edge test, \Theta(n log n) time for an update, \Theta(n

We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. In O(n log n) time we can compute a triangulation with O(n) new points, and no obtuse triangles, that approximates the MWST. We can also approximate the MWST with triangulations having no sharp angles. We generalize some of our results to higher dimensional triangulation problems. No previous polynomial time triangulation algorithm was known to approximate the MWST within a factor better than O(log n). 1 Introduction Optimal triangulation has furnished a number of problems of longstanding interest in computational geometry. These problems have applications to cartography, spatial data analysis, and finite element methods. Optimization criteria for which e#cient algorithms are known include maximizing the minimum angle [20, 24], minimizing the maximum angle [6], minimizing the minimum angle [7], mi...

this paper we present a simple, practical algorithm that computes the greedy triangulation in expected time O(n) and space O(n), for n points drawn independently from a uniform distribution over some fixed convex shape C.

Given a finite set of points in a plane, a triangulation is a maximal set of nonintersecting line segments connecting the points. The weight of a triangulation is the sum of the Euclidean lengths of its line segments. Given a set of points in a plane, the minimum weight triangulation problem is to find a triangulation whose weight is minimum. No polynomial time algorithm is known to solve this problem, and it is also unknown whether the problem is NP-hard. The current best polynomial time approximation algorithm produces a triangulation that can be O(log n) times the weight of the optimal triangulation. We propose an algorithm that triangulates a set P , of n points in a plane in O(n 3 ) time and that never does worse than the greedy triangulation. The algorithm produces an optimal triangulation if the points in P are the vertices of a convex polygon. The algorithm has the flavor of a heuristic proposed by Lingas and an analysis similar to his can be performed for our algorithm also,...