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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This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log n). This leads to a simple algorithm of the same complexity for triangulating polygons. More generally, if S is presented as a plane graph with k connected components, then the expected running time of the algorithm is O(n log n k log n). As a by-product our algorithm creates a search structure of expected linear size that allows point location queries in the resulting trapezoidation in logarithmic expected time. The analysis of the expected performance is elementary and straightforward. All expectations are with respect to "coinflips" generated by the algorithm and are not based on assumptions about the geometric distribution of the input. Large Portions of the research reported here were conducted while the author visit...

We show how to construct an O( p n)-separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O( p n)-separator of that subgraph. We also show how to construct an O(n ffl )-way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n 1=2+ffl )-separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n= log n) processors on a CRCW PRAM. Keywords: Computational geometry, algorithmic graph theory, planar graphs, planar separators, polygon triangulation, parallel algorithms, PRAM model. 1 Introduction Let G = (V; E) be an n-node graph. An f(n)-separator is an f(n)-sized subset of V whose removal disconnects G into two subgraphs G 1 and G 2 each...

Given a two-dimensional, non-overlapping layout of convex and non-convex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial two-dimensional packing applications, compaction can improve the material utilization of already tightly packed layouts. Efficient algorithms for compacting a layout of non-convex polygons are not previously known. This dissertation offers the first systematic study of compaction of non-convex polygons. We start by formalizing the compaction problem as that of planning a motion that minimizes some linear objective function of the positions. Based on this formalization, we study the complexity of compaction and show it to be PSPACE-hard. The major contribution of this dissertation is a position-based optimization model that allows us to calculate directly new polygon positions that constitute a locally optimum solution of the objective via linear programming. This model yields the first ...

The power of purely functional programming in the construction of data structures has received much attention, not only because functional languages have many desirable properties, but because structures built purely functionally are automatically fully persistent: any and all versions of a structure can coexist indefinitely. Recent results illustrate the surprising power of pure functionality. One such result was the development of a representation of double-ended queues with catenation that supports all operations, including catenation, in worst-case constant time [19].

We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm and, hence, positively answers his question of whether a simpler randomized algorithm for the problem exists. The new algorithm can be viewed as a combination of Chazelle's algorithm and of non-optimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991), with the essential innovation that sampling is performed on subchains of the initial polygonal chain, rather than on its edges. It is also essential, as in Chazelle's algorithm, to include a bottom-up preprocessing phase previous to the top-down construction phase. 1 Introduction Polygon triangulation is a classic problem in comp...

A preliminary version of this paper has appeared as an extended abstract at CGI'98; see [26]. y

Geometric algorithms and data structures are often easiest to understand visually, in terms of the geometric objects they manipulate. Indeed, most papers in computational geometry rely on diagrams to communicate the intuition behind the results. Algorithm animation uses dynamic visual images to explain algorithms. Thus it is natural to present geometric algorithms, which are inherently dynamic, via algorithm animation. The accompanying videotape presents a video review of geometric animations; the review was premiered at the 1992 ACM Symposium on Computational Geometry. The video review includes single-algorithm animations and sample graphic displays from "workbench" systems for implementing multiple geometric algorithms. This report contains short descriptions of each video segment. vi Preface This booklet and the accompanying videotape contain animations of a variety of computational geometry algorithms. Computational geometry has existed as a field for almost two decades, and int...

The Graham scan is a fundamental backtracking technique in computational geometry which was originally designed to compute the convex hull of a set of points in the plane and has since found application in several different contexts. In this note we show how to use the Graham scan to triangulate a simple polygon. The resulting algorithm triangulates an n vertex polygon P in O(kn) time where k-1 is the number of concave vertices in P. Although the worst case running time of the algorithm is O(n 2 ), it is easy to implement and is therefore of practical interest. 1. Introduction A polygon P is a closed path of straight line segments. A polygon is represented by a sequence of vertices P = (p 0 ,p 1 ,...,p n-1 ) where p i has real-valued x,y-coordinates. We assume that no three vertices of P are collinear. The line segments (p i ,p i+1 ), 0 i n-1, (subscript arithmetic taken modulo n) are the edges of P. A polygon is simple if no two nonconsecutive edges intersect. A simple polygon part...