Showing papers on "Constrained Delaunay triangulation published in 1992"

Randomized incremental construction of Delaunay and Voronoi diagrams

Abstract: In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “on-line” than earlier similar methods, takes expected timeO(nℝgn) and spaceO(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.

Quality mesh generation in three dimensions

Abstract: We show how to triangulate a three dimensional polyhedral region with holes. Our triangulation is optimal in the following two senses. First, our triangulation achieves the best possible aspect ratio up to a constant. Second, for any other triangulation of the same region into m triangles with bounded aspect ratio, our triangulation has size n = O(m). Such a triangulation is desired as an initial mesh for a finite element mesh refinement algorithm. Previous three dimensional triangulation schemes either worked only on a restricted class of input, or did not guarantee well-shaped tetrahedra, or were not able to bound the output size. We build on some of the ideas presented in previous work by Bern, Eppstein, and Gilbert, who have shown how to triangulate a two dimensional polyhedral region with holes, with similar quality and optimality bounds.

Triangulation of trimmed surfaces in parametric space

Abstract: The paper presents a new approach for triangulating trimmed surfaces. The basic idea is to perform the triangulation completely in parametric space. The trimmed regions of the surfaces are first mapped into parametric space and approximated by 2D polygonal regions, which are then pretriangulated by a restricted Delaunay triangulation algorithm. The generated triangles are subdivided further until each edge of the triangles is smaller than the allowed length that results from the surface definition and the specified tolerance. All the triangles are finally mapped back into Euclidean space so that the coordinate triples for the triangle vertices can be calculated. This approach makes the triangulation more reliable and faster, and it is also easy to avoid cracks between patches and surfaces. Thus, the algorithm is particularly suitable for the generation of a valid triangulation model for many engineering applications, for example stereolothography.

Delaunay triangulation in computational fluid dynamics

Abstract: A method, which utilises the Delaunay criterion, is described by which computational grids consisting of assemblies of triangles or tetrahedra can be constructed. An algorithm is briefly outlined to construct the triangulation and its dual, the Voronoi diagram. Issues related to how to construct boundary conforming grids from such a triangulation are addressed, and details are presented of how grid points within the domain can be generated automatically. The point generation algorithm utilises either the given boundary point distribution, or, for grid adaption, a background mesh. Computational aspects of constructing the triangulation in both 2 and 3 dimensions are covered. Examples of meshes and flow computations for a range of aerospace geometries are presented.

Delaunay's mesh of a convex polyhedron in dimension d. application to arbitrary polyhedra

Abstract: This paper presents a method for creating a Delaunay triangulation connected to a set of specified points. The theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach, valid for dimensions 2 and 3, which is simple, robust and well adapted to computation. Convex polyhedral and arbitrary polyhedral situations are introduced.

Adaptive approximation by piecewise linear polynomials on triangulations of subsets of scattered data

Abstract: Given a set V of data points in $R^2 $ with corresponding data values, the problem of adaptive piecewise polynomial approximation is to choose a subset of points of V, to create a triangulation of this subset, and to define a piecewise linear surface over the triangulation such that the deviation of this surface from the data set is no more than a prescribed error tolerance. A typical numerical scheme starts with some initial triangulation and adds more points (and triangles) as necessary until the resulting piecewise linear surface satisfies the error bound. In this paper two ingredients of such schemes are discussed. The first problem is that of constructing a suitable triangulation of a subset of points. The use of data-dependent triangulations that depend on the given function values at the data points is discussed, and some data-dependent criteria for optimizing a triangulation are presented and compared to the Delaunay criterion leading to the well-known Delaunay triangulation traditionally used for...

Algorithm for Delaunay triangulation and convex-hull computation using a sparse matrix

Abstract: A direct algorithm for computing the Delaunay triangulation of 2D data points is presented. The algorithm is based on a sparse-matrix data structure and on a circular-triangulation strategy, called shelling, that guarantees that the triangulation is complete and correct, and allows the dynamic update of the internal sparse-matrix data structure during triangulation. An edge list is used to govern the shelling procedure, and to output edges of the convex hull of the data set. The algorithm is numerically stable, i.e. degeneracies such as collinear or coincident points are automatically handled.

An upper bound for conforming Delaunay triangulations

Abstract: A plane geometric graph C in R2conforms 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(m2n) points that conforms to G. The algorithm that constructs the points is also described.

Numerical stability of algorithms for 2D Delaunay triangulations

Abstract: We show that two Delaunay triangulation algorithms, a diagonal-flipping algorithm and an incremental algorithm, can be implemented in approxiamte arithmetic. The two algorithms have worst-case running time O(n2) on a set of n sites. The correctness assertion is that the algorithms produce a triangulation of the set of sites so that each triangle has an “almost empty” circumcircle, i.e., a circumscribing pseudocircle slightly contracted from the circumcircle contains no sites in its interior.

New Results for the Minimum Weight Triangulation Problem

Abstract: 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(n3) time and that never does worse than the greedy triangulation. We investigate issues of local optimality pertaining to known triangulation algorithms and suggest an interesting new approach to studying triangulation algorithms. We restate the minimum weight triangulation problem as a graph problem and show the NP-hardness of a closely related graph problem. Finally, we show that the constrained problem of computing the minimum weight triangulation, given a set of points in a plane and enough edges to form a triangulation, is NP-hard. These results are an advance towards a proof that the minimum weight triangulation problem is NP-hard.

Mesh generation/refinement using fractal concepts and iterated function systems

Abstract: A novel method of mesh generation is proposed which is based on the use of fractal concepts to derive contractive, affine transformations. The transformations are constructed in such a manner that the attractors of the resulting maps are a union of the points, lines and surfaces in the domain. In particular, the mesh nodes may be generated recursively as a sequence of points which are obtained by applying the transformations to a coarse background mesh constructed from the given boundary data. A Delaunay triangulation or similar edge connection approach can then be performed on the resulting set of nodes in order to generate the mesh. Local refinement of an existing mesh can also be performed using the procedure. The method is easily extended to three dimensions, in which case the Delaunay triangulation is replaced by an analogous 3D tesselation.

An intersection algorithm based on Delaunay triangulation

Abstract: A robust method for finding points of intersection of line segments in a 2-D plane is presented. The plane is subdivided by Delaunay triangulation to localize areas where points of intersection exist and to guarantee the topological consistency of the resulting arrangement. The subdivision is refined by inserting midpoints recursively until the areas containing points of intersection are sufficiently localized. The method is robust in the sense that it does not miss points of intersection that are easily detectable when costly line-pair checking is performed. The algorithm is adaptive in the sense that most of the computational cost is incurred for the areas where finding points of intersection is difficult. >

Results on triangulation and high quality mesh generation

Abstract: We present several results on the triangulation problem in 2D and 3D. Given a polygonal or polyhedral object, a triangulation is a decomposition of the object into a collection of non-overlapping triangles or tetrahedra. Since the resulting pieces form a mesh of the object, the process of triangulation is also called mesh generation. Two main types of triangulations are distinguished by whether they contain Steiner points: vertices of the triangulation that are not vertices of the input. Non-Steiner triangulation has been well studied in 2D and is possible for every polygon. However, not every polyhedron in 3D admits a non-Steiner triangulation. Here we show that determining whether a given polyhedron can be triangulated without the use of Steiner points is an NP-complete problem, and hence likely to be computationally intractable. Another consideration is the shape of the pieces. For applications such as the finite element method, excessively skinny triangles or tetrahedra may hurt the convergence or stability of numerical computations. The quality mesh generation problem requires the triangulation to satisfy some shape bound. (In general, this will necessitate the use of Steiner points.) For instance, one may want all pieces' aspect ratios to be less than some global maximum, where the aspect ratio of an object is its length divided by its width. We present a 2D algorithm for triangulating polygons such that all triangles have a bounded aspect ratio. The algorithm is based on successive refinement of a Delaunay triangulation, and produces a mesh that is size-optimal, meaning that the number of triangles is within a constant factor of the minimum possible for the given input and aspect ratio bound. We also describe a 3D version of the Delaunay refinement algorithm that has neither size nor shape guarantees, but performs well in practice. Our work makes several contributions to the study of triangulation and mesh generation. In 2D, our Delaunay refinement algorithm helps to narrow the gap between triangulation theory and practice. In 3D, we add to the theoretical understanding of triangulation and towards the ultimate goal of a practical, reliable, high-quality mesh generation algorithm.

Triangulation of random and scattered data

Abstract: A rapid and efficient method for triangulating random points is based on a "circular" triangulation strategy that allows the deletion of data points from the data set during triangulation. In one embodiment, data is initially preprocessed by sorting and is put into a sparse matrix, while in another embodiment, data is preprocessed directly into a uniform grid prior to the triangulation strategy. A circular queue is used to govern the triangulation process and allows dynamic update of the internal matrix or grid data structure. A substantial decrease in complexity is provided by the triangulation strategy as the number of points to be searched for triangle points decreases as the triangles are created. The method is stable and fast and is not sensitive to difficult cases such as collinear or nearly collinear points.

A mesh generator for tetrahedral elements using Delaunay triangulation

Abstract: A tetrahedral mesh generator is developed. The generator is based on the Delaunay triangulation which is implemented by using the insertion polyhedron algorithm. Some new methods for dealing with the problems associated with the three-dimensional Delaunay triangulation and the insertion polyhedron algorithm are presented: degeneracy, the crossing situation, identification of the internal elements, and internal point generation. The generator works for both convex and nonconvex domains, including those with high aspect-ratio subdomains. Some examples (electromagnetic design and semiconductor device design) illustrate the capability of the generator. >

Applying Two-dimensional Delaunay Triangulation to Stereo Data Interpolation

Abstract: Interpolation of 3D segments obtained through a trinocular stereo process is achieved by using a 2D Delaunay triangulation on the image plane of one of the vision system cameras. The resulting two-dimensional triangulation is backprojected into the 3D space, generating a surface description in terms of triangular faces. The use of a constrained Delaunay triangulation in the image plane guarantees the presence of the 3D segments as edges of the surface representation.

Three-dimensional unstructured grid generation via incremental insertion and local optimization

Abstract: Algorithms for the generation of 3D unstructured surface and volume grids are discussed. These algorithms are based on incremental insertion and local optimization. The present algorithms are very general and permit local grid optimization based on various measures of grid quality. This is very important; unlike the 2D Delaunay triangulation, the 3D Delaunay triangulation appears not to have a lexicographic characterization of angularity. (The Delaunay triangulation is known to minimize that maximum containment sphere, but unfortunately this is not true lexicographically). Consequently, Delaunay triangulations in three-space can result in poorly shaped tetrahedral elements. Using the present algorithms, 3D meshes can be constructed which optimize a certain angle measure, albeit locally. We also discuss the combinatorial aspects of the algorithm as well as implementational details.

Robust and efficient implementation of the Delaunay tree

Abstract: In this paper, we present some practical results concerning the implementation of the algorithm described in (DMT) which computes dynamically the Delaunay triangulation of a set of sites in the plane in logarithmic expected update time More precisely, we show that the hypotheses of non degenerate positions can be dropped and the problem of precision can be treated correctly

An integrated approach for automated generation of two/three dimensional finite element grids using spatial occupancy enumeration and delaunay triangulation

Abstract: The paper describes the development of an automated triangulation algorithm for arbitrary domains. The method involves octree decomposition and subdomain meshing using Delaunay triangulation. Extension to three dimensional objects involving curved boundary surfaces is briefly described.

Automatic generation of 3D meshes for complicated solids

Abstract: To approach full automatic mesh generation, three problems have to be solved: (1) full automatic generation of geometry specifying the mesh from the solid model; (2) a constrained triangulation algorithm for adding a new point to an existing mesh; and (3) a description and realization of the user's mesh distribution desires which may be implied by the problem itself, as in the case of self-adaptive finite element analysis. The authors merge a novel mesh control idea (see J.M. Zhou, 1989) with the constrained Delaunay triangulation algorithm and an initial triangulation algorithm to achieve efficient, flexible, and reliable triangulation of complicated solids. A new strategy for adaptive mesh refinement is also included. >

Delaunay criteria for triangulating surfaces

Abstract: Computer aided design and manufacturing processes frequently map planar triangulations onto surfaces. Due to their properties relevant to finite element analysis, Delaunay triangulations have become popular for triangulating arbitrary planar domains that are subsequently mapped onto surfaces. Although surface triangulations cannot enjoy all the properties of planar triangulations, utilization of as many of the properties of Delaunay triangulations as possible in algorithms for triangulating surfaces can have advantages as well as disadvantages. This paper focuses on the advantages and disadvantages of using Delaunay criteria in triangulating surfaces.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Delaunay triangulation and computational fluid dynamics meshes

Abstract: In aerospace computational fluid dynamics (CFD) calculations, the Delaunay triangulation of suitable quadrilateral meshes can lead to unsuitable triangulated meshes. Here, we present case studies which illustrate the limitations of using structured grid generation methods which produce points in a curvilinear coordinate system for subsequent triangulations for CFD applications. We discuss conditions under which meshes of quadrilateral elements may not produce a Delaunay triangulation suitable for CFD calculations, particularly with regard to high aspect ratio, skewed quadrilateral elements.

Three-dimensional surface reconstruction using Delaunay triangulation in the image plane

Abstract: In this paper a technique for 3D surface reconstruction from stereo segments by using a constrained 2D Delaunay triangulation in the image plane is described. The proposed approach provides an appropriate tessellation of stereo data, being consistent with the visibility constraints of the imaging system. It results in a scene surface representation suitable for various applications of computer vision, as 3D object recognition and obstacle detection. Results and comparisons between our approach and Delaunay tetrahedralization are illustrated on real data together with some considerations about computational complexity and running time efficiency.

Application of the delaunay triangulation to geometric intersection problems

Abstract: The paper presents a new robust method for finding intersections of line segments in the plane. This method first constructs the Delaunay triangulation spanning the end points of line segments, and next recursively inserts the midpoints in the line segments that are not realized by Delaunay edges, until the descendants of the line segments become realized by Delaunay edges or the areas containing points of intersection are sufficiently localized. The method is robust in the sense that in any imprecise arithmetic it gives a topologically consistent arrangement as the output, and is stable in the sense that it does not miss intersections that can be easily detected by naive pairwise check with the precision at hand.

On the Relationship among Constrained Geometric Structures

Abstract: In this paper, we show the inclusion relation among several constrained geometric structures. In particular, we examine the constrained relative neighborhood graph in relation with other constrained geometric structures such as the constrained minimum spanning tree, constrained Gabriel graph, straight-line dual of bounded Voronoi diagram and the constrained Delaunay triangulation. We modify a linear time algorithm for computing the relative neighborhood graph from the Delaunay triangulation and show that the constrained relative neighborhood graph can also be computed in linear-time from the constrained Delaunay triangulation in the (ℜ2, L p ) metric space.