Showing papers on "Delaunay triangulation published in 1991"

Optimality of the Delaunay triangulation in Rd

Abstract: In this paper we present new optimality results for the Delaunay triangulation of a set of points in ℝ d . These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ℝ d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.

Efficient Delaunay triangulation using rational arithmetic

Abstract: Many fundamental tests performed by geometric algorithms can be formulated in terms of finding the sign of a determinant. When these tests are implemented using fixed precision arithmetic such as floating point, they can produce incorrect answers; when they are implemented using arbitrary-precision arithmetic, they are expensive to compute. We present adaptive-precision algorithms for finding the signs of determinants of matrices with integer and rational elements. These algorithms were developed and tested by integrating them into the Guibas-Stolfi Delaunay triangulation algorithm. Through a combination of algorithm design and careful engineering of the implementation, the resulting program can triangulate a set of random rational points in the unit circle only four to five times slower than can a floating-point implementation of the algorithm. The algorithms, engineering process, and software tools developed are described.

Mesh relaxation: A new technique for improving triangulations

Abstract: SUMMARY Given a list of points defining a domain boundary, a three-stage process is often used to triangulate a domain. First, an appropriate distribution of interior points is generated. Next the points are connected to form triangles. And, finally, the connectivity data are used to reposition the interior points using the Laplacian smoothing technique, thereby usually improving the shapes of the triangles. This paper describes a new technique for mesh improvement-adjusting the connection structure during the second stage of this process. The new scheme, which we call mesh relaxation, consists of a procedure for iteratively making the mesh topology more regular by edge swapping. For each interior edge, a relaxation index is computed that depends on the degrees of its end points and adjacent points. Any edge for which this index exceeds a prescribed threshold will be swapped, i.e. replaced by a new edge connecting the adjacent points of the original edge. After all edge swaps are completed, Laplacian smoothing is applied to the mesh. Examples show that, when the mesh point density varies smoothly and due care is taken in the vicinity of the boundary, mesh relaxation can dramatically increase the regularity of the mesh and produce improved triangle shapes.

Geometrical analysis of the structure of simple liquids : percolation approach

Abstract: The problem of searching for quantitative laws governing the structure of simple liquids is formulated as a site percolation problem on the Voronoi network. The sites of this four-coordinated network correspond to the figures formed by the four neighbouring atoms (Delaunay simplices). Three quantitative characteristics of the form of the Delaunay simplices are introduced to enable one to colour the sites of the Voronoi network corresponding to the simplices of a specific form and to study the percolation of colouring through the network sites. The clusters of contiguous Delaunay simplices of the specific form have been studied and the percolation thresholds for various colouring types have been obtained for instantaneous configurations of the Lennard-Jones liquid (obtained by the Monte Carlo procedure) as well as for the configurations with removed thermal excitations (F structure). Percolation of all the types of colouring introduced turns out to be correlated, i.e., the Delaunay simplices of a given for...

Delaunay versus max‐min solid angle triangulations for three‐dimensional mesh generation

Abstract: The Delaunay triangulation has been used in several methods for generating finite element tetrahedral meshes in three-dimensional polyhedral regions. Other types of three-dimensional triangulations are possible, such as a triangulation satisfying a local max-min solid angle criterion. In this paper, we present experimental results to show that max-min solid angle triangulations are better than Delaunay triangulations for finite element tetrahedral meshes, since the former type of triangulations contains tetrahedra of better shape than the latter type. We also describe how mesh points are generated and triangulated in our tetrahedral mesh generation method.

The expected extremes in a delaunay triangulation

Abstract: We give an expected-case analysis of Delaunay triangulations. To avoid edge effects we consider a unit-intensity Poisson process in Euclidean d-space, and then limit attention to the portion of the triangulation within a cube of side n1/d. For d equal to two, we calculate the expected maximum edge length, the expected minimum and maximum angles, and the average aspect ratio of a triangle. We also show that in any fixed dimension the expected maximum vertex degree is Θ(log n/loglog n). Altogether our results provide some measure of the suitability of the Delaunay triangulation for certain applications, such as interpolation and mesh generation.

Constructing discrete medial axis of 3-d objects

Abstract: In this paper, an algorithm to construct the approximate medial axis of an object is proposed. The algorithm is based on the Delaunay triangulation of points on the object boundaries. Because the medial axis constructed by this algorithm consists of a set of discrete points, we call it the discrete medial axis. Based on the classification of these discrete points, the structure of the medial axis surfaces of a three-dimensional object are discussed in detail. The correctness of the algorithm is substantiated by a brief theoretical analysis.

A new method for the characterization of topographic surfaces

Abstract: A method is described for the extraction of morphological information from a terrain approximated by a Delaunay triangulation, in order to find a combinatorial simpler surface description while maintaining its basic features. Characteristic regions (i.e., regions with concave, convex, planar or saddle shape) are considered the basic descriptive elements of the surface morphology, and are defined by taking into account the type of adjacency between triangles. Adjacencies between regions define the surface characteristic lines, which are classified as ridges, ravines or generic creases, and characteristic points, which are classified as maxima, minima or saddle points. A graph-like data structure is constructed on these shape features, called the Characteristic Region Configuration Graph, which represents die surface in an effective and concise way.

An efficient advancing front algorithm for Delaunay triangulation

Abstract: There has been some recent interest in fluid dynamics calculations on unstructured meshes. One method of unstructured mesh generation involves Delaunay triangulation. This method has certain advantages but it can be expensive to implement. Furthermore, there can be problems with crossing grid lines near boundaries. A method shown here avoids many of the robustness and efficiency problems previously associated with Delaunay triangulation. As an added feature, a simple algorithm is shown which allows removal of diagonal edges from cells that are nearly rectangular. This can result in significant savings in the cost per iteration of a flow solver using this grid.

A quadratic time algorithm for the minmax length triangulation

Abstract: It is shown 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/sup 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 modifications the algorithm works for arbitrary normed metrics. >

Geometric learning algorithms

Abstract: Emergent computation in the form of geometric learning is central to the development of motor and perceptual systems in biological organisms and promises to have a similar impact on emerging technologies including robotics, vision, speech, and graphics. This paper examines some of the trade-offs involved in different implementation strategies, focusing on the tasks of learning discrete classifications and smooth nonlinear mappings. The trade-offs between local and global representations are discussed, a spectrum of distributed network implementations are examined, and an important source of computational inefficiency is identified. Efficient algorithms based on k -d trees and the Delaunay triangulation are presented and the relevance to biological networks is discussed. Finally, extensions of both the tasks and the implementations are given.

On sorting triangles in a delaunay tessellation

Abstract: In a two-dimensional Delaunay-triangulated domain, there exists a partial ordering of the triangles (with respect to a vertex) that is consistent with the two-dimensional visibility of the triangles from that vertex. An equivalent statement is that a polygon that is star-shaped with respect to a given vertex can be extended, one triangle at a time, until it includes the entire domain. Arbitrary planar triangulations do not possess this useful property which allows incremental processing of the triangles.

A self-organizing neural network approach for automatic mesh generation

Abstract: An automatic mesh generator, SOFT (self-organizing finite-element tessellation), based on self-organizing neural networks has been demonstrated. With user-supplied mesh density function and boundary mesh, this mesh generator provides a graded mesh, with asymptotic characteristics quite similar to weighted Dirichlet tessellation and dual Delaunay triangulation. Local mesh restrictions such as fixed boundary and/or internal meshes are easily incorporated in this mesh generator. Although the algorithm is applicable to general n-dimensional meshes, two-dimensional rectangular and triangular meshes are presented for simplicity. >

Graded tetrahedral finite element meshes

Abstract: Vertices in the body centred cubic (bcc) lattice are used to create a tetrahedral spatial decomposition. With this spatial decomposition an octree approach is combined with Delaunay triangulations to decompose solids into tetrahedral finite element meshes. Solids must have their surfaces triangulated and the vertices in the triangulation are finite element nodes. Local densities of interior tetrahedra are controlled by the densities of surface triangles. Accuracy of the decomposition into finite elements depends on the accuracy of the surface triangulation which can be constructed with state of the art computer aided design systems.

Piecewise planar surface models from sampled data

Abstract: Interactive visualization of three dimensional data requires construction of a geometric model for rendering by a graphics processor. We present an automated method for transforming dense, uniformly sampled data grids to an irregular triangular mesh that represents a piecewise planar approximation to the sampled data. The mesh vertices comprise surface-specific points, which characterize important surface features. We obtain surface-specific points by a novel application of linear and non-linear filters, and thresholding. We define a procedure for constructing a triangulation, derived from a Delaunay triangulation, that conforms to the sampled data. In our example application, modeling a terrain surface over a large area, an 80% reduction in polygons maintains an acceptable fit. This method also extends to the tessellation of images. Applications include scientific visualization and construction of virtual environments.

Unstructured and adaptive mesh generation for high Reynolds number viscous flows

Abstract: A method for generating and adaptively refining a highly stretched unstructured mesh suitable for the computation of high-Reynolds-number viscous flows about arbitrary two-dimensional geometries was developed. The method is based on the Delaunay triangulation of a predetermined set of points and employs a local mapping in order to achieve the high stretching rates required in the boundary-layer and wake regions. The initial mesh-point distribution is determined in a geometry-adaptive manner which clusters points in regions of high curvature and sharp corners. Adaptive mesh refinement is achieved by adding new points in regions of large flow gradients, and locally retriangulating; thus, obviating the need for global mesh regeneration. Initial and adapted meshes about complex multi-element airfoil geometries are shown and compressible flow solutions are computed on these meshes.

Constructing 3-D discrete medial axis

Abstract: In solid modeling, each represent~ tion scheme is suitable for some operations and applications. Some modeling systems work on two or more different representation schemes in order to perform every operation in the most suitable scheme. The problem of conversion between different representation schemes arises. There exists another option for a solid modeling system, however, That is to use one representation, say boundary representation, and create an associated structure which can remedy the weakness of the representation. This is the motivation of the work presented in this paper: we propose the medial axis of the object to associate with its B~ep to improve the applicability of solid modeler and reduce the computation complexity of certain algorithms. *Department of Mechanical and Aerospace Engineering tDepartment of Computer Science Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing M=hinery. To copy otherwise, or to sepublish, requires a fee and/or specific permission. In this paper, an algorithm to construct the approximate medial axis of an object is proposed. The algorithm is based on the Delaunay triangulation of the pointa on the object boundaries. Because the medial axis constructed by this algorithm consists of a set of discrete points, we call it discrete medial axis. Bssed on the classification of these discrete points, the structure of the medial axis surfaces of a threedimensional object are discussed in detail. The correctness of the algorithm is substantiated by a brief theoretical analysis.

Domain Delaunay Tetrahedrization of arbitrarily shaped curved polyhedra defined in a solid modeling system

Abstract: An algorithm is presented for constructing a topologically and geometrically valid Domain Delaunay Z’etrahedr.zatiorz (DDT) of an arbitrarily shaped solid model with quadric curved faces (including objects with holes and nonmanifold objects). The algorithm operates on the boundary representation (B-rep) of the solid, and makes extensive use of properties of the Delaunay triangulation. This algorithm also includes a mechanism for transferring neighborhood information from the solid model to the elements of the tetrahedral model. Neighborhood information is used for identifying tetrahedral to be included in the DDT, and — in combination with geometric criteria — for ensuring that the DDT approximates satisfactorily the curved faces of the solid.

Computing correct Delaunay triangulations

Abstract: In recent years the practical computation of Delaunay triangulations, resp. Voronoi diagrams has received a lot of attention in the literature. While the Delaunay triangulation is an important basic tool in geometric optimization algorithms, it is nontrivial to achieve a numerically stable computer implementation. In this technical note we assume that all generating points are grid points of a regularM byM lattice in the plane. Depending onM we derive the necessary word length a binary computer must have for integer representation in order to obtain exact Delaunay triangulations. This analysis is carried out for theL 1-,L 2- andL ∞-metric.