Showing papers on "Delaunay triangulation published in 1995"

A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation

Abstract: 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 finite 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.

Geophysical parametrization and interpolation of irregular data using natural neighbours

Abstract: SUMMARY An approach is presented for interpolating a property of the Earth (for example temperature or seismic velocity) specified at a series of ‘reference’ points with arbitrary distribution in two or three dimensions. The method makes use of some powerful algorithms from the field of computational geometry to efficiently partition the medium into ‘Delaunay’ triangles (in 2-D) or tetrahedra (in 3-D) constructed around the irregularly spaced reference points. The field can then be smoothly interpolated anywhere in the medium using a method known as natural-neighbour interpolation. This method has the following useful properties: (1) the original function values are recovered exactly at the reference points; (2) the interpolation is entirely local (every point is only influenced by its natural-neighbour nodes); and (3) the derivatives of the interpolated function are continuous everywhere except at the reference points. In addition, the ability to handle highly irregular distributions of nodes means that large variations in the scale-lengths of the interpolated function can be represented easily. These properties make the procedure ideally suited for ‘gridding’ of irregularly spaced geophysical data, or as the basis of parametrization in inverse problems such as seismic tomography. We have extended the theory to produce expressions for the derivatives of the interpolated function. These may be calculated efficiently by modifying an existing algorithm which calculates the interpolated function using only local information. Full details of the theory and numerical algorithms are given. The new theory for function and derivative interpolation has applications to a range of geophysical interpolation and parametrization problems. In addition, it shows much promise when used as the basis of a finite-element procedure for numerical solution of partial differential equations.

Automatic reconstruction of surfaces and scalar fields from 3D scans

Abstract: We present an efficient and uniform approach for the automatic reconstruction of surfaces of CAD (computer aided design) models and scalar fields defined on them, from an unorganized collection of scanned point data. A possible application is the rapid computer model reconstruction of an existing part or prototype from a three dimensional (3D) points scan of its surface. Color, texture or some scalar material property of the physical part, define natural scalar fields over the surface of the CAD model. Our reconstruction algorithm does not impose any convexity or differentiability restrictions on the surface of the original physical part or the scalar field function, except that it assumes that there is a sufficient sampling of the input point data to unambiguously reconstruct the CAD model. Compared to earlier methods our algorithm has the advantages of simplicity, efficiency and uniformity (both CAD model and scalar field reconstruction). The simplicity and efficiency of our approach is based on several novel uses of appropriate sub-structures (alpha shapes) of a three-dimensional Delaunay Triangulation, its dual the three-dimensional Voronoi diagram, and dual uses of trivariate Bernstein-Bezier forms. The boundary of the CAD model is modeled using implicit cubic Bernstein-Bezier patches, while the scalar field is reconstructed with functional cubic Bernstein-Bezier patches. CR

Unstructured grid generation using iterative point insertion and local reconnection

Abstract: A procedure is presented for efficient generation of high-quality two- or three-dimensional unstructured grids of triangular or tetrahedral elements. The present procedure uses an iterative point creation and insertion scheme wherein points are created using advancing-front type point placement. Initially, the connectivity for these generated points is obtained by directly subdividing the elements which contain them, without regard to quality. This connectivity is then improved by iteratively using local reconnection subject to a quality criterion. For two dimensions, a min-max criterion is used and for three dimensions, a Delaunay in-sphere criterion followed by a min-max type criterion is used. The overall procedure is applied repetitively until a complete field grid is generated with a desired point distribution. Grid quality and performance statistics are presented for a variety of two- and three-dimensional configurations. The combined quality and efficiency attributes of this procedure appear to be a substantial improvement over existing methods.

A general surface approach to the integration of a set of range views

Abstract: This paper presents a new and general solution to the problem of range view integration. The integration problem consists in computing a connected surface model from a set of registered range images acquired from different viewpoints. The proposed method does not impose constraints on the topology of the observed surfaces, the position of the viewpoints, or the number of views that can be merged. The integrated surface model is piecewise estimated by a set of triangulations modeling each canonical subset of the Venn diagram of the set of range views. The connection of these local models by constrained Delaunay triangulations yields g non-redundant surface triangulation describing all surface elements sampled by the set of range views. Experimental results show that the integration technique can be used to build connected surface models of free-form objects. No integrated models built from objects of such complexity have yet been reported in the literature, It is assumed that accurate range views are available and that frame transformations between all pairs of views can be reliably computed. >

Bubble mesh: automated triangular meshing of non-manifold geometry by sphere packing

Abstract: This paper presents a new computational method for fully automated triangular mesh generation, consistently applicable to wire-frame, surface, solid, and nonmanifold geometries. The method, called bubble rrzeshing, is based on the observation that a pattern of tightly packed spheres mimics a Voronoi diagram, from which a set of well-shaped Delaunay triangles and tetrahedral can be created by connecting the centers of the spheres. Given a domain geometry and a node-spacing function, spheres are packed on geometric entities, namely, vertices, edges, faces, and volumes, in ascending order of dimension. Once the domain is filled with spheres, mesh nodes are placed at the centers of these spheres and are then connected by constrained Delaunay triangulation and tet rahedrizat ion. To obtain a closely packed configuration of spheres, the authors devised a technique for physically based mesh relaxation with adaptive population control, The process of mesh relaxation significantly reduces the number of ill-shaped triangles and tetrahedral.

A comparison of sequential Delaunay triangulation algorithms

Abstract: This paper presents an experimental comparison of a number of different algorithms for computing the Deluanay triangulation. The algorithms examined are: Dwyer’s divide and conquer algorithm, Fortune’s sweepline algorithm, several versions of the incremental algorithm (including one by Ohya, Iri, and Murota, a new bucketing-based algorithm described in this paper, and Devillers’s version of a Delaunay-tree based algorithm that appears in LEDA), an algorithm that incrementally adds a correct Delaunay triangle adjacent to a current triangle in a manner similar to gift wrapping algorithms for convex hulls, and Barber’s convex hull based algorithm. Most of the algorithms examined are designed for good performance on uniformly distributed sites. However, we also test implementations of these algorithms on a number of non-uniform distibutions. The experiments go beyond measuring total running time, which tends to be machine-dependent. We also analyze the major high-level primitives that algorithms use and do an experimental analysis of how often implementations of these algorithms perform each operation.

Construction of three-dimensional improved-quality triangulations using local transformations

Abstract: Three-dimensional Delaunay triangulation are the most common form of three-dimensional triangulation known, but they are not very suitable for tetrahedral finite element meshes because they tend to contain poorly shaped sliver tetrahedra. In this paper, we present an algorithm for constructing improved-quality triangulation with respect to a tetrahedron shape measure. This algorithm uses combination of two or more local transformations to improve a given triangulation toward an optimal triangulation. Experimental results on finite element meshes show that this algorithm is much more effective than previous methods at removing slivers from Delaunay triangulation and producing nearly optimal triangulation. A variation of this algorithm for improving a pseudo locally optimal non-Delaunay triangulationn toward a Delaunay triangulation is also presented.

A Delaunay based numerical method for three dimensions: generation, formulation, and partition

Abstract: We present new geometrical and numerical analysis structure theorems for the Delaunay diagram of point sets in R~ for a fixed d where the point sets arise naturally in numerical methods. In particular, we show that if the largest ratio of the circum-radius to the length of smallest edge over all simplexes in the Delaunay diagram of P, DT(P), is bounded, (called the bounded radius-edge ratio property), then DT(P) is a subgraph of a density graph, the Delaunay spheres form a k-ply system for a constant k, and that we get optimal rates of convergence for approximate solutions of Poisson’s equation constructed using control volume techniques. The density graph result implies that DT(P) has a partition of cost O(rrl ‘Id) that can be efficiently found by the geometric separator algorithm of Miller, Teng, Thurston, and Vavasis and therefore the numerical linear system defined on D7’(P) using the finite-volume method can be solved efficiently on a parallel machine (either by a director an iterative method). The constant ply structure of Delaunay spheres leads to a linear-space point location structure for these Delaunay diagrams with O(log n) time per query. Moreover, we present a new parallel algorithm for computing the Delaunay diagram for these point sets in any fixed dimension in O(log n) random parallel time and n processors. Our results show that the bounded radius-edge ratio property is desirable for well-shaped triangular meshes for numerical methods such as finite element, finite difference, and in particular, finite volume methods.

On levels of detail in terrains

Abstract: In many applications it is important that one can view a scene at di erent levels of detail. A prime example is ight simulation: a high level of detail is needed when ying low, whereas a low level of detail su ces when ying high. More precisely, one would like to visualize the part of the scene that is close at a high level of detail, and the part that is far away at a low level of detail. We propose a hierarchy of detail levels for a polyhedral terrain (or, triangulated irregular network) that allows this: given a view point, it is possible to select the appropriate level of detail for each part of the terrain in such a way that the parts still t together continuously. The main advantage of our structure is that it uses the Delaunay triangulation at each level, so that triangles with very small angles are avoided. This is the rst method that uses the Delaunay triangulation and still allows to combine di erent levels into a single representation.

Computation of 3D skeletons using a generalized Delaunay triangulation technique

Abstract: The skeletal representation of 3D solids based on the medial axis transform has many applications in engineering. However, these applications are seldom realized, owing to the lack of viable computational techniques for generating skeletons. Such a computational technique, based on a notion of the generalized Voronoi diagram of a set of mixed-dimensional entities, is presented. It is shown that the generalized Voronoi diagram of a set of specific mixed dimensional set derived from the set of boundary entities of a polyhedron is, in fact, the exact skeleton of the polyhedron. Rather than the generalized Voronoi diagram being directly computed, its dual, an abstract Delaunay triangulation, is computed, from which the skeleton can be derived. An approach based on the Voronoi diagram of a well chosen representative point set on the boundary is also discussed as a special case; it is shown that the limitations of this approach are overcome by the generalization developed. Overall, it is argued that this generalization of the Voronoi diagram and the notion of the abstract generalized Delaunay triangulation are useful, and that they provide a viable approach to the computation of skeletons. Finally, details of the implementation, results, and an evaluation are presented.

A split Lagrangian-Eulerian method for simulating transient viscoelastic flows

Abstract: A novel numerical method for simulating time-dependent flow of viscoelastic fluids derived from dumbbell models is described. The constitutive equation is solved in a co-deforming frame, where the natural time-derivative is the upper-convected derivative. Mesh reconnection is achieved using a variant of Delaunay triangulation. The velocity and pressure are found via a finite element solution of the momentum equations. The method is tested by applying it to the benchmark problem of a sphere falling along the axis of a cylindrical tube.

Delaunay triangulation in three dimensions

Abstract: Triangulation in two and higher dimensions began with Dirichlet, Voronoi, Thiessen, and Delaunay. A number of textbooks and papers have extensively covered the properties of triangulations and algorithms for their construction. Most dealt with theoretical aspects of the algorithms and gave upper bounds on their complexity. Here we present a new algorithm and its implementation. Instead of providing a theoretical analysis, we present implementation details, and tests and examples. The algorithm is a generalization of our previous method. >

Dihedral bounds for mesh generation in high dimensions

Abstract: We show that any set of n points in IR has a Steiner Delaunay triangulation with O(ndd/2e) simplices, none of which has an obtuse dihedral angle. This result improves a naive bound of O(n). No bound depending only on n is possible if we require the maximum dihedral angle to measure at most 90◦−2 or the minimum dihedral to measure

Linear-size nonobtuse triangulation of polygons

Abstract: We give an algorithm for triangulatingn-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than ?/2. The number of triangles in the triangulation is onlyO(n), improving a previous bound ofO(n2), and the running time isO(n log2n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm.

Apparatus and method for geometric morphing

Abstract: A method of geometric morphing between a first object having a first shape and a second object having a second shape. The method includes the steps of generating a first Delaunay complex corresponding to the first shape and a second Delaunay complex corresponding to the second shape and generating a plurality of intermediary Delaunay complexes defined by a continuous family of mixed shapes corresponding to a mixing of the first shape and the second shape. The method further includes the steps of constructing a first skin corresponding to the first Delaunay complex and a second skin corresponding to the second Delaunay complex and constructing a plurality of intermediary skins corresponding to the plurality of intermediary Delaunay complexes. The first skin, second skin and plurality of intermediary skins may be visually displayed on an output device.

Mesh generator and generating method

Abstract: A mesh generator includes a mesh generation processing unit for setting a two-dimensional triangular mesh satisfying the conditions of the Delaunay partitioning on a semiconductor device to be analyzed, a triangular element deletion unit for deleting a predetermined mesh node and a mesh edge linking with the mesh node from the set mesh, a vertex selection unit for setting a new triangular mesh satisfying the conditions of the Delaunay partitioning in a polygon formed in a region from which the mesh node and the mesh edge are deleted without adding a new mesh node, and a triangular element generation unit.

Implicit reconstruction of solids from cloud point sets

Abstract: Chek T. Lirnl George M. Turkiyyah2 hark A. G’ante# Duane W. storti~ University of Washington Seattle, WA 98195 {ctlim@u, george@ce,ganter@u, storti@u}.Washington.edu This paper describes a new technique that combines numerical optimization methods with triangulation methods for generating mathematical representations of solids from 3D point data. The solid representation obtained takes the form of an algebraic function whose level surface closely approximates the surface described by the data, The algebraic function is obtained via Implicit Solid Modeling, a constructive scheme for approximating Boolean volume set operations on implicitly defined primitive volumes, and is comprised of a blended union of spherical primitives. The parameters of the algebraic function are the spatial locations and radii of the spheres as well as the parameters that describe the blending of these primitives, Fitting an implicit solid model to a data set is formulated as a sequence of non-linear optimization problems of an increasing number of variables. The cost function we employ in these optimizations is a weighted combination of discrepancies in location (distance from points to boundary of reconstructed object), discrepancies in surface normals, and desired curvature characteristics of the reconstructed solid. Since a set of trivariate data points without any connectivity information is ambiguous, an infinite number of solids, in principle, can be constructed to fit them. Different characteristics of the solid can be specified through the cost function to create the most desirable interpretation of the data. The starting point of the optimization—corresponding to the starting configuration of the primitives—is determined by performing a 3D Delaunay triangulation on the data set, and is based on the locations and sizes of the resulting tetrahedral. The effectiveness of the algorithm is demonstrated through the reconstruction of several sample data sets, including a molar and a femur. Tradeoffs between accuracy and compactness of the representations are also examined. 1Department of Mechanical Engineering, FU-10 ‘Department of Civil Engineering, FX-10. 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 of Computing Machinery.To copy otherwise, or to republish, requires a fee andlor specific permission. Solid Modeling ’95, Salt Lake City, Utah USA

Numerical stability of algorithms for 2d delaunay triangulations

Abstract: We consider the correctness of 2-d Delaunay triangulation algorithms implemented using floating-point arithmetic. The α-pseudocircle through points a, b, c consists of three circular arcs connecting ab, bc, and ac, each arc inside the circumcircle of a, b, c and forming angle α with the circumcircle; a triangulation is α-empty if the α-pseudocircle through the vertices of each triangle is empty. We show that a simple Delaunay triangulation algorithm—the flipping algorithm—can be implemented to produce O(n∈)-empty triangulations, where n is the number of point sites and ∈ is the relative error of floating-point arithmetic; its worst-case running time is O(n2). We also discuss floating-point implementation of other 2-d Delaunay triangulation algorithms.

A triangulated spatial model for cartographic generalisation of areal objects

Abstract: Cartographic generalisation involves interaction between individual operators concerned with processes such as object elimination, detail reductions amalgamation, typification and displacement. Effective automation of these processes requires a means of maintaining knowledge of the spatial relationships between map objects in order to ensure that constraints of topology and of proximity are obeyed in the course of the individual generalisation transformations. Triangulated spatial models, based on the constrained Delaunay triangulation, have proven to be of particular value in representing the proximal and topological relations between map objects and hence in performing many of the essential tasks of fully automated cartographic generalisation. These include the identification of nearby objects; determination of the structure of space between nearby objects; execution of boundary simplification, merge and collapse operations; and the detection and resolution, by displacement, of topological inconsistencies arising from individual operators. In this paper we focus on the use of a triangulated model for operations specific to execution of merge operations between areal objects. The model is exploited to identify the regions of space between nearby objects and to execute merge operations in which the triangulation is used variously to adopt intervening space and to move adjacent rectangular objects to touch each other. Methods for updating the triangulation are described.

Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices

Abstract: We give a uniform description, in terms of Coxeter diagrams, of the Voronoi domains of the root and weight lattices of any semisimple Lie algebra. This description provides a classification not only of all the facets of these Voronoi domains but simultaneously a classification of their dual or Delaunay cells and their facets. It is based on a much more general theory that we develop here providing the same sort of information in the setting of chamber geometries defined by arbitrary reflection groups. These generalized kaleidoscopes include the classical spherical, Euclidean, and hyperbolic kaleidoscopes as special cases. We prove that under certain conditions the Delaunay cells are Voronoi cells for the vertices of the Voronoi complex. This leads to the description in terms of Wythoff polytopes of the Voronoi cells of the weight lattices.

Large mesh generation from boundary models with parametric face representation

Abstract: The triangulation of boundary representation geometries (BRep geometries) is neccessary for display generation, stereo-lithography applications and also finite element mesh (FE mesh) generation. The accuracy of the tesselation is of great significance not only for stereo-lithography and rendering algorithms but also for FE mesh generation, since even minor simplifications of the geometry of the solid can lead to large errors in the FE computation. We therefore turn our attention to a new incremental algorithm for the accurate triangulation of meshes of trimmed parametric surfaces. Instead of triangulating each face seperately, the faces of the solid are glued and triangulated together according to the neighbourhood information of the BRep. By this procedure, glueing together faces with different triangulations is avoided. Another advantage of the algorithm is that the error between the linear approximation of the boundary and the boundary itself is controlled step by step until it lies within a predefined tolerance : Because the algorithm is fully incremental, later improvements of this tolerance can be easily added. The techniques we used to produce a robust implementation of this algorithm under finite precision arithmetic are reported. CR

On computing voronoi diagrams for sorted point sets

Abstract: We show that the Voronoi diagram of a finite sequence of points in the plane which gives sorted order of the points with respect to two perpendicular directions can be computed in linear time. In contrast, we observe that the problem of computing the Voronoi diagram of a finite sequence of points in the plane which gives the sorted order of the points with respect to a single direction requires Ω(n log n) operations in the algebraic decision tree model. As a corollary from the first result, we show that the bounded Voronoi diagrams of simple n-vertex polygons which can be efficiently cut into the so called monotone histograms can be computed in o(n log n) time.

On Steiner tree problem with 45/spl deg/ routing

Abstract: We consider Steiner minimal trees (SMT) in the plane where the orientations of interconnection are restricted to be either horizontal, vertical or of slopes +1 and -1. We derive a ratio, 1/4-2/spl radic/2 which we conjecture to be the Steiner ratio for 45/spl deg/-SMT. We provide a method to find an optimal 45/spl deg/-SMT for 3 or 4 points by analyzing its topological connections. For arbitrary point set of size n we present an O(n log n) time heuristic based on the notions of 45/spl deg/ minimum spanning tree and Delaunay triangulation. Empirical results compared with the rectilinear and Euclinear cases are given.

Evaluation of parallelization strategies for an incremental Delaunay triangulator in e3

Abstract: The paper deals with the parallelization of Delaunay triangulation, a widely used space partitioning technique. Two parallel implementations of a three-dimensional incremental construction algorithm are presented. The first is based on the decomposition of the spatial domain, while the second relies on the master-slaves approach. Both parallelization strategies are evaluated, stressing practical issues rather than theoretical complexity. We report on the exploitation of two different parallel environments: a tightly coupled distributed memory MIMD architecture and a network of workstations co-operating under the Linda environment Then, a third hybrid solution is proposed, specifically addressed to the exploitation of higher parallelism. It combines the other two solutions by grouping the processing nodes of the multicomputer into clusters and by exploiting parallelism at two different levels.