Showing papers on "Delaunay triangulation published in 1987"
TL;DR: An algorithm for computing Delaunay triangulations of arbitrary collections of points in the plane using FORTRAN 77 for the generation of finite element meshes and the construction of contour plots is described.
234 citations
TL;DR: An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation of sites in the plane reduces its expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square.
Abstract: An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its ź(n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn≤216, the range of the experiments. It is conjectured that the average number of edges it creates--a good measure of its efficiency--is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theLp metric for 1
211 citations
01 Oct 1987
TL;DR: It is shown that the constrained Delaunay triangulation (CDT) can be built in optimal &Ogr;(n log n) time using a divide-and-conquer technique, which matches the time required to build an arbitrary (unconstrained) Delaunays and an arbitrary constrained (nonDelaunay) triangulations.
Abstract: Given a set of n vertices in the plane together with a set of noncrossing edges, the constrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimal O(n log n) time using a divide-and-conquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (nonDelaunay) triangulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that should make them useful for the finite-element method. Applications also include motion planning in the presence of polygonal obstacles in the plane and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.
210 citations
TL;DR: A number of mathematical results relevant to the problem of constructing a triangulation, i.e., a simplical tessellation of the convex hull of an arbitrary finite set of points in n-space are established.
179 citations
174 citations
12 Oct 1987
TL;DR: It is shown that there is a constant c(≤ 1+√5/2 π ≈ 5.08) independent of S and N such that DT(a, b)/d( a, b) ≪ c.
Abstract: Let S be any set of N points in the plane and let DT(S) be the graph of the Delaunay triangulation of S. For all points a and b of S, let d(a, b) be the Euclidean distance from a to b and let DT(a, b) be the length of the shortest path in DT(S) from a to b. We show that there is a constant c(≤ 1+√5/2 π ≈ 5.08) independent of S and N such that DT(a, b)/d(a, b) ≪ c.
143 citations
TL;DR: In this article, Sturm characterized Delaunay's surfaces variationally; indeed, as the solutions to an isoperimetric problem in the calculus of variations, and revealed how those surfaces make their appearance in gas dynamics; soap bubbles and stems of plants provide simple examples.
Abstract: In 1841 the astronomer/mathematician C. Delaunay isolated a certain class of surfaces in Euclidean space, representations of which he described explicitly [1]. In an appendix to that paper, M. Sturm characterized Delaunay’s surfaces variationally; indeed, as the solutions to an isoperimetric problem in the calculus of variations. That in turn revealed how those surfaces make their appearance in gas dynamics; soap bubbles and stems of plants provide simple examples. See Chapter V of the marvellous book [8] by D’Arcy Thompson for an essay on the occurrence and properties of such surfaces in nature.
99 citations
TL;DR: The algorithm runs in polynomial time and produces a triangulation within a ratio of O (log n ) to the cost of an optimal triangulations of a set of n points in the Euclidian plane.
77 citations
TL;DR: In this paper, two shape parameters for Delaunay simplices are proposed which allow one to recognize slightly distorted tetrahedral and octahedral simplices, and these simplex types are found to be the basic building units in dense packings of hard and soft spheres.
Abstract: Two novel shape parameters for Delaunay simplices are proposed which allow one to recognize slightly distorted tetrahedral and octahedral simplices. These simplex types are found to be the basic building units in dense packings of hard and soft spheres.
73 citations
01 Oct 1987
TL;DR: This paper defines a new Voronoi diagram for the endpoints of a set of line segments in the plane which do not intersect (except possibly at their endpoints), and obtains an optimal algorithm to construct the Delaunay triangulation of that set.
Abstract: In this paper, we first define a new Voronoi diagram for the endpoints of a set of line segments in the plane which do not intersect (except possibly at their endpoints), which is called a bounded Voronoi diagram. In this Voronoi diagram, the line segments themselves are regarded as obstacles. We present an optimal T(n log n) algorithm to construct it, where n is the number of input line segments.We then show that the straight-line dual of the bounded Voronoi diagram of a set of non-intersecting line segments is the Delaunay triangulation of that set. And the straight-line dual can be obtained in time proportional to the number of input line segments when the corresponding bounded Voronoi diagram is available. Consequently, we obtain an optimal T(n log n) algorithm to construct the Delaunay triangulation of a set of n non-intersecting line segments in the plane. Our algorithm improves the time bound O(n2) of the previous best algorithm.
65 citations
TL;DR: A technique is discussed for obtaining a contour tree efficiently as a byproduct of an operational contouring system, which may then be used to obtain contour symbolism or interval statistics as well as for further geomorphological study.
Abstract: A technique is discussed for obtaining a contour tree efficiently as a byproduct of an operational contouring system. This tree may then be used to obtain contour symbolism or interval statistics as well as for further geomorphological study. Alternatively, the tree may be obtained without the computational expense of detailed contour interpolation. The contouring system proceeds by assuming a Voronoi neighbourhood or domain about each data point and generating a dual-graph Delaunay triangulation accordingly. Since a triangulation may be traversed in a tree order, individual triangles may be processed in a guaranteed top-to-bottom sequence on the map. At the active edge of the map under construction a linked list is maintained of the contour ‘stubs’ available to be updated by the next triangle processed. Any new contour segment may extend an existing stub, open two new stubs or close (connect) two previous stubs. Extending this list of edge links backwards into the existing map permits storage of...
TL;DR: Different models and data structures for encoding triangular grids and Hierarchical surface models are presented, which are based on nested triangulations of the surface domain and provide variable resolution surface representations.
Abstract: The problem of representing 2 1/2 dimensional surfaces defined at a set of randomly located points by means of triangular grids is considered. Such representations approximate a surface as a network of planar, triangular faces with vertices at the data points. In the paper we describe different models and data structures for encoding triangular grids. Since Delaunay triangulation provides a common basis for many models of 2 1/2 D surfaces, we review its basic properties and we describe the most important approaches to its construction. Hierarchical surface models are also presented, which are based on nested triangulations of the surface domain and provide variable resolution surface representations. An algorithm is described for building a hierarchical description of a nested triangulation at different levels of abstraction. Finally, the 3D surface reconstruction problem is briefly discussed.
01 Oct 1987
TL;DR: It is proved that it is always possible to find piecewise-linear homeomorphisms between rectangular regions and described then in terms of a joint triangulation of the domain and the range rectangular regions.
Abstract: In rubber-sheeting applications in cartography, it is useful to seek piecewise-linear homeomorphisms (PLH maps) between rectangular regions which map an arbitrary sequence of n points {p1, p2, …,pn} from the interior of one rectangle to a corresponding sequence {q1, q2, …, qn} of n points in the interior of the second region. This paper proves that it is always possible to find such PLH maps and describes then in terms of a joint triangulation of the domain and the range rectangular regions.One naive approach to finding a PLH map is to triangulate (in any fashion) the domain rectangle on its n points and four corners and to define a piecewise affine map on each triangle up11p12p13 to be the unique affine map that sends the three vertices p11, p12, p13 of the triangle to the three corresponding vertices q11, q12, q13 of the image triangle uq11q12q13. Such piecewise affine maps send triangles to triangles, agree on shared edges, and thus extend globally, and will be called triangulation maps. The shortcoming of building transformations in this fashion is that the resulting triangulation map need not be one-to-one, although there is a simple test to determine if such a map is one-to-one (see Theorem 2 below). If the map is one-to-one, then the image triangles will form a triangulation of the range space; and we will have a joint triangulation. If the map is not one-to-one, then there will be folding over of triangles. It may be possible to alleviate this folding by choosing a different triangulation of the n domain points, or it may be the case that no triangulation of the n domain points will work. (See figures 5 and 6 below). We show that it will be possible, in all cases, to rectify the folding by adding appropriate additional triangulation vertex pairs {pn+1, pn+2, …, pn+m} and {qn+1, qn+2, …, qn+m} and retriangulating (see Theorem 1 below). This paper examines conditions for triangulation maps to be homeomorphisms and explores different ways of modifying triangulations and triangulation maps to make them joint triangulations and homeomorphisms.The paper concludes with a section on alternative constructive approaches to the open problem of finding joint triangulations on the original sequences of vertex pairs without augmenting those sequences of pairs.The existence proofs in this paper do not solve computational geometry problems per se; instead they permit us to formulate new computational geometry problems. The problems we pose are of interest to us because of a particular application in automated cartography.
01 Oct 1987
TL;DR: Two new algorithms finding relative neighborhood graph RNG(V) for a set V of n points are presented and their complexity analysis is based on some general facts pertaining to properties of equilateral triangles in the metric space.
Abstract: Two new algorithms finding relative neighborhood graph RNG(V) for a set V of n points are presented. The first algorithm solves this problem for input points in (R2,Lp) metric space in time O(n a(n,n)) if the Delaunay triangulation DT(V) is given. This time performance is achieved due to attractive and natural application of FIND-UNION data structure to represent so-called elimination forest of edges in DT(V). The second algorithm solves the relative neighborhood graph problem in (Rd,Lp), 1
01 Oct 1987
TL;DR: It is shown that maximal planar graphs inscribable in a sphere are 1-tough, and these appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations.
Abstract: We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graph with set of sites S is 1-tough if for any set P ⊆ S, c(S - P) ≤ |S|, where c(S - P) is the number of components of the subgraph induced by the complement of P and |P| is the number of sites in P. We also show that, under the same conditions, the number of interior components of S - P is at most |P| - 2. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations. We give examples to show that these bounds can be attained, and we state and prove several corollaries. In particular, we show that maximal planar graphs inscribable in a sphere are 1-tough.
TL;DR: It is proved that the greedy triangulation heuristic for minimum weight triangulations of convex polygons yields solutions within a constant factor from the optimum within time O(n2) time andO(n) space.
Abstract: We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's Ω(n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n
2) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).
TL;DR: A new heuristic for minimum weight triangulation of planar point sets is proposed, and a polygon whose vertices are all points from the input set is constructed.
Abstract: A new heuristic for minimum weight triangulation of planar point sets is proposed. First, a polygon whose vertices are all points from the input set is constructed. Next, a minimum weight triangulation of the polygon is found by dynamic programming. The union of the polygon triangulation with the polygon yields a triangulation of the input n-point set. A nontrivial upper bound on the worst-case performance of the heuristic in terms of n and another parameter is derived. Under the assumption of uniform point distribution it is observed that the heuristic yields a solution within the factor of $O(\log n)$ from the optimum almost certainly, and the expected length of the resulting triangulation is of the same order as that of a minimum length triangulation. The heuristic runs in time $O(n^3 )$ .
TL;DR: This method is derived from the well known Watson's algorithm and uses Lawson's circle criterion in the local retriangulation process for triangulation of large sets of scattered points in R 2.
TL;DR: In this paper, Hsiang and Yu generalized Delaunay's theorem to constant mean curvature rotation hypersurfaces in Rn+1, and further generalized it to rotational W-hypersurfaces of σ Γ type.
Abstract: In 1841 Delaunay proved that if one rolls a conic section on a line in a plane and then rotates about that line the trace of a focus, one obtains a constant mean curvature surface of revolution in R3. Conversely, all such surfaces, except spheres, are constructed in this way. In 1981, Hsiang and Yu generalized Delaunay's theorem to constant mean curvature rotation hypersurfaces in Rn+1. In 1982, Hsiang further generalized Delaunay's theorem to rotational W-hypersurfaces of σ Γtype in Rn+1. These are hypersurfaces such that the /th-basic symmetric polynomial of the principal curvatures (k^x)), namely,
TL;DR: In this paper, the motion of two massive particles is considered within the framework of the first post-Newtonian approximation, and the system Hamiltonian is constructed and normalized through first order using a canonical transformation method of implicit variables.
Abstract: The motion of two massive particles is considered within the framework of the first post-Newtonian approximation. The system Hamiltonian is constructed and normalized through first order using a canonical transformation method of implicit variables. Closed-form solutions for the Delaunay elements in the phase space are obtained. The bridge between the phase space and the state space of the Lagrangian of the motion is provided by a velocity-dependent Legendre transformation. By explicit inversion of this transformation, expressions for the Keplerian elements in the state space are obtained from the Delaunay element solutions.
TL;DR: In this article, the dependence of the properties of Voronoi polyhedra and Delaunay simplexes on the phase state of the material was considered and the dominance of atomic configurations in the form of a tetrahedron and a quadrant of an octahedron in all close-packed disordered systems was established.
Abstract: The authors summarize the work of various authors on the study of computer models for disordered systems using the language of Voronoi polyhedra and Delaunay simplexes. We consider the dependence of the properties of Voronoi polyhedra and Delaunay simplexes on the phase state of the material. We have established the predominance of atomic configurations in the form of a tetrahedron and a quadrant of an octahedron in all close-packed disordered systems. We have demonstrated the absence of crystalline and icosahedral configurations of atoms in such systems. We emphasize the need for eliminating thermal excitations (transition to the essential structure) in formulating the structural principles of a liquid.
TL;DR: In recent years, with the rise of computational geometry and an increasing number of people working on the subject, many formerly neglected problems were solved e~iciently and elegantly, but at the same time, even more arose and became of interest: a few are hereafter offered.
Abstract: In recent years, with the rise of computational geometry and an increasing number of people working on the subject (see Lee and Preparata [5] for a survey), many formerly neglected problems were solved e~iciently and elegantly, but at the same time, as is natural for every expanding field, even more arose and became of interest: a few are hereafter offered. Although the minimum spanning tree problem (MST) has long been solved (at least in theory), related questions still remain unanswered. For example, let us consider n points in the plane, with which we wish to form a \"reasonable\" polygon. In practice these points are on the boundary of an unknown object, and the polygon is \"reasonable\" if the order of the points is roughly the same on the boundary of the other object and on the polygon. A traveling salesman tour (TST) would be a nice solution, but finding it is an NP-hard problem, and so we would like an easier criterion for constructing a good polygon. Suppose that such a polygon is part of the Delaunay triangulation of the points. It contains (n -2) Delaunay triangles, and the (graph-theoretic) dual of these triangles is a subtree of the Voronoi diagram, and covers ( n 2) Voronoi nodes (Figure 1). A good criterion of \"reasonableness\" might be to minimize the cost (cumulated edge-length) of this tree: whence the question of choosing k = n 2 nodes among N = 2n e 2 (number of nodes of the Voronoi diagram, e being the number of edges on the convex hull of the n data points), in order to minimize the cost of their MST (problem proposed by J. D. Boissonnat). Another problem in connection with Delaunay graphs, appealing but deceptive in its apparent simplicity, was proposed to me by J. D. Boissonnat and H. Crapo: this is the question of proving that a planar Delaunay triangulation always has a Hamiltonian cycle--which, if true, might lead to nice heuristics for the TST problem. For the definition of a Delaunay triangulation see, for example, Shamos and Hoey [8]. Algorithms for solving geometrical problems in the plane abound, and their worst-case complexity has been fairly well studied, but many difficulties arise
01 Jan 1987
TL;DR: The development of a complete electromagnetic analysis system is discussed in this thesis and a technique of measuring error of the finite element solution is presented.
Abstract: The development of a complete electromagnetic analysis system is discussed in this thesis.
The work discusses the development of two new computational ideas and their incorporation into an adaptive analysis system. The first is the automatic discretization of a problem, described in an object data base, into a set of triangles in two dimensions and tetrahedra in three dimensions. This process is based on Delaunay triangulation, which generates an optimal mesh for a given set of data points. This work expands on the basic Delaunay algorithms to allow the incorporation of boundary planes and surfaces.
Secondly, a technique of measuring error of the finite element solution is presented. This technique directly tests the approximate solution, obtained through finite element analysis, with the problem description.
Adaptive systems are presented in both two and three dimensions which merge the mesh generator and the error criteria into an automatic system. The two dimensional solution technique is based on the vector potential (')A while the three dimensional system uses the reduced scalar potential (phi).
Several magnetic recording examples are presented to demonstrate the performance of the adaptive technique. In addition, an eddy current analysis is developed and compared with experimental results.
01 Jul 1987
TL;DR: In this paper, an application of the extended Delaunay methods is made to the ideal resonance problem, where the theory of integration proposed in a preceding paper works in a simple problem, and discuss how to proceed in more complicated situations.
Abstract: In this paper, an application of the extended Delaunay methods is made to the ideal resonance problem We show how the theory of integration proposed in a preceding paper works in a simple problem, and discuss how to proceed in more complicated situations
TL;DR: A method is given for discretising the basic equations governing the behaviour of semiconductor devices in a systematic manner, which will produce the standard generalisation of the Scharfetter-Gummel discretisation to two dimensions currently in use.
Abstract: In this paper we derive the basic equations governing the behaviour of semiconductor devices. We then examine several reformulations of the equations in terms of different dependent variables. The necessity of generating, automatically, local mesh refinements is then discussed, and the advantages of a Delaunay triangulation are described. In the final section a method is given for discretising the equations in a systematic manner, which will produce the standard generalisation of the Scharfetter-Gummel discretisation to two dimensions currently in use. The paper concludes with a brief annotated bibliography.
01 Jan 1987
TL;DR: A method of generating hybrid structured-unstructured grids suitable for viscous flow calculations around high lift aerodynamic geometries is presented and some attention will be focused on ways in which a flow algorithm based on an hybrid grid can be modified for increased efficiency.
Abstract: A method of generating hybrid structured-unstructured grids suitable for viscous flow calculations around high lift aerodynamic geometries is presented. The unstructured technique provides the flexibility to discretise the complex multiply connected domain whilst maintaining regular shaped triangles. The approach which will be described is based on the Voronoi construction and the dual, the Delaunay triangulation. In the near-wall regions, where viscous effects are dominant and where flow algorithm modifications, including the incorporation of a turbulence model may be required, it proves advantageous to construct a regular structured grid. In addition to details given on the grid generation, some attention will be focused on ways in which a flow algorithm based on an hybrid grid can be modified for increased efficiency.