Showing papers on "Delaunay triangulation published in 1980"
TL;DR: This paper provides a unified discussion of the Delaunay triangulation and two algorithms are presented for constructing the triangulations over a planar set ofN points.
Abstract: This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set ofN points. The first algorithm uses a divide-and-conquer approach. It runs inO(N logN) time, which is asymptotically optimal. The second algorithm is iterative and requiresO(N
2) time in the worst case. However, its average case performance is comparable to that of the first algorithm.
1,460 citations
TL;DR: The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree and the Delaunay (Voronoi) triangulation.
1,213 citations
TL;DR: In this paper, the authors describe the minimal triangulations of polygonal domains and the restricted triangulation problem of finding a T of minimal weight among those containing s (S is a subset of E where no two edges of S cross each other).
Abstract: Publisher Summary This chapter describes the minimal triangulations of polygonal domains. If V is a set of n distinct points (vertices) M1, M2,. . . , Mn in the plane, no three points are collinear. This assumption is not essential but simplifies the explanations. Let E be the family of ½n(n − 1) line segments (edges) joining the vertices of V. A triangulation T of V is a maximal subset of E in which no two edges cross each other. Minimal weight triangulation (MWT) is a triangulation on V for which s(T) is minimal. The restricted triangulation problem consists of finding a T of minimal weight among those containing s (S is a subset of E where no two edges of S cross each other. Then there exists some triangulation T such that S ⊂ T.). Triangulation of a convex polygon and triangulation of a simple polygon domain are discussed in the chapter.
186 citations
TL;DR: The problems of developing an efficient Delaunay triangulation algorithm are described and a number of modifications to existing algorithms suggested.
Abstract: The problems of developing an efficient Delaunay triangulation algorithm are described and a number of modifications to existing algorithms suggested. The process of triangle creation is related to that of isarithmic mapping. Estimates of relative time savings are given using a grid versus a triangular data base.
77 citations
65 citations
TL;DR: In this paper, a new measure based on the ratio of the length of the sides and the content of the subsimplices of the triangulation is introduced, which is appropriate for computing fixed points using simplicial subdivisions.
Abstract: We consider measures for triangulations ofRn. A new measure is introduced based on the ratio of the length of the sides and the content of the subsimplices of the triangulation. In a subclass of triangulations, which is appropriate for computing fixed points using simplicial subdivisions, the optimal one according to this measure is calculated and some of its properties are given. It is proved that for the average directional density this triangulation is optimal (within the subclass) asn goes to infinity. Furthermore, we compare the measures of the optimal triangulation with those of other triangulations. We also propose a new triangulation of the affine hull of the unit simplex. Finally, we report some computational experience that confirms the theoretical results.
28 citations
TL;DR: This paper describes the design of an O( N 2 ) heuristic algorithm for the Directed Steiner Minimal Tree (DSMT) problem in the plane that employs the Delaunay triangulation, a minimum spanning arborescence algorithm, and a triangulations of theArborescence to develop solutions to the DSMT problem.
4 citations
01 Jan 1980
TL;DR: In this paper, the relative neighbourhood graph (RNG) of a set of n points on the plane is defined and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT).
Abstract: The relative neighbourhood graph (RNG) of a set of n points on the plane is defined. The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT). It is shown that the RNG is a superset of the MST and a subset of the DT. Two algorithms for obtaining the RNG of n points on the plane are presented. One algorithm runs in 0(n 2) time and the other runs in 0(n 3) time but works also for the d-dimensional case. Finally, several open problems concerning the RNG in several areas such as geometric complexity, computational perception, and geometric probability, are outlined. Relative neighbourhood graph Minimal spanning tree Triangulations Delaunay triangulation Dot patterns Computational perception Pattern recognition Algorithms Geometric complexity Geometric probability
2 citations
TL;DR: The concretisation of an abstract isomorphism between two Lie-algebras SO(4,2) leads to the transformation of the Keplerian problem from cartesian coordinates into Delaunay similar elements in the eccentric anomaly as mentioned in this paper.
Abstract: The concretisation of an abstract isomorphism between two Lie-algebras SO(4,2) leads to the transformation of the Keplerian problem from cartesian coordinates into Delaunay similar elements in the eccentric anomaly.
1 citations
01 Jan 1980
TL;DR: In this paper, a general time element, valid with any arbitrary independent variables, and used with Cartesian coordinates for the integration of the elliptic motion in orbits is examined, and the reduction of the in track error resulting from using Cartesian coordinate with the time element is reported.
Abstract: A general time element, valid with any arbitrary independent variables, and used with Cartesian coordinates for the integration of the elliptic motion in orbits, is examined. The derivation of the time element from a set of canonical elements of the Delaunay type, developed in the extended phase space, is presented. The application of the method using an example of a transfer orbit for a geosynchronous mission is presented. The eccentric and elliptic anomaly are utilized as the independent variable. The reduction of the in track error resulting from using Cartesian coordinates with the time element is reported.