Boris N. Delaunay. Sur la Sph ere Vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7:793--800, 1934.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an edge. Now assume that no four points in V lle on a circle. As a result, if we connect all neighbors with straight line segments we get a triangulation of the pointset. This triangulation is called a Delaunay triangulation after Delaunay who studied such triangulations already a long time ago ([18]) Theorem 3.3 The Delaunay triangulation of a set of points can be constructed in time O(n 16g n) The Delaunay triangulation has a number of interesting properties. The most important one is probably that the circle with the three vertices of a triangle of the triangulation on its boundary ....

Delaunay, B., Sur la sphere vide. Bull. Acad. Sci. USSR. (VII), Classe Sci. Mat. Nat., 793-800 (1934)

....computational geometry. For more on Voronoi diagrams, we refer the reader to [10, 14] Equally well known is the dual graph of the Voronoi diagram, obtained by connecting two points from K by an edge if their associated Voronoi cells are neighbors, i.e. share a common edge. In his 1934 paper [6], B. N. Delaunay proved that this dual graph of the Voronoi diagram is a triangulation of the convex hull [K] now called the Delaunay triangulation. The Delaunay triangulation corresponding to the point set from Figure 5 is displayed in Figure 6. Delaunay triangulations have numerous ....

Delaunay, B., Sur la sph`ere vide, Bull. Acad. Sci. USSR (VII), Classe Sci. Mat. Nat., 1934, 793-- 800.

....7 8 9 10 11 12 13 14 15 16 17 18 E 433 19 20 Fig. 18. Delaunay triangulation. s = 2, this assumption means that no four knots from K are co circular. A consequence of this slight restriction is that any given point in IR s can belong to at most s 1 Voronoi cells. In his 1934 paper [132], B. N. Delaunay proved that the dual graph of the Voronoi diagram is a triangulation of the convex hull [K] now called the Delaunay triangulation. Note that in our case [K] IR s , by an earlier assumption on K. The Delaunay triangulation corresponding to the point set from Figure 13 is shown ....

Delaunay, B., Sur la sph`ere vide, Bull. Acad. Sci. USSR (VII), Classe Sci. Mat. Nat., 1934, 793--800.

....and computational geometry. For more on Voronoi diagrams, we refer the reader to [10, 14] Equally well known is the dual graph of the Voronoi diagram, obtained by connecting two points from K by an edge if their associated Voronoi cells are neighbors, i.e. share a common edge. In his 1934 paper [6], B. N. Delaunay proved that this dual graph of the Voronoi diagram is a triangulation of the convex hull [K] now called the Delaunay triangulation. The Delaunay triangulation corresponding to the point set from Figure 5 is displayed in Figure 6. Delaunay triangulations have numerous ....

Delaunay, B., Sur la sphere vide, Bull. Acad. Sci. USSR (VII), Classe Sci. Mat. Nat., 1934, 793--800.

....following three steps: 1. Construct a triangulation of the nodes. 2. Compute a set of nodal gradient values. 3. Construct a C 1 triangle based interpolant of the data values and nodal gradients. In the unconstrained case the first step usually consists of constructing a Delaunay triangulation [6] (chosen to avoid small angles and hence long thin triangles) the nodal gradients are chosen to produce a well behaved interpolant that accurately reproduces test functions [17] and the interpolation method is chosen for e#ciency [5] or flexibility [16] since the quality of the surface is ....

B. Delaunay, Sur la sphere vide, Bull. Acad. Sci. USSR (VII), (1934), pp. 793--800.

....equal to the intersection of all the half spaces determined by the mid perpendiculars of all segments p i p j , p i 6= p j . Voronoi faces always lie in such perpendicular bisectors. The interest of Voronoi polytopes for the numerical integration of pdes is due to the following Theorem (Delaunay [14]) 3 Theorem 1 (Delaunay) Let S be a point set in IR d , V(S) be the Voronoi diagram of S. Then, the straight line dual 1 of V(S) is a tessellation of the convex hull of S. If the Voronoi boxes are to be used for the discretization of pdes with the bm, each edge perpendicular to a Voronoi ....

B. Delaunay, "Sur la sph`ere vide," Bull. Acad. Sci. USSR(VII), pp. 793--800, 1934.

....of a set V of vertices is a set T of triangles whose vertices collectively are V , whose interiors do not intersect each other, and whose union is the convex hull of V , if every triangle intersects V only at the triangle s vertices. The Delaunay triangulation D of V , introduced by Delaunay [20] in 1934, is the graph defined as follows. Any circle in the plane is said to be empty if it encloses no vertex of V . Vertices are permitted on the circle. Let u and v be any two vertices of V . A circumcircle (circumscribing circle) of the edge uv is any circle that passes through u and v. Any ....

Boris N. Delaunay. Sur la Sph ere Vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7:793--800, 1934.

....potential difference. The total and reduced potential calculations are performed by the finite element method on irregular tetrahedronal mesh. The method of reproduced sections is used to construct a finite element mesh. The basic mesh is constructed with the algorithm of Delaunay triangulation [2]. This method permits to automate the process of tetrahedronal mesh generation. The magnetic field of currents is represented as a superposition of the fields from tetrahedrons on which the currents is divided. The vector of current density is of a constant value and the same direction in every ....

B. Delaunay, "Sur la sphere vide," Izvestiya Akademii Nauk, USSR, Math and Nat Sci Div, No. 6, p. 793, 1934.

....union of the simplices in T equals the convex hull of S. Given a triangulation T for S, we say that T is a Delaunay triangulation for S if S is the set of 0 Gammasimplices in T , and for each d Gammasimplex in T there does not exist a point of S in the interior of the circumsphere of the simplex [2]. A larger class of triangulations that includes the Delaunay triangulations can be defined. Again, let S be a finite set of points in R d , and for each point p in S let w p be a real valued weight assigned to p. Given p in S and a point x in R d , the power distance of x from p, denoted by ....

B. Delaunay, Sur la sph`ere vide, Bull. Acad. Sci. USSR (VII), Classe Sci. Mat. Nat., 793-800 (1934).

....or 4 points co circular) this is guaranteed to produce a valid triangulation. This is shown in Figure 3.1(d) Note that the convex hull is a subset of the Delaunay triangulation. This is named after B. Delaunay who proved that the dual of the Voronoi diagram of P is a triangulation of P [Delaunay 34] a) Voronoi Diagram (b) Power Diagram Figure 3.2: Voronoi and Power Diagrams Power Diagrams Power diagrams are a generalization of the Voronoi diagrams. Voronoi diagrams are defined for simple points (or circles of equal radii) whereas power diagrams are defined for circles of possibly ....

B. Delaunay. Sur la sph`ere vide. Bull. Acad. Sci. USSR: Class. Sci. Math. Nat., 7:793--800, 1934.

....represent positions where a commodity is available. The region corresponding to a given site would then represent those points from which this is the site of choice for getting the commodity. Or, we might want to find for each site, the other site to which it is closest. The Delaunay triangulation [De] is also constructed from a set of sites. In the plane, we build a triangulation of a point set by connecting pairs of sites. This technique can be used to generate an exponential number of different triangulations. For many applications, some of these triangulations are preferable to others. For ....

....range searching extensions of k d trees. Recent results here apply deep results from probability theory (dealing with the Vapnik Chervonenkis dimension) to obtain theoretical improvements [EW] Other results give methods of extending range searching techniques to more sophisticated queries [DE]. 4. Conclusions and further results This paper has shown that the fields of computational geometry and computer graphics have had significant impact on one another. Indeed, a major frustration in writing this paper is that few of the exciting interactions can be explored. The two problems we ....

Delaunay, B., "Sur la sphere vide", Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk, vol. 7, 1934, pp. 793-800.

....in the algorithm. 1.1 Requirements of the underlying numerical method Definition 1 A tessellation T of a set of points S is a Delaunay tessellation if there exists a point free circumsphere for each tessellation element. We use the term Delaunay tessellation and not Delaunay triangulation [7, 8] because our meshes include element types other than tetrahedra. Definition 2 A Delaunay tessellation of a set of points S is adequate for the control volume discretization method if the corresponding Voronoi diagram fulfills the following conditions: a) No Voronoi point is outside the boundary ....

B. Delaunay, "Sur la sph`ere vide," Bull. Acad. Sci. USSR(VII), pp. 793--800, 1934.

....shortest path between any two vertices in the graph is at most a constant factor longer than a straight line. A variety of low dilation graphs that could be used for this purpose are surveyed by Eppstein (1996) For our implementation, we have chosen a planar graph called a Delaunay triangulation (Delaunay 1934; Fortune 1992) Delaunay triangulations have several desirable properties. ffl They provide a structure that makes it possible to quickly determine edge weights in the graph. ffl Local modifications of the triangulation can be made inexpensively. ffl Several free implementations are available ....

Delaunay, B. N. (1934). Sur la sphere vide. Bull. Acad. Science USSR VII: Class. Sci. Math., pp. 793--800.

No context found.

Boris N. Delaunay. Sur la Sph ere Vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7:793--800, 1934.

....angle, measured from above, of less than 180 # . In other words, the apex of t lies above the witness hyperplane of s, and vice versa. For convenience, say that every hyperface included in only one d simplex is locally regular as well. It follows from a generalization of the Delaunay Lemma [7] that a triangulation that fills X , respects X , and has no submerged vertices is a weighted CDT if and only if every hyperface is a constraining simplex or is locally semiregular. Each flip algorithm waits until a non constraining hyperface of the triangulation is no longer locally regular, then ....

Boris N. Delaunay. Sur la Sph ere Vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7:793--800, 1934.

....validity are presently lacking. Another widely used approach to tetrahedral mesh generation is Delaunay triangulation. A triangulation is Delaunay if the spheres circumscribing the tetrahedra in the mesh do not contain any vertices other than those defining the circumscribed tetrahedra themselves [14]. Algorithms for the efficient Delaunay triangulation of the convex hull of a point set are well documented in the literature [5, 60, 4] However, three dimensional Delaunay triangulation is not without difficulty, even when attention is restricted to convex domains. A common problem is the ....

.... A Delaunay triangulation of a set of points in three dimensional space is a tetrahedral decomposition of the convex hull of the point set, where the vertices of the tetrahedra belong to the pointset,such that the interior of the circumspheres of the tetrahedra does not contain points from the set [14]. The triangulation is unique if situations in which more than four points lie on the surface of the circumsphere of any tetrahedra do not occur. Of utmost concern when meshing three dimensional volumes by the Delaunay method is the appearance of tetrahedra with vanishingly small volume, also ....

B. Delaunay. Sur la sphere vide. In Classe Sci. Mat. Nat.,volume VII, pages 793--800. Bull. Acad. Sci. USSR, 1934.

No context found.

Delaunay, B., "Sur la Sph'ere Vide", Izvestia Akademii Nauk SSSR, Vol. 7, No. 6, Oct. 1934, pp. 793-800.

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