Delaunay, B. Sur la sphere vide, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934), Izv. Akad. Nauk SSSR, pp. 793--800  Home/Search   Document Not in Database   Summary   Related Articles   Check

....triangulation of S (see for example [21,67] Assuming general position of the points, this triangulation is unique and decomposes the convex hull of S into tetrahedra. The triangulation is named after the Russian geometer Boris Delaunay (also Delone) who introduced it in his seminal paper [14] in 1934. As explained below, the Delaunay triangulation of S is dual to another complex defined by S known as the Voronoi diagram [79, 80] Both complexes are related in an interesting way to the family of all , shapes of S. The relationship between , shapes and Delaunay triangulations is of ....

....then its projection is not part of Dv. Therefore, it is possible to attach a tetrahedron underneath the edge and its two incident triangles. In other words, flipping edges in R 2 corresponds to attaching tetrahedra in R 3. Clearly, this process has to end sometimes. A classical lemma by Delaunay [14] implies that this happens only when Tv = Dv, that is, when T is a Delaunay triangulation. Also note, that the upper bound theorem implies that no more that O(n 2) tetrahedra can be attached in the worst case, which makes Lawsoh s flip method a quadratic algorithm. Pj Figure 4.1: Edge flipping ....

B Delaunay. Sur la sphere vide. Izvestiya Akademii Nauk SSSR, Otdelenie Matematicheskii i Estestven- nyka Nauk, 7:793 800, 1934.

....projectors, as shown in Figure 7 (c) The intersections are defined by points on the boundary. To compensate for various distortions, we fill the interior with original feature points from the registration process. Finally, we feed the points in the intersection to a Delaunay triangulation [2] program to create a 2D mesh in the projector s screen space, shown in Figure 7 (d) The texture coordinates are translated and normalized between 0 and 1 within each patch. 5.2 Photometric Calibration Photometric calibration is an integral portion of PixelFlex. The somewhat arbitrary placement ....

B. Delaunay. Sur la sphere vide. izv. akak. nauuk sssr. Otdelenie Matematicheskii i Estestvennyka Nauk, 7:793 -- 800, 1934.

....triangles from worst to best) are ruled out by the NP completeness of the following decision problem [Llo77] given a collection of points and edges, decide whether a subset of the edges defines a triangulation of the points. Most positive results are related to the Delaunay triangulation [Del34]. It has been shown that among all triangulations of a given finite point set, the Delaunay triangulation optimizes various criteria. The Delaunay triangulation maximizes the minimum angle [Sib78] minimizes the maximum circumscribing circle [D AS89] and minimizes the maximum smallest enclosing ....

B. Delaunay. Sur la sph`ere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934), 793--800.

....buffer; 2. choose a projector pixel V, select a small sample of neighboring pixels, and search for the sample in the neighborhood of the predicted location in the camera s image; 3. perform the Kalman Filter update for the corresponding projector pixel; and 4. use 2D Delaunay Triangulation [Delau34] to update the mesh. We repeat this process continuously while the system is being used. Notice that in the time update (Equation 3) we add a small amount of process variance Q to compensate for slow changes in the system. Q should not be zero, or the Kalman filter will cease to update its ....

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

....of two simplices is either empty or a face of both. Applications of triangulations can be found in finite element analysis [Cave74] surface interpolation [Laws77] shape reconstruction [Bois88] and other research areas. An important type of triangulation is the Delaunay triangulation [Dela34]. It is dual to the socalled Voronoi diagram [Voro08] The popularity of the two dimensional Delaunay triangulation is partly due to the fact that it optimizes various quality measures, including the smallest angle [Sibs78] the largest circumscribed circle [D AS89] the largest minimum enclosing ....

B. Delaunay. Sur la sph`ere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934), 793--800.

....is the NP completeness of the following decision problem [Llo77] given a collection of points and edges, decide whether or not there is a subset of the edges that defines a triangulation of the points. Most positive results are related to the Delaunay triangulation defined for finite point sets [Del34]. It has been shown that among all triangulations of a given point set, the Delaunay triangulation optimizes various criteria. These include the maxmin angle [Sib78] the minmax circumscribed circle [D AS89] the minmax smallest enclosing circle [D AS89, Raj91] and the minimum integral of the ....

B. Delaunay. Sur la sph`ere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934), 793--800.

....is a circle through a and b so that all other points of S lie outside the circle. The collection of Delaunay 2 Though min min and max max versions are possible, they are usually trivial and uninteresting. 6 edges defines a plane geometric graph D(S) known as the Delaunay triangulation of S [Dela34]. In the non degenerate case, which excludes four or more points on a common circle, D(S) is indeed a triangulation. In fact, it is a triangulation that lexicographically maximizes the nondecreasing vector of smallest angles of triangles. In the degenerate cases, some faces of D(S) are convex ....

....This operation was incorporated into a plane sweep scheme (Section 2.1) to incrementally compute a locally optimal triangulation T (S) Laws77] that is, one that has no edge flip to improve its quality. It was found that this locally optimality actually implies that T (S) is a completion of D(S) [Dela34, Sibs78]. Since any two completions of D(S) have the same value for their smallest angles, T (S) is actually a max min angle triangulation (see [Edel87, page 301 303] There are various possibilities to implement the above scheme. One variant is to first compute an arbitrary triangulation, T (S) and ....

B. Delaunay. Sur la sph`ere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934), 793--800.

.... shape of the triangles and tetrahedra are important because it relates to the convergence and stability of numerical methods such as the nite element analysis, see Strang and Fix [20] Probably the most common tetrahedral meshes are Delaunay triangulations, which are named after Boris Delaunay [7] and are also known as duals to Voronoi diagrams, which are named after Georges Voronoi [22] They are supported by fast algorithms both for construction and for maintenance under local changes. In this paper we make essential use of a somewhat larger class of tetrahedral meshes referred to as ....

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

....be the CDT, because edge e violates the circle condition. The converse is also true, giving an example of a local optimization leading to global optimality. Lemma 3. If no quadrilaterals are reversed, the triangulation is the CDT. Proof: The idea behind this proof can be traced back to Delaunay [58]. We show that the circumcircle of each triangle abc contains no vertex d. Therefore each edge is a correct Delaunay edge. Suppose such a point d does exist for some triangle abc. The proof of Lemma 2 shows that we may choose d visible to a. For a circle C with center at coordinates (x; y) and ....

B. Delaunay. Sur la sph`ere vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934) 793--800.

....algorithm. This algorithm adds the points of S by x coordinate order. Upon each addition, the algorithm walks around the convex hull of the already added points, starting from the rightmost previous point and adding edges until the slope reverses, as shown in Figure 8. The following theorem [45] guarantees the success of edge flipping: a triangulation in which no quadrilateral is reversed must be a completion of the Delaunay triangulation. 5.2 Constrained Delaunay triangulation There is another way, besides conforming Delaunay triangulation, to extend Delaunay triangulation to ....

B. Delaunay. Sur la sph`ere vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk, 7:793--800, 1934.

....in [17] 4 Triangulating a Point Set There has been great interest in finding efficient algorithms for constructing triangulations for a point set which optimize certain criterion (or multiple criteria at the same time. One of the most important constructions is Delaunay Triangulation(DT) [7], which is the dual of Voronoi Diagram. In the following, S denotes the input point set (S ae R 2 ) Definition 4 (Voronoi Cell) V p = fq 2 R 2 j j pq jj pr j; 8r 2 Sg. where j xy j denotes the Euclidean distance between points x; y 2 R 2 . VS denotes the set fV p j p 2 Sg. If C is a ....

....of DT, it also gives rise to an important algorithm called the Flip Algorithm. An inner edge e is called locally delaunay, if it satisfies the empty circle criterion. It can be shown that if all the edges of a triangulation are locally delaunay, then that triangulation is actually the DT [7]. The flip algorithm starts with any triangulation and considers each edge e in turn. The edge e is replaced by the other diagonal of the quadrangle formed by the two triangles containing e if it is not locally delaunay. The proof of termination within O(n 2 ) flips uses the lifting argument and ....

B. Delaunay. Sur la sphere vide. Otdelenie Matematicheskii i Estestvennyka Nauk 7, VII Seria:793--800, 1934.

....the state space automatically based on a small set of randomly chosen landmark states. In her approach, each landmark state defines a region and states other than landmark states are members of the region defined by the nearest landmark state. This approach is a version of Delaunay triangulation [29,39], a family of methods that decompose the state space through Voronoi diagrams. Just like in Dayan Hinton s and Dean Lin s approach, Kaelbling s approach applies dynamic programming at multiple levels: At the lower lever, local controllers are learned for moving from one region to another. On the ....

B.N. Delaunay. Sur la sphere vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk (Bulletin of Academy of Sciences of the USSR), 7:793--800, 1934.

....a triangle. Many criteria have been proposed as to what constitutes a good triangulation. The criteria are either of topographical or geometrical characteristic. Typical geometrical characteristics are: maximizing the smallest angle or minimizing the total edge length. The Delaunay triangulation [3] satisfies the property of maximizing the smallest angle, but there is no refinement process known, leading to a strict hierarchical Delaunay triangulation. In our applications (visualizing large terrains [8] we are interested in triangles without small angles, because small angles have some ....

B. Delaunay. Sur la sphere vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk, 7:793--800, 1934.

.... and shape of the triangles and tetrahedra is important because it relates to the convergence and stability of numerical methods such as the nite element analysis, see Strang and Fix [20] Probably the most common tetrahedral meshes are Delaunay triangulations, which are named after Boris Delaunay [7] and also known as duals to Voronoi tilings, which are named after Georges Voronoi [22] They are supported by fast algorithms both for the construction and for the maintenance under local changes. In this paper we make essential use of a somewhat larger class of tetrahedral meshes referred to as ....

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

....3 and higher dimensions followed soon. 1 Delaunay triangulations are considered particularly useful for automatic three dimensional mesh generation, 2 a key problem in scientific computing, and a robust and efficient implementation is therefore in high demand. A classical lemma by Delaunay [5] implies that a triangulation is (globally) Delaunay if and only if each triangle is locally Delaunay. In three dimensions, a triangle f a;b;c is called locally Delaunay if it either lies on the convex hull, or it is the shared face of two tetrahedra t a;b;c;d and t a;b;c;e , and the circumsphere ....

B. Delaunay. Sur la sph`ere vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk, 7:793--800, 1934.

....If the curved bounding surfaces of the cells are approximated by triangles, the resulting cells can be tetrahedralized so that the cycle remains. 4.8. 2 The Delaunay Triangulation An acyclicity theorem for the obstructs relation for meshed polyhedra resulting from a Delaunay triangulation (DT) [19] has been proven by Edelsbrunner [26] A DT of a set S of points in E 3 is a triangulation such that a sphere circumscribed about the four vertices of any tetrahedron in the triangulation contains no other points in S. Therefore, in order to eliminate cycles from a meshed data set or to be sure ....

DELAUNAY, B. Sur la sph`ere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934), 793--800.

....the degree requirements for edges and vertices are considered in [40, 98] Tetrahedral Meshes. The most common unstructured mesh is the triangular mesh in R 2 and the tetrahedral mesh in R 3 , see figure 5. By far the most popular such mesh is the Delaunay complex named after Boris Delaunay [23]. For a finite point set S R 3 the Delaunay complex is unique and decomposes the convex hull of the set. In the non degenerate case all elements are tetrahedra, and a tetrahedron spanned by 4 points in S belongs to the complex iff its circumsphere encloses no point of S. An important but ....

B. Delaunay. Sur la sph`ere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934), 793--800.

....triangulation of S (see for example [11, 35] Assuming general position of the points, this triangulation is unique and decomposes the convex hull of S into tetrahedra. The triangulation is named after the Russian geometer Boris Delaunay (also Delone) who introduced it in his seminal paper [6] in 1934. As explained below, the Delaunay triangulation of S is dual to another complex defined by S known as the Voronoi diagram [41, 42] Both complexes are related in an interesting way to the family of all ff shapes of S. The relationship between ff shapes and Delaunay triangulations will be ....

....called locally Delaunay if it belongs to the Delaunay triangulation of T 0 [ T 00 . This is the case iff the point p v lies outside the sphere b T 0 . Local Delaunayhood is a necessary condition for oe T to belong to the Delaunay triangulation, but it is not sufficient. Nevertheless, Delaunay [6] proved that if all triangles are locally Delaunay then the triangulation is a Delaunay triangulation. The incremental flip algorithm needs to restore Delaunayhood whenever a new point is added to the triangulation. For this, it identifies triangles that are not locally Delaunay and tries to flip ....

B Delaunay. Sur la sph`ere vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk, 7:793--800, 1934.

.... a pocket becomes a void inaccessible from the outside before it disappears . It is clear that considerations based on relative distance are required to make this intuition concrete and algorithmically useful. Such considerations are expressed in terms of Voronoi cells [18] and Delaunay simplices [5]. These are key concepts in this paper and they play a crucial role in defining, delimiting, and algorithmically constructing pockets. The algorithm is implemented and sample applications to proteins whose coordinates are available from the protein databank are given. Outline. Section 2 discusses ....

B. Delaunay. Sur la sph`ere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7 (1934), 793--800.

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

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