K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9(5), 223--228, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... [14] to construct the Voronoi diagram of circular arcs (not necessarily geodesic) on the sphere, as a spherical analog of the method by Alt and Schwarzkopf [1] Brown s most important discovery, however, is arguably the connection between 3 dimensional convex hulls and planar Voronoi diagrams [4, 5]. Let L be a set of points on the sphere, and consider a stereographic projection mapping the sphere to a plane. Let V be the resulting set of points in the plane. Brown showed that the Voronoi diagram of V (in the plane) has the same combinatorial structure as that part of the convex hull of L ....

K. Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters 9 (1979), pp. 223-228.

....hull of a set of points is dual to the intersection of a set of half spaces. Geometric transformations can use spaces of different dimensions : H. Edelsbrunner and F. Aurenhammer (and others) have worked a lot on results obtained owing to geometric transformations in Computational Geometry (see [Bro79] who has applied this tool to Voronoi diagram for the first time, ES86] and [Aur91] for a survey) Their approach is as follows : if we take the unit paraboloid in IE of equation ( Pi) x d 1 = x 1 x 2 : x then the intersection between Pi and the dual hyperplane p of a point ....

....The number of sites interior to the sphere is the number of associated hyperplanes in the arrangement, that pass above this intersecting point. More generally, we get : The order k Voronoi diagram of S is dual to the k level of the arrangement of the hyperplanes dual to the sites of S. In [Bro79] K.Q. Brown proved the duality between Voronoi diagram and convex hull, using inversions. In [ES86, Aur91] the proofs of such properties where purely analytic. We propose in [DMT92] a much more geometric interpretation, that avoids all calculations (see also [BCDT91] We now give a quick clue ....

K.Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9:223--228, 1979.

....(for example [4, 39] compete with divide and conquer algorithms (for example [8,42] Newer research studies randomization [41] One approach uses a lifting map (see below) to transform the triangulation problem in R to the problem of constructing the convex hull in R . This idea goes back to [7]; details on the con structions of convex hulls in d dimensions can be found in [21] Another approach is based on local transformations or flips [28,46,52] A variant of this method will be discussed in section 4.1 of this thesis. Lifting map. Identify 3 with the xlx2x3 space in 4, that is, ....

K Q Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9(5):223 228, 1979.

....d . The space of M obius transformations is generated by inversions. Examples include re ections, rotations, translations, dilations, and the well known stereographic lifting map from c R d to S d R d 1 relating d dimensional Delaunay triangulations to (d 1) dimensional convex hulls [13]. M obius transformations are conformal, meaning they locally preserve angles. There are many other conformal maps in the plane in fact, conformal maps are widely used in two dimensional mesh generation algorithms but M obius transformations are the only continuous conformal maps in dimensions ....

K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9(5):223-228, 1979.

....about looking at extra buckets. The advantages of our scheme are that it is well tuned to the case when sites are distributed in the unit square and it avoids the extra overhead of managing a priority queue, especially avoiding duplicate insertions. 1.6. Convex Hull Based Algorithms Brown [6] was the first to establish a connection between Voronoi diagrams in dimension d and convex hulls in dimension d 1. Edelsbrunner and Seidel [10] later found a correspondence between Delaunay triangles of a set of sites in dimension 2 and downward facing faces of the convex hull of those sites ....

K. Brown. Voronoi diagrams from convex hulls. IPL, pages 223--228, 1979.

.... Inserting or deleting the solid point changes the farthest point Voronoi diagram (solid and dashed lines) we maintain the convex hulls of the points in each Voronoi region (dotted lines) Proof: Since the farthest point Delaunay triangulation is the projection of a three dimensional convex hull [10], we can maintain it using Mulmuley s dynamic convex hull algorithm [31] The Voronoi diagram is dual to the Delaunay triangulation, so each change in the Voronoi diagram can be found from a corresponding change in the Delaunay triangulation. # Along with the farthest point Voronoi diagram ....

K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Proc. Lett., 9:223--226, 1979.

.... the maximum smallest enclosing circle [D AS89, Raj91] Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms as edge flipping [Laws72, Laws77] divide and conquer [ShHo75, GuSt85] geometric transformation [Brow79], plane sweep [For87] and randomized incrementation [GuKS90] Recently, Edelsbrunner, Tan, and Waupotitsch devised a polynomial time algorithm that minimizes the maximum angle [EdTW92] This algorithm constructs a minmax angle triangulation by iteratively inserting a new edge, removing old edges ....

K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9 (1979), 223-- 228.

....by determining the region in the farthest point Voronoi diagram containing that point. Lemma 8. We can maintain the farthest point Voronoi diagram in expected time O(log n) per update. 8 Proof: Since the farthest point Delaunay triangulation is the projection of a three dimensional convex hull [8], we can maintain it using Mulmuley s dynamic convex hull algorithm [26] The Voronoi diagram is dual to the Delaunay triangulation, so each change in the Voronoi diagram can be found from a corresponding change in the Delaunay triangulation. # Along with the farthest point Voronoi diagram ....

K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Proc. Lett., 9:223--226, 1979.

....by determining the region in the farthest point Voronoi diagram containing that point. Lemma 4.1. We can maintain the farthest point Voronoi diagram in expected time O(log n) per update. Proof. Since the farthest point Delaunay triangulation is the projection of a three dimensional convex hull [9], we can maintain it using Mulmuley s dynamic convex hull algorithm [23] The Voronoi diagram is dual to the Delaunay triangulation, so each change in the Voronoi diagram can be found from a corresponding change in the Delaunay triangulation. # Along with the farthest point Voronoi diagram ....

K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Proc. Lett., 9:223--226, 1979.

.... and the minimum integral of the gradient squared [Rip90] Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms as edge flipping [Laws72, Laws77] divide andconquer [ShHo75, GuSt85] geometric transformation [Brow79], plane sweep [For87] and randomized incrementation [GuKS90] Recently, polynomial time algorithms have also been found for the minmax angle and the minmax edge length criteria [EdTW92, EdTa91] The method of [EdTW92] is most relevant to this paper. It constructs a minmax angle triangulation by ....

K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9 (1979), 223-- 228.

....triangulations; it is related to many positive results known in the area. Besides the O(n 2 ) time edge flip method (Section 2. 3) Delaunay triangulation can be computed in Theta(n log n) time by diverse algorithmic paradigms such as divide and conquer [GuSt85, ShHo75] geometric transformation [Brow79], and plane sweep [Fort87] Furthermore, it can be computed in time O(n log n) with high probability by randomized incrementation [GKS92] Among all triangulations of a given point set S, the Delaunay triangulation optimizes criteria such as the max min angle [Sibs78] the min max circumscribed ....

K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9 (1979), 223--228.

....Delaunay triangulation in R d can be computed from a convex hull in R d 1 . To determine the Delaunay triangulation of a set of points: lift the points to a paraboloid and compute their convex hull. The set of ridges of the lower convex hull is the Delaunay triangulation of the original points [Brown 1979]. The intersection of halfspaces about the origin is equivalent to the convex hull of the points in dual space [Preparata and Shamos 1985] To determine the intersection of halfspaces: locate an interior point by linear programming [Teller 1994] translate the interior point to the origin, ....

....for Delaunay triangulations. They express their algorithm in terms of triangulations and the in sphere test. By the correspondence between Delaunay triangulation and convex hull, each triangle is a facet of the convex hull and the in sphere test determines the visible facets for the lifted point [Brown 1979]. 3. COMPARISON OF QUICKHULL WITH THE RANDOMIZED ALGORITHMS If Quickhull and the randomized algorithms perform essentially the same steps, why do we prefer Quickhull Quickhull uses less space than most of the randomized incremental algorithms and runs faster for distributions with non extreme ....

Brown, D. 1979. Voronoi diagrams from convex hulls. Information Processing Letters 9, 223--228.

....distance to an input point by the squared distance minus a real valued weight. For more information on Delaunay triangulations and Voronoi diagrams, see the surveys by Fortune [88] and Aurenhammer [3] There is a nice relationship between Delaunay triangulation and threedimensional convex hulls [34, 70]. Lift each point of the input to a paraboloid in three space by mapping the point with coordinates (x; y) to the point (x; y; x 2 y 2 ) The convex hull of the lifted points can be divided into lower and upper parts; a face belongs to the lower convex hull if it is supported by a plane that ....

K.Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9 (1979) 223--228.

....triangulation algorithms as well. Bowyer [30] and Watson [136] gave incremental algorithms with reasonable expected case performance. Barber [15] implemented a randomized algorithm in arbitrary dimension. This algorithm can be used to compute Delaunay triangulations through a well known reduction [31] which lifts the Delaunay triangulation of points in IR d to the convex hull of points in IR d 1 . 27 7.2 Advancing Front We have already mentioned an advancing front approach to placing Steiner points for Delaunay triangulation. A pure advancing front mesh generator [77, 79, 97, 101] ....

K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett., 9:223--228, 1979.

.... This transformation has been used, e.g. in data structures for circular range queries [8] It is a curious fact that both the nearest and farthest point Voronoi diagrams can be obtained as the planar projections of the upper and lower portions of the convex hull of the transformed point set [1, 2]. Our result provides another relation between the upper and lower hulls, in terms of the angles of their corresponding planar triangles. As a consequence of our results, the sharpest angle determined by three vertices of a convex polygon can be determined in linear time, using the linear time ....

K.Q. Brown, Voronoi diagrams from convex hulls, Inform. Process. Lett. 9:223--228, 1979.

....intersection, we can determine an internal point of the convex hull and use it as the origin for the duality transform. The origin is known to be contained in the intersection of the half spaces. For (ii) given a set of n points in the plane we apply an inversive transformation given by Brown [4] to the input points to transform the problem into nding the convex hull of n points in 3 space. iii) can be obtained in O(log n) time from the Voronoi diagram. iv) can be obtained by running a minimal spanning tree algorithm on the edges of Delaunay triangulation which is the dual graph of ....

K.Q. Brown. Voronoi diagram from convex hulls. Informat. Process Lett., 9:223 - 228.

....important geometrical concept that is used for solving a large number of problems in many areas. Accordingly, several algorithms have been devised and implemented for constructing it in two and higher dimensions (see Bentley, Weide and Yao (1980) Bowyer (1981) Brostow, Dussault and Fox (1978) Brown (1979), Finney (1979) Green and Sibson (1978) Lee and Schachter (1980) Maus (1984) Ohya, Iri and Murota (1984) Seidel (1986) Shamos (1978) Shamos and Hoey (1975) Tanemura, Ogawa and Ogita (1983) Watson (1981) Witzgall (1973b) and many of its statistical and geometrical properties have been ....

Brown K. Q. (1979), Voronoi diagrams from convex hulls, Inform. Process.

....no point in its interior and its center lies inside the convex hull of the points. Euclidean minimal spanning tree: Given a set of points, find a minimal spanning tree where the edge weights are proportional to the Euclidean distance between the points. Remark 1. 5: Using a technique of Brown [18], construction of a Voronoi diagram (in plane) is (n; 1) reducible to convex hull in three dimensions. Later in the chapter, we describe an algorithm for computing the 3 D convex hull. 1.6.3 Triangulation and Visibility Given a simple polygon (i.e. one without self intersecting edges) with n ....

K.Q. Brown. Voronoi diagram from convex hulls. Informat. Process Lett., 9:223 -- 228.

....Follows immediately from Lemma 5. 4 Summary Lemmas 2, 5 and 6 imply the following Theorem 1 The convex hull of n points in 3 space can be computed on a p processor coarse grained multicomputer with n p p 2 ffl , ffl 0, in time O( n log 2 n p Gamma n;p ) By standard duality transform [12] we obtain: Corollary 1 The Voronoi diagram of a set of n points in the Euclidean plane can be computed on a p processor coarse grained multicomputer with n p p 2 ffl , ffl 0, in time O( n log 2 n p Gamma n;p ) Note that in order to compute the Voronoi diagram the algorithm can be ....

K.Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, Vol. 9, 1979, pages 223--228.

....points each. Therefore every moving point generates Omega Gammae n Gamma k) d d 2 e ) topological events. 4 Maintaining the Topological Structure in Parallel Constructing and maintaining Voronoi diagrams is well understood in theory and practice using a single processor real RAM (cf. [8, 9, 25, 28]) In opposite to that, only in two dimensions there exist parallel algorithms for constructing Voronoi diagrams [14, 26] in O(logn) randomized time on a O(n) processor CREW PRAM. However, in higher dimensions there is no efficient algorithm known for computing the Voronoi diagram in parallel. ....

K. Brown, Voronoi diagrams from convex hulls, Info. Proc. Letters, Vol. 9, No. 5, 1979, pp 223 -- 228

.... implements general position even for point sets that are in arbitrary position. Lifting map. One can use a lifting map to transform, generally speaking, the Delaunay triangulation problem in IR d to the problem of constructing the convex hull in IR d 1 . This idea goes back to Brown [3]; details on the construction of convex hulls in d dimensions can be found texts like Edelsbrunner [9] In this paper, we will use the lifting map for d = 2; 3. Identify IR d with the x 1 x 2 : x d space in IR d 1 , that is, the subspace x d 1 = 0. The lifting map is a geometric ....

K. Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9(5):223--228, 1979.

....global communication phases with at most O(n=p) data sent received by each processor. With respect to [15] the algorithm presented here allows for an arbitrary input distribution. In particular, it allows for inputs created by mapping a 2D Voronoi diagram problem to a 3D convex hull problem [12] (which could not be handled in [15] The techniques used in this paper are very different from the ones presented in [15] and [18] The randomization methods presented are very different from the ones previously reported, e.g. in [34] and related papers. Using Valiant s terminology for the BSP ....

.... the following Theorem 1 The convex hull of n points in 3 space can be computed on a p processor coarse grained multicomputer with n=p p 2 ffl , ffl 0, in time O( n log 2 n p Gamma n;p ) By standard transformation of 2D Voronoi diagram construction to 3D convex hull computation [12] we obtain: Corollary 1 The Voronoi diagram of a set of n points in the Euclidean plane can be computed on a p processor coarse grained multicomputer with n=p p 2 ffl , ffl 0, in time O( n log 2 n p Gamma n;p ) Note that in order to compute the Voronoi diagram, the algorithm can be ....

K.Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, Vol. 9, 1979, pages 223--228.

....Delaunay triangulation in R d can be computed from a convex hull in R d 1 . To determine the Delaunay triangulation of a set of points: lift the points to a paraboloid and compute their convex hull. The set of ridges of the lower convex hull is the Delaunay triangulation of the original points [9]. The intersection of halfspaces about the origin is equivalent to the convex hull of the points in dual space [39] To determine the intersection of halfspaces: locate an interior point by linear programming [43] translate the interior point to the origin, transform halfspaces into points by ....

....for Delaunay triangulations. They express their algorithm in terms of triangulations and the in sphere test. By the correspondence between Delaunay triangulation and convex hull, each triangle is a facet of the convex hull and the in sphere test determines the visible facets for the lifted point [9]. 3. Comparison of Quickhull with the randomized algorithms If Quickhull and the randomized algorithms perform essentially the same steps, why do we prefer Quickhull Quickhull uses less space than most of the randomized incremental algorithms and runs faster for distributions with non extreme ....

D.F. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9:223-- 228, 1979.

....Delaunay triangulation) If G is a plane graph, and f is a face of G, the operation of stellating the face f consists of adding a new vertex in the interior of f and connecting all vertices incident on f to the new vertex. The following lemma, which is closely related to a result in [5], is an easy consequence of standard properties of stereographic projection [9] Lemma 2.1 A plane graph G is Delaunay realizable, with face f as its unbounded face, if and only if the graph obtained from G by stellating f is inscribable. Our proof also makes use of the following numerical ....

K. Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9(5):223--228, December 1979.

....e 2 E(G) and 0 w 0 (e i ) ffi for 1 i k, then there is an inscription I 0 of H realizing w 0 with d(I(v) I 0 (v) ffl for all v 2 V (G) The following lemma describes the connection between Delaunay tessellations and inscribable graphs. It is a different formulation from that in [3]. The operation of stellating a face f in a plane graph G consists of adding a vertex inside the face f and then connecting all vertices incident on f to the new vertex. Lemma 4.3 A plane graph G is realizable as DT(S) for some set S, with f as the unbounded face, if and only if the graph G 0 ....

K. Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9(5):223--228, December 1979.

....edges incident on v is equal to 1. W3) For each noncoterminous cutset C E(G) the total weight of all edges in C is strictly greater than 1. The following lemma describes the connection between Delaunay tessellations and graphs of inscribable type, using a different formulation from that in [3]. The proof is an immediate consequence of basic properties of stereographic projection [6] The operation of stellating a face f in a plane graph G consists of adding a vertex inside the face f and then connecting all vertices incident on f to the new vertex. Lemma 2.2 A plane graph G is ....

K. Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9(5):223--228, December 1979.

....triangulation may not be simplices, if more than d 2 points lie in a certain hypersphere. We will still use the word triangulation in this degenerate case. The fact that the Delaunay triangulation is a polyhedral complex is a corollary of the lifting property of Delaunay triangulations [4, 9]: Proposition 1.2 (lifting property) Let S be a finite point set in E d . Then, the Delaunay triangulation of S coincides with the orthogonal projection of the lower envelope of the point set S ae E d 1 obtained lifting S into the paraboloid of equation x d 1 = P d i=1 x 2 i . I,e: ....

K.Q. Brown, Voronoi Diagrams from Convex Hulls, Inf. Process. Lett. 9 (1979), 223--228.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9(5), 223--228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett., 9(5):223--228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inf. Process. Lett. 9 (5), pages 223--228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9(5), 223--228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. IPL, 9:223-228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inf. Process. Lett. 9 (5), pages 223--228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9(5), 223--228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inf. Process. Lett. 9 (5), pages 223--228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inf. Process. Lett. 9 (5), pages 223--228, 1979.

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K. Q. Brown. Voronoi Diagrams from Convex Hulls. Information Processing Letters 9:223--228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Inform. Process. Lett. 9(5), 223--228, 1979.

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K.Q. Brown. Voronoi Diagrams from Convex Hulls. Information Processing Letters, Vol. 9, pages 223-- 228, 1979.

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K.Q. Brown, "Voronoi diagrams from convex hulls", Information Processing Letters, 9(5):223 -- 228, 1979.

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K.Q. Brown. Voronoi diagrams from convex hull. Inform. Process. Lett., 9:223--228, 1979.

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K.Q. Brown, Voronoi diagrams from convex hulls, Information Processing Letters 9:223--228, 1979.

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K. Brown, Voronoi diagrams from convex hulls, Info. Proc. Letters, Vol. 9, No. 5, pp 223 -- 228, 1979.

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K. Brown, Voronoi diagrams from convex hulls, Info. Proc. Letters, Vol. 9, No. 5, 1979, pp 223 -- 228

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K. Q. Brown. Voronoi diagrams from convex hull. ipl, 9:223--228, 1979.

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K.Q. Brown. Voronoi Diagrams from Convex Hulls. Information Processing Letters, Vol. 9, pages 223-- 228, 1979.

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K. Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9(5):223--228, December 1979.

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