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Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 743 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 187 (15 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
On Bregman Voronoi Diagrams
 in "Proc. 18th ACMSIAM Sympos. Discrete Algorithms
, 2007
"... The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi ..."
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Cited by 61 (28 self)
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The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a byproduct, Bregman Voronoi diagrams allow one to define informationtheoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other
A discrete Laplace–Beltrami operator for simplicial surfaces
 Discrete Comput. Geom
"... Abstract We define a discrete LaplaceBeltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finiteelements Laplacian (the so called "cotan formula ..."
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Cited by 59 (3 self)
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Abstract We define a discrete LaplaceBeltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finiteelements Laplacian (the so called "cotan formula") except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete LaplaceBeltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Keywords: Laplace operator, Delaunay triangulation, Dirichlet energy, simplicial surfaces, discrete differential geometry 1 Dirichlet energy of piecewise linear functions Let S be a simplicial surface in 3dimensional Euclidean space, i.e. a geometric simplicial complex in Ê 3 whose carrier S is a 2dimensional submanifold,
Straight Skeletons for General Polygonal Figures in the Plane
, 1996
"... : A novel type of skeleton for general polygonal figures, the straight skeleton S(G) of a planar straight line graph G, is introduced and discussed. Exact bounds on the size of S(G) are derived. The straight line structure of S(G) and its lower combinatorial complexity may make S(G) preferable to th ..."
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Cited by 42 (2 self)
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: A novel type of skeleton for general polygonal figures, the straight skeleton S(G) of a planar straight line graph G, is introduced and discussed. Exact bounds on the size of S(G) are derived. The straight line structure of S(G) and its lower combinatorial complexity may make S(G) preferable to the widely used Voronoi diagram (or medial axis) of G in several applications. We explain why S(G) has no Voronoi diagram based interpretation and why standard construction techniques fail to work. A simple O(n) space algorithm for constructing S(G) is proposed. The worstcase running time is O(n 3 log n), but the algorithm can be expected to be practically efficient, and it is easy to implement. We also show that the concept of S(G) is flexible enough to allow an individual weighting of the edges and vertices of G, without changes in the maximal size of S(G), or in the method of construction. Apart from offering an alternative to Voronoitype skeletons, these generalizations of S(G) have ap...
A generic framework for parcellation of the cortical surface into gyri using geodesic Voronoi diagrams
, 2003
"... In this paper, we propose a generic automatic approach for the parcellation of the cortical surface into labeled gyri. These gyri are defined from a set of pairs of sulci selected by the user. The selected sulci are first automatically identified in the data, then projected onto the cortical surface ..."
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Cited by 38 (6 self)
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In this paper, we propose a generic automatic approach for the parcellation of the cortical surface into labeled gyri. These gyri are defined from a set of pairs of sulci selected by the user. The selected sulci are first automatically identified in the data, then projected onto the cortical surface. The parcellation stems from two nested Vorono diagrams computed geodesically to the cortical surface. The first diagram provides the zones of influence of the sulci. The boundary between the two zones of influence of each selected pair of sulci stands for a gyrus seed. A second diagram yields the gyrus parcellation. The distance underlying the Vorono diagram allows the method to interpolate the gyrus boundaries where the limiting sulci are interrupted. The method is illustrated with twelve different hemispheres.
Surface reconstruction based on a dynamical system
 In Proc. 23rd Ann. Conf. European Association for Computer Graphics (Eurographics), Computer Graphics Forum
, 2002
"... We present an efficient algorithm that computes a manifold triangular mesh from a set of unorganized sample points in 3. The algorithm builds on the observation made by several researchers that the Gabriel graph of the sample points provides a good surface description. However, this surface descrip ..."
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Cited by 38 (3 self)
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We present an efficient algorithm that computes a manifold triangular mesh from a set of unorganized sample points in 3. The algorithm builds on the observation made by several researchers that the Gabriel graph of the sample points provides a good surface description. However, this surface description is only onedimensional. We associate the edges of the Gabriel graph with index 1 critical points of a dynamical system induced by the sample points. Exploiting also the information contained in the critical points of index 2 provides a twodimensional surface description which can be easily turned into a manifold. 1.
The VisibilityVoronoi complex and its applications
 In Proc. 21st Annu. ACM Sympos. Comput. Geom. (SCG
, 2005
"... We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 ..."
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Cited by 35 (4 self)
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We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning naturallooking paths for a robot translating amidst polygonal obstacles in the plane. A naturallooking path is short, smooth, and keeps — where possible — an amount of clearance c from the obstacles. The VV (c)diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configurationspace polygonal obstacles and constructs a data structure called the VVcomplex. The VVcomplex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV (c)diagram for that cvalue. The preprocessing time is O(n 2 log n), where n is the total number of obstacle vertices, and the data structure can be queried directly for any cvalue by merely performing a Dijkstra search. We have implemented a Cgalbased software package for computing the VV (c)diagram in an exact manner for a given clearance value, and used it to plan naturallooking paths in various applications.
Towards a LeagueIndependent Qualitative Soccer Theory for RoboCup
, 2005
"... The paper discusses a topdown approach to model soccer knowledge, as it can be found in soccer theory books. The goal is to model soccer strategies and tactics in a way that they are usable for multiple RoboCup soccer leagues, i.e. for different hardware platforms. We investigate if and how socc ..."
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Cited by 34 (13 self)
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The paper discusses a topdown approach to model soccer knowledge, as it can be found in soccer theory books. The goal is to model soccer strategies and tactics in a way that they are usable for multiple RoboCup soccer leagues, i.e. for different hardware platforms. We investigate if and how soccer theory can be formalized such that specification and execution is possible. The advantage is clear: theory abstracts from hardware and from specific situations in leagues.
Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes
, 2007
"... We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a c ..."
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Cited by 27 (7 self)
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We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the nondegeneracy of the Hessian of the total scalar curvature of generalized convex polytopes with positive singular curvature. This Hessian is shown to be equal to the Hessian of the volume of the dual generalized polyhedron. We prove the nondegeneracy by applying the technique used in the proof of AlexandrovFenchel inequality. Our construction of a convex polytope from a given metric is implemented in a computer program.