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Anisotropic Voronoi Diagrams and GuaranteedQuality Anisotropic Mesh Generation
 in SCG ’03: Proceedings of the nineteenth annual symposium on Computational geometry
, 2003
"... We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteedquality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of aniso ..."
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Cited by 60 (2 self)
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We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteedquality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of anisotropic Voronoi diagrams of arbitrary dimensionalitymost notably circumstances in which a site can see its entire Voronoi cell. In two dimensions, the anisotropic Voronoi diagram dualizes to a triangulation under these same circumstances. We use these properties to develop an algorithm for anisotropic triangular mesh generation in which no triangle has an angle smaller than 20 # , as measured from the skewed perspective of any point in the triangle.
Spheres, Molecules, and Hidden Surface Removal
, 1996
"... We devise techniques to manipulate a collection of loosely interpenetrating spheres in threedimensional space. Our study is motivated by the representation and manipulation of molecular con gurations, modeled by a collection of spheres. We analyze the sphere model and point toitsfavorable properties ..."
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Cited by 47 (12 self)
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We devise techniques to manipulate a collection of loosely interpenetrating spheres in threedimensional space. Our study is motivated by the representation and manipulation of molecular con gurations, modeled by a collection of spheres. We analyze the sphere model and point toitsfavorable properties that make it more easy to manipulate than an arbitrary collection of spheres. For this special sphere model we present e cient algorithms for computing its union boundary and for hidden surface removal. The e ciency and practicality of our approach are demonstrated by experiments on actual molecule data.
Arrangements and their Applications in Robotics: Recent Developments
, 1995
"... this paper addresses and survey previous work on these problems. We state the basic new results in Section 3. We exemplify the usefulness of these results by applying them to problems involving robot motion planning (Section 4) and visibility and aspect graphs (Section 5). Section 6 deals with new r ..."
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Cited by 23 (9 self)
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this paper addresses and survey previous work on these problems. We state the basic new results in Section 3. We exemplify the usefulness of these results by applying them to problems involving robot motion planning (Section 4) and visibility and aspect graphs (Section 5). Section 6 deals with new results on Minkowski sums of convex polyhedra in three dimensions, which have applications in robot motion planning and in other related areas. The paper concludes in Section 7, with further applications of the new results and with some open problems.
Maintaining the Extent of a Moving Point Set
, 1997
"... Let S be a set of n moving points in the plane. We give new efficient and compact kinetic data structures for maintaining the diameter, width, and smallest area or perimeter bounding rectangle of the points. When the points in S move with pseudoalgebraic motions, these structures process O(n 2+ffl ..."
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Cited by 20 (6 self)
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Let S be a set of n moving points in the plane. We give new efficient and compact kinetic data structures for maintaining the diameter, width, and smallest area or perimeter bounding rectangle of the points. When the points in S move with pseudoalgebraic motions, these structures process O(n 2+ffl ) events. We also give constructions showing that\Omega\Gamma n 2 ) combinatorial changes are possible in these extent functions even when the points move on straight lines with constant velocities. We give a similar construction and upper bound for the convex hull, improving known results. 1 Introduction Suppose S is a set of n moving points in the plane. In this paper we investigate how to maintain various descriptors of the extent of the point set, such as diameter, width, smallest enclosing rectangle, etc. These extent measures give an indication of how spread out the point set S is and are useful in various virtual reality applications such as clipping, collision checking, etc. As...
MOTION
, 2004
"... Motion is ubiquitous in the physical world, yet its study is much less developed than that of another common physical modality, namely shape. While we have several standardized mathematical shape descriptions, and even entire disciplines devoted to that area–such as ComputerAided Geometric Design ( ..."
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Cited by 16 (1 self)
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Motion is ubiquitous in the physical world, yet its study is much less developed than that of another common physical modality, namely shape. While we have several standardized mathematical shape descriptions, and even entire disciplines devoted to that area–such as ComputerAided Geometric Design (CAGD)—the
Reverse facility location problems
 IN PROC. 17TH CANADIAN CONFERENCE ON COMPUTATIONAL GEOMETRY (CCCG’05
, 2005
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The partition technique for overlays of envelopes
 SIAM J. Comput
"... We obtain a neartight bound of O(n 3+ε), for any ε> 0, on the complexity of the overlay of the minimization diagrams of two collections of surfaces in four dimensions. This settles a longstanding problem in the theory of arrangements, most recently cited by Agarwal and Sharir [3, Open Problem 2 ..."
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Cited by 15 (8 self)
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We obtain a neartight bound of O(n 3+ε), for any ε> 0, on the complexity of the overlay of the minimization diagrams of two collections of surfaces in four dimensions. This settles a longstanding problem in the theory of arrangements, most recently cited by Agarwal and Sharir [3, Open Problem 2], and substantially improves and simplifies a result previously published by the authors [17]. Our bound is obtained by introducing a new approach to the analysis of combinatorial structures arising in geometric arrangements of surfaces. This approach, which we call the ‘partition technique’, is based on kfold divide and conquer, in which a given collection F of n surfaces is partitioned into k subcollections Fi of n/k surfaces each, and the complexity of the relevant combinatorial structure in F is recursively related to the complexities of the corresponding structures in each of the Fi’s. We introduce this approach by applying it first to obtain a new simple proof for the known nearquadratic bound on the complexity of an overlay of two minimization diagrams of collections of surfaces in R 3, thereby simplifying the previously available proof [2]. The main new bound on overlays has numerous algorithmic and combinatorial applications, some of which are presented in this paper. 1
Constructing TwoDimensional Voronoi Diagrams via DivideandConquer of Envelopes in Space
"... We present a general framework for computing twodimensional Voronoi diagrams of different site classes under various distance functions. The computation of the diagrams employs the Cgal software for constructing envelopes of surfaces in 3space, which implements a divideandconquer algorithm. A st ..."
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Cited by 9 (4 self)
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We present a general framework for computing twodimensional Voronoi diagrams of different site classes under various distance functions. The computation of the diagrams employs the Cgal software for constructing envelopes of surfaces in 3space, which implements a divideandconquer algorithm. A straightforward application of the divideandconquer approach for Voronoi diagrams yields highly inefficient algorithms. We show that through randomization, the expected running time is nearoptimal (in a worstcase sense). We believe this result, which also holds for general envelopes, to be of independent interest. We describe the interface between the construction of the diagrams and the underlying construction of the envelopes, together with methods we have applied to speed up the (exact) computation. We then present results, where a variety of diagrams are constructed with our implementation, including power diagrams, Apollonius diagrams, diagrams of line segments, Voronoi diagrams on a sphere, and more. In all cases the implementation is exact and can handle degenerate input.
Geometric optimization and sums of algebraic functions
, 2009
"... We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of nonnegative, constant descriptioncomplexity algebraic functions. We first give an FPTAS for optimizing such a sum of algebraic functions, and then we appl ..."
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Cited by 9 (1 self)
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We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of nonnegative, constant descriptioncomplexity algebraic functions. We first give an FPTAS for optimizing such a sum of algebraic functions, and then we apply it to several geometric optimization problems. We obtain the first FPTAS for two fundamental geometric shape matching problems in fixed dimension: maximizing the volume of overlap of two polyhedra under rigid motions, and minimizing their symmetric difference. We obtain the first FPTAS for other problems in fixed dimension, such as computing an optimal ray in a weighted subdivision, finding the largest axially symmetric subset of a polyhedron, and computing minimumarea hulls. 1
Arrangements on parametric surfaces II: Concretizations and applications
 IN COMPUTER SCIENCE
, 2010
"... We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The fundamental ..."
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Cited by 4 (4 self)
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We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.