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Collision detection for deforming necklaces
 IN SYMP. ON COMPUTATIONAL GEOMETRY
, 2002
"... In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macromolecules, muscles, rope, and other ‘linear ’ objects in the physical world. In this paper, we exploit this linearity ..."
Abstract

Cited by 39 (11 self)
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In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macromolecules, muscles, rope, and other ‘linear ’ objects in the physical world. In this paper, we exploit this linearity to develop geometric structures associated with necklaces that are useful in physical simulations. We show how these structures can be implemented efficiently and maintained under necklace deformation. In particular, we study a bounding volume hierarchy based on spheres built on a necklace. Such a hierarchy is easy to compute and is suitable for maintenance when the necklace deforms, as our theoretical and experimental results show. This hierarchy can be used for collision and selfcollision detection. In particular, we achieve an upper bound of O(nlog n) in two dimensions and O(n 2−2/d) in ddimensions, d ≥ 3, for collision checking. To our knowledge, this is the first subquadratic bound proved for a collision detection algorithm using predefined hierarchies. In addition, we show that the power diagram, with the help of some additional mechanisms, can be also used to detect selfcollisions of a necklace in certain ways complementary to the sphere hierarchy.
Kinetic Medians and kdTrees
, 2002
"... We propose algorithms for maintaining two variants of kd trees of a set of moving points in the plane. A pseudo kdtree allows the number of points stored in the two children to di#er by a constant factor. ..."
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Cited by 27 (8 self)
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We propose algorithms for maintaining two variants of kd trees of a set of moving points in the plane. A pseudo kdtree allows the number of points stored in the two children to di#er by a constant factor.
Buddy Tracking  Efficient Proximity Detection Among Mobile Friends
, 2004
"... Global positioning systems (GPS) and mobile phone networks are making it possible to track individual users with an increasing accuracy. It is natural to ask whether one can use this information to maintain social networks. Here each user wishes to be informed whenever one of a list of other users, ..."
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Cited by 26 (1 self)
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Global positioning systems (GPS) and mobile phone networks are making it possible to track individual users with an increasing accuracy. It is natural to ask whether one can use this information to maintain social networks. Here each user wishes to be informed whenever one of a list of other users, called the user's friends, appears in the user's vicinity. In contrast to more traditional positioning based algorithms, the computation here depends not only on the user's own position on a static map, but also on the dynamic position of the user's friends. Hence it requires both communication and computation resources. The computation can be carried out either between the individual users in a peertopeer fashion or by centralized servers where computation and data can be collected at one central location. In the peertopeer model, a novel algorithm for minimizing the number of location update messages between pairs of friends is presented. We also present an efficient algorithm for the centralized model, based on region hierarchy and quadtrees. The paper provides an analysis of the two algorithms, compares them with a naive approach, and evaluates them using the IBM City Simulator system.
MOTION
"... 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 17 (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
Advances in Indexing for Mobile Objects
"... In this paper we discuss the latter line of research, and we review recent advances and remaining challenges.Since moving objects can be viewed as points moving along algebraic curves, the problem has a natural representation in the context of computational geometry. Hence, many results we present h ..."
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In this paper we discuss the latter line of research, and we review recent advances and remaining challenges.Since moving objects can be viewed as points moving along algebraic curves, the problem has a natural representation in the context of computational geometry. Hence, many results we present here draw heavily on therich literature of geometric indexing structures, kinetic data structures, and geometric approximation techniques.
Kinetic Medians and ��Trees
"... Abstract. We propose algorithms for maintaining two variants of ��trees of a set of moving points in the plane. A pseudo ��tree allows the number of points stored in the two children to differ by a constant factor. An overlapping ��tree allows the bounding boxes of two children to overlap. We sho ..."
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Abstract. We propose algorithms for maintaining two variants of ��trees of a set of moving points in the plane. A pseudo ��tree allows the number of points stored in the two children to differ by a constant factor. An overlapping ��tree allows the bounding boxes of two children to overlap. We show that both of them support range search operations in Ç Ò time, where ¯ only depends on the approximation precision. As the points move, we use eventbased kinetic data structures to update the tree when necessary. Both trees undergo only a quadratic number of events, which is optimal, and the update cost for each event is only polylogarithmic. To maintain the pseudo ��tree, we develop algorithms for computing an approximate median level of a line arrangement, which itself is of great interest. We show that the computation of the approximate median level of a set of lines or line segments can be done in an online fashion smoothly, i.e., there are no expensive updates for any events. For practical consideration, we study the case in which there are speedlimit restrictions or smooth trajectory requirements. The maintenance of the pseudo ��tree, as a consequence of the approximate median algorithm, can also adapt to those restrictions. 1
Abstract An Nguyen £
, 2004
"... In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macromolecules, muscles, ropes, and other linear objects in the physical world. We exploit this linearity to develop geometr ..."
Abstract
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In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macromolecules, muscles, ropes, and other linear objects in the physical world. We exploit this linearity to develop geometric structures associated with necklaces that are useful for collision detection in physical simulations. We show how these structures can be implemented efficiently and maintained under necklace deformation. In particular, we study a bounding volume hierarchy based on spheres which can be used for collision and selfcollision detection of deforming and moving necklaces. As our theoretical and experimental results show, such a hierarchy is easy to compute and, more importantly, is also easy to maintain when the necklace deforms. Using this hierarchy, we achieve a collision detection upper bound of ¦¨§�©�������©� � in two dimensions and ¦¨§�©����������� � in �dimensions, ���� �. To our knowledge, this is the first subquadratic bound proved for a collision detection algorithm using predefined hierarchies. In addition, we show that the power diagram, with the help of some additional mechanisms, can be used to
CS268: Geometric Algorithms Handout # 3 Design and Analysis
, 2007
"... 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 ( ..."
Abstract
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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