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20
Deformable spanners and applications
 In Proc. of the 20th ACM Symposium on Computational Geometry (SoCG’04
, 2004
"... For a set S of points in R d,ansspanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)spanner with O(n/ε d) edges, where ε is a spe ..."
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Cited by 49 (6 self)
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For a set S of points in R d,ansspanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)spanner with O(n/ε d) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), wellseparated pair decomposition, and approximate kcenters. 1
A Path Planning Approach for Computing LargeAmplitude Motions of Flexible Molecules
, 2005
"... Motivation: Motion is inherent in molecular interactions. Molecular flexibility must be taken into account in order to develop accurate computational techniques for predicting interactions. Energybased methods currently used in molecular modeling (i.e. molecular dynamics, Monte Carlo algorithms) ar ..."
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Cited by 42 (6 self)
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Motivation: Motion is inherent in molecular interactions. Molecular flexibility must be taken into account in order to develop accurate computational techniques for predicting interactions. Energybased methods currently used in molecular modeling (i.e. molecular dynamics, Monte Carlo algorithms) are, in practice, only able to compute local motions while accounting for molecular flexibility. However, largeamplitude motions often occur in biological processes. We investigate the application of geometric path planning algorithms to compute such large motions in flexible molecular models. Our purpose is to exploit the efficacy of a geometric conformational search as a filtering stage before subsequent energy refinements.
Collision detection for deforming necklaces
, 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 ..."
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Cited by 34 (9 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, 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
An efficient, errorbounded approximation algorithm for simulating quasistatics of complex linkages
 In ComputerAided Design
, 2006
"... Design and analysis of articulated mechanical structures, commonly referred to as linkages, is an integral part of any CAD/CAM system. The most common approaches formulate the problem as purely geometric in nature, though dynamics or quasistatics of linkages should also be considered. Existing opti ..."
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Cited by 15 (9 self)
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Design and analysis of articulated mechanical structures, commonly referred to as linkages, is an integral part of any CAD/CAM system. The most common approaches formulate the problem as purely geometric in nature, though dynamics or quasistatics of linkages should also be considered. Existing optimal algorithms that compute forward dynamics or quasistatics of linkages have a linear runtime dependence on the number of joints in the linkage. When forces are applied to a linkage, these techniques need to compute the accelerations of all the joints and can become impractical for rapid prototyping of highly complex linkages with a large number of joints. We introduce a novel algorithm that enables adaptive refinement of the forward quasistatics simulation of complex linkages. This algorithm can cull away joints whose contribution to the overall linkage motion is below a given userdefined threshold, thus limiting the computation of the joint accelerations and forces to those that contribute most to the overall motion. It also allows a natural tradeoff between the precision of the resulting simulation and the time required to compute it. We have implemented our algorithm and tested its performance on complex benchmarks consisting of up to 50,000 joints. We demonstrate that in some cases our algorithm is able to achieve up to two orders of magnitude of performance improvement, while providing a highprecision, errorbounded approximation of the quasistatics of the simulated linkage.
RealTime Knot Tying Simulation
"... While rope is arguably a simpler system to simulate than cloth, the realtime simulation of rope, and knot tying in particular, raise unique and difficult issues in contact detection and management. Some practical knots can only be achieved by complicated crossings of the rope, yielding multiple sim ..."
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Cited by 13 (0 self)
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While rope is arguably a simpler system to simulate than cloth, the realtime simulation of rope, and knot tying in particular, raise unique and difficult issues in contact detection and management. Some practical knots can only be achieved by complicated crossings of the rope, yielding multiple simultaneous contacts, especially when the rope is pulled tight. This paper describes a simulator allowing a user to grasp and smoothly manipulate a virtual rope and to tie arbitrary knots, including knots around other objects, in realtime. One component of the simulator precisely detects selfcollisions in the rope, as well as collisions with other objects. Another component manages collisions to prevent penetration, while making the rope slide with some friction along itself and other objects, so that knots can be pulled tight in believable manner. An additional module uses recent results from knot theory to identify which topological knots have been tied, also in realtime. This work was motivated by surgical suturing, but simulation in other domains, such as sailing and rock climbing, could benefit from it.
Algorithm and data structures for efficient energy maintenance during Monte Carlo simulation of proteins
 Journal of Computational Biology
, 2004
"... Monte Carlo simulation (MCS) is a common methodology to compute pathways and thermodynamic properties of proteins. A simulation run is a series of random steps in conformation space, each perturbing some degrees of freedom of the molecule. A step is accepted with a probability that depends on the c ..."
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Cited by 9 (2 self)
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Monte Carlo simulation (MCS) is a common methodology to compute pathways and thermodynamic properties of proteins. A simulation run is a series of random steps in conformation space, each perturbing some degrees of freedom of the molecule. A step is accepted with a probability that depends on the change in value of an energy function. Typical energy functions sum many terms. The most costly ones to compute are contributed by atom pairs closer than some cutoff distance. This paper introduces a new method that speeds up MCS by exploiting the facts that proteins are long kinematic chains and that few degrees of freedom are changed at each step. A novel data structure, called the ChainTree, captures both the kinematics and the shape of a protein at successive levels of detail. It is used to efficiently detect selfcollision (steric clash between atoms) and/or find all atom pairs contributing to the energy. It also makes it possible to identify partial energy sums left unchanged by a perturbation, thus allowing the energy value to be incrementally updated. Computational tests on four proteins of sizes ranging from 68 to 755 amino acids show that MCS with the ChainTree method is significantly faster (as much as 10 times faster for the largest protein) than with the widely used grid method. They also indicate that speedup increases with larger proteins.
Local polyhedra and geometric graphs
 In Proc. 14th ACMSIAM Sympos. on Discrete Algorithms
, 2003
"... We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest ed ..."
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Cited by 8 (0 self)
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We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest edges differ by at most a polynomial factor. A polyhedron is local if all its faces are simplices and its edges form a local geometric graph. We show that any boolean combination of any two local polyhedra in IR d each with n vertices, can be computed in O(n log n) time, using a standard hierarchy of axisaligned bounding boxes. Using results of de Berg, we also show that any local polyhedron in IR d has a binary space partition tree of size O(n log d1 n). Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions.
BioCD: An efficient algorithm for selfcollision and distance computation between highly articulated molecular models
, 2005
"... This paper describes an efficient approach to (self) collision detection and distance computations for complex articulated mechanisms such as molecular chains. The proposed algorithm called BioCD is particularly designed for samplingbased motion planning on molecular models described by long kinemat ..."
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Cited by 8 (3 self)
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This paper describes an efficient approach to (self) collision detection and distance computations for complex articulated mechanisms such as molecular chains. The proposed algorithm called BioCD is particularly designed for samplingbased motion planning on molecular models described by long kinematic chains possibly including cycles. The algorithm considers that the kinematic chain is structured into a number of rigid groups articulated by preselected degrees of freedom. This structuration is exploited by a twolevel spatiallyadapted hierarchy. The proposed algorithm is not limited to particular kinematic topologies and allows good collision detection times. BioCD is also tailored to deal with the particularities imposed by the molecular context on collision detection. Experimental results show the effectiveness of the proposed approach which is able to process thousands of (self) collision tests per second on flexible protein models with up to hundreds of degrees of freedom.