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33
Fast proximity queries with swept sphere volumes
, 1999
"... We present novel algorithms for fast proximity queries using swept sphere volumes. The set of proximity queries includes collision detection and both exact and approximate separation distance computation. We introduce a new family of bounding volumes that correspond to a core primitive shape grown ..."
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Cited by 105 (20 self)
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We present novel algorithms for fast proximity queries using swept sphere volumes. The set of proximity queries includes collision detection and both exact and approximate separation distance computation. We introduce a new family of bounding volumes that correspond to a core primitive shape grown outward by some offset. The set of core primitive shapes includes a point, line, and rectangle. This family of bounding volumes provides varying tightness of t to the underlying geometry. Furthermore, we describe efficient and accurate algorithms to perform different queries using these bounding volumes. We present a novel analysis of proximity queries that highlights the relationship between collision detection and distance computation. We also present traversal techniques for accelerating distance queries. These algorithms have been used to perform proximity queries for applications including virtual prototyping, dynamic simulation, and motion planning on complex models. As compared to earlier algorithms based on bounding volume hierarchies for separation distance and approximate distance computation, our algorithms have
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 ..."
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Cited by 37 (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.
BoxTrees and Rtrees with NearOptimal Query Time
, 2001
"... A boxtree is a boundingvolume hierarchy that uses axisaligned boxes as bounding volumes. The query complexity of a boxtree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing boxtrees with ..."
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Cited by 31 (6 self)
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A boxtree is a boundingvolume hierarchy that uses axisaligned boxes as bounding volumes. The query complexity of a boxtree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing boxtrees with small worstcase query complexity with respect to queries with axisparallel boxes and with points. We also prove lower bounds on the worstcase query complexity for boxtrees, which show that our results are optimal or close to optimal. Finally, we present algorithms to convert boxtrees to Rtrees, resulting in Rtrees with (almost) optimal query complexity. 1
Algorithms for Minimum Volume Enclosing Simplex in R³
 SIAM J. Comput
, 1999
"... We develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in R 3 . If the convex hull of the points has n vertices, then our algorithm takes (n 4 ) time. Combining our exact but slow algorithm with a simple but crude approximation technique, we al ..."
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Cited by 13 (0 self)
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We develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in R 3 . If the convex hull of the points has n vertices, then our algorithm takes (n 4 ) time. Combining our exact but slow algorithm with a simple but crude approximation technique, we also develop an "approximation algorithm. The algorithm computes in O(n + 1=" 6 ) time a simplex whose volume is within (1 + ") factor of the optimal, for any " > 0. 1 Introduction Approximating a geometric body by a combinatorially simpler shape is a problem with many applications. In computer graphics and robotics, for instance, checking for collision between complex geometric models is frequently a computational bottleneck. Therefore, collision detection packages commonly use simple bounding objects, such as axisaligned bounding boxes [4, 14, 16], discrete oriented polytopes [9, 13], or spheres [10], to quickly eliminate pairs whose bounding objects are collisionfree. Since interse...
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 11 (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.
Analyzing Bounding Boxes for Object Intersection
, 1998
"... Heuristics that exploit bounding boxes are common in algorithms for rendering, modeling and animation. While experience has shown that bounding boxes improve the performance of these algorithms in practice, the previous theoretical analysis has concluded that bounding boxes perform poorly in the ..."
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Cited by 10 (1 self)
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Heuristics that exploit bounding boxes are common in algorithms for rendering, modeling and animation. While experience has shown that bounding boxes improve the performance of these algorithms in practice, the previous theoretical analysis has concluded that bounding boxes perform poorly in the worst case. This paper reconciles this discrepancy by analyzing intersections among n geometric objects in terms of two parameters: ff, an upper bound on the aspect ratio or elongatedness of each object; and oe, an upper bound on the scale factor or size disparity between the largest and smallest objects. Letting K o and K b be the number of intersecting object pairs and bounding box pairs, respectively, we analyze a ratio measure of the bounding boxes' efficiency, ae = K b =(n +K o ). The analysis proves that ae = O(ff p oe log 2 oe), and ae = \Omega\Gamma ff p oe). One important consequence is that if ff and oe are small constants (as is often the case in practice) then K b = O(K o ) + O(n), so an algorithm that uses bounding boxes has time complexity proportional to the number of actual object intersections. This theoretical result validates the efficiency that bounding boxes have demonstrated in practice. Another consequence of our analysis is a proof of the outputsensitivity of an algorithm for reporting all intersecting pairs in a set of n convex polyhedra with constant ff and oe. The algorithm takes time O(n log d\Gamma1 n + K o log d\Gamma1 n) for dimension d = 2; 3. This running time improves on the performance of previous algorithms, which make no assumptions about ff and oe. A preliminary version of this paper was presented at the 9th Symposium on Discrete Algorithms, San Francisco, 1998. y Computer Science, Washington Univ., St. Lo...
Easy Realignment of kDOP Bounding Volumes
, 2003
"... In this paper we reconsider pairwise collision detection for rigid motions using a kDOP bounding volume hierarchy. This data structure is particularly attractive because it is equally efficient for rigid motions as for arbitrary point motions (deformations). ..."
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Cited by 8 (1 self)
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In this paper we reconsider pairwise collision detection for rigid motions using a kDOP bounding volume hierarchy. This data structure is particularly attractive because it is equally efficient for rigid motions as for arbitrary point motions (deformations).
On Rtrees with low stabbing number
 In Proc. Annual European Symposium on Algorithms
, 2002
"... . The Rtree is a wellknown boundingvolume hierarchy that is suitable for storing geometric data on secondary memory. Unfortunately, no good analysis of its query time exists. We describe a new algorithm to construct an Rtree for a set of planar objects that has provably good query complexity ..."
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Cited by 7 (2 self)
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. The Rtree is a wellknown boundingvolume hierarchy that is suitable for storing geometric data on secondary memory. Unfortunately, no good analysis of its query time exists. We describe a new algorithm to construct an Rtree for a set of planar objects that has provably good query complexity for point location queries and range queries with ranges of small width. For certain important special cases, our bounds are optimal. We also show how to update the structure dynamically, and we generalize our results to higherdimensional spaces. 1 Introduction Researchers in computational geometry have developed data structures for many types of queries on geometric data: pointlocation structures, rangesearching structures, nearestneighbor searching structures, and so on. The asymptotic worstcase behavior of these data structures is usually quite goodor at least close to the theoretical lower bounds. In practice, however, other kinds of data structures are often used. One reason...
Geometric intersection
 Handbook of Discrete and Computational Geometry, chapter 33
, 1997
"... Detecting whether two geometric objects intersect and computing the region of intersection are fundamental problems in computational geometry. Geometric intersection problems arise naturally in a number of applications. Examples include geometric packing and covering, wire and component layout in VL ..."
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Cited by 6 (0 self)
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Detecting whether two geometric objects intersect and computing the region of intersection are fundamental problems in computational geometry. Geometric intersection problems arise naturally in a number of applications. Examples include geometric packing and covering, wire and component layout in VLSI, map overlay
Combinatorial and experimental methods for approximate point pattern matching
 Algorithmica
, 2003
"... Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary ..."
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Cited by 6 (0 self)
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Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high degree polynomials, and require robust calculations of intersection points of high degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average case analysis and a detailed empirical study of our methods.