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46
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 114 (10 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Deformable free space tilings for kinetic collision detection
 International Journal of Robotics Research
, 2000
"... We present kinetic data structures for detecting collisions between a set of polygons that are not only moving continuously but whose shapes can also change continuously with time. We construct a planar subdivision of the common exterior of the polygons, called a pseudotriangulation, that certifies ..."
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Cited by 69 (10 self)
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We present kinetic data structures for detecting collisions between a set of polygons that are not only moving continuously but whose shapes can also change continuously with time. We construct a planar subdivision of the common exterior of the polygons, called a pseudotriangulation, that certifies their disjointness. We show different schemes for maintaining pseudotriangulations as a kinetic data structure, and we analyze their performance. Specifically, we first describe an algorithm for maintaining a pseudotriangulation of a point set, and show that the pseudotriangulation changes only quadratically many times if points move along algebraic arcs of constant degree. We then describe an algorithm for maintaining a pseudotriangulation of a set of convex polygons. Finally, we extend our algorithm to maintaining a pseudotriangulation of a set of simple polygons.
Range Searching
, 1996
"... Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 69 (1 self)
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Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.
Geometric Applications of a Randomized Optimization Technique
, 1999
"... We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated ..."
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Cited by 54 (11 self)
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We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal kpoint subsets, matching point sets under translation, computing rectilinear pcenters and discrete 1centers, and solving linear programs with k violations.
Kinetic Data Structures
, 1999
"... Modeling the physical world in the computer raises problems that intertwine discrete and continuous aspects. For example, physical objects move along continuous trajectories; yet every so often discrete events occur, such as collisions between objects. In a model of objects in space, there are many ..."
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Cited by 32 (1 self)
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Modeling the physical world in the computer raises problems that intertwine discrete and continuous aspects. For example, physical objects move along continuous trajectories; yet every so often discrete events occur, such as collisions between objects. In a model of objects in space, there are many discrete attributes that one may want to compute: the closest pair, the convex hull, the minimum spanning tree, etc. When the objects are in motion, the values of these attributes change over time, and it becomes necessary to keep track of them as the objects move. In this thesis, we introduce a general approach, and an analysis framework, for solving this type of problems. To keep track of a discrete attribute, we create a new type of data structure, called a kinetic data structure. A kinetic data structure is made of a proof of correctness of the attribute which is animated through time by a discrete event simulation.
Collision prediction for polyhedra under screw motions
 ACM Symposium in Solid Modeling and Applications
, 2003
"... The prediction of collisions amongst N rigid objects may be reduced to a series of computations of the time to first contact for all pairs of objects. Simple enclosing bounds and hierarchical partitions of the spacetime domain are often used to avoid testing objectpairs that clearly will not colli ..."
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Cited by 32 (3 self)
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The prediction of collisions amongst N rigid objects may be reduced to a series of computations of the time to first contact for all pairs of objects. Simple enclosing bounds and hierarchical partitions of the spacetime domain are often used to avoid testing objectpairs that clearly will not collide. When the remaining pairs involve only polyhedra under straightline translation, the exact computation of the collision time and of the contacts requires only solving for intersections between linear geometries. When a pair is subject to a more general relative motion, such a direct collision prediction calculation may be intractable. The popular brute force collision detection strategy of executing the motion for a series of small time steps and of checking for static interferences after each step is often computationally prohibitive. We propose instead a less expensive collision prediction strategy, where we approximate the relative motion between pairs of objects by a sequence of screw motion segments, each defined by the relative position and orientation of the two objects at the beginning and at the end of the segment. We reduce the computation of the exact collision time and of the corresponding face/vertex and edge/edge collision points to the numeric extraction of the roots of simple univariate analytic functions. Furthermore, we propose a series of simple rejection tests, which exploit the particularity of the screw motion to immediately decide that some objects do not collide or to speedup the prediction of collisions by about 30%, avoiding on average 3/4 of the rootfinding queries even when the object actually collide.
Quick Collision Detection of Polytopes in Virtual Environments
"... The problem of collision detection is fundamental to interactive applications suchascomputer animation and virtual environments. In these elds, prompt recognition of possible impacts is important for computing realtime response. We present a simple exact collision detection algorithm for convex pol ..."
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Cited by 29 (2 self)
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The problem of collision detection is fundamental to interactive applications suchascomputer animation and virtual environments. In these elds, prompt recognition of possible impacts is important for computing realtime response. We present a simple exact collision detection algorithm for convex polytopes. The algorithm nds quickly a separating plane between two polytopes if they are noncolliding, or else reports collision if it cannot possibly nd a separating plane. In the case of noncollision, the separating plane found for one time frame is cached as a witness for the next time frame, an idea borrowed from [10]; this use of time coherence further speeds up the algorithm in dynamic applications. Both temporal and geometric coherences are exploited to make this algorithm run in expected constant time empirically.
Smallest Enclosing Cylinders
, 2000
"... This paper addresses the complexity of computing the smallestradius infinite cylinder that encloses an input set of n points in 3space. We show that the problem can be solved in time O(n4 logO(1) n) in an algebraic complexity model. We also achieve a time of O(n4 L · µ(L)) in a bit complexity mo ..."
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Cited by 17 (1 self)
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This paper addresses the complexity of computing the smallestradius infinite cylinder that encloses an input set of n points in 3space. We show that the problem can be solved in time O(n4 logO(1) n) in an algebraic complexity model. We also achieve a time of O(n4 L · µ(L)) in a bit complexity model where L is the maximum bit size of input numbers and µ(L) is the complexity of multiplying two L bit integers. These and several other results highlight a general linearization technique which transforms nonlinear problems into some higherdimensional but linear problems. The technique is reminiscent of the use of Plücker coordinates, and is used here in conjunction with Megiddo’s parametric searching. We further report on experimental work comparing the practicality of an exact with that of a numerical strategy.
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