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73
Four Results on Randomized Incremental Constructions
 Comput. Geom. Theory Appl
, 1993
"... We prove four results on randomized incremental constructions (RICs): ffl an analysis of the expected behavior under insertion and deletions, ffl a fully dynamic data structure for convex hull maintenance in arbitrary dimensions, ffl a tail estimate for the space complexity of RICs, ffl a lower ..."
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Cited by 92 (17 self)
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We prove four results on randomized incremental constructions (RICs): ffl an analysis of the expected behavior under insertion and deletions, ffl a fully dynamic data structure for convex hull maintenance in arbitrary dimensions, ffl a tail estimate for the space complexity of RICs, ffl a lower bound on the complexity of a game related to RICs. 1
On Geometric Assembly Planning
, 1992
"... This dissertation addresses the problem of generating feasible assembly sequences for a mechanical product from a geometric model of the product. An operation specifies a motion to bring two subassemblies together to make a larger subassembly. An assembly sequence is a sequence of operations that co ..."
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Cited by 71 (12 self)
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This dissertation addresses the problem of generating feasible assembly sequences for a mechanical product from a geometric model of the product. An operation specifies a motion to bring two subassemblies together to make a larger subassembly. An assembly sequence is a sequence of operations that construct the product from the individual parts. I introduce the nondirectional blocking graph, a succinct characterization of the blocking relationships between parts in an assembly. I describe efficient algorithms to identify removable subassemblies by constructing and analyzing the NDBG. For an assembly A of n parts and m partpart contacts equivalent to k contact points, a subassembly that can translate a small distance from the rest of A can be identified in O(mk 2 ) time. When rotations are allowed as well, the time bound is O(mk 5 ). Both algorithms are extended to find connected subassemblies in the same time bounds. All free subassemblies can be identified in outputdependent ...
Voronoi Diagrams in Higher Dimensions under Certain Polyhedral Distance Functions
, 1995
"... The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R^d, the maximum complexity of its Voronoi diagram under the L1 metric, and also under a simplicial dist ..."
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Cited by 57 (25 self)
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The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R^d, the maximum complexity of its Voronoi diagram under the L1 metric, and also under a simplicial distance function, are both shown to be \Theta(n dd=2e ). The upper bound for the case of the L1 metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axisparallel hypercubes in R^d. This complexity is \Theta(n dd=2e ), for d 1, and it improves to \Theta(n bd=2c ), for d 2, if all the hypercubes have the same size. Under the L 1 metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R³ is shown to be \Theta(n 2 ). We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in d...
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
Controlled Simplification of Genus for Polygonal Models
, 1997
"... Genusreducing simplifications are important in constructing multiresolution hierarchies for levelofdetailbased rendering, especially for datasets that have several relatively small holes, tunnels, and cavities. We present a genusreducing simplification approach that is complementary to the exis ..."
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Cited by 45 (1 self)
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Genusreducing simplifications are important in constructing multiresolution hierarchies for levelofdetailbased rendering, especially for datasets that have several relatively small holes, tunnels, and cavities. We present a genusreducing simplification approach that is complementary to the existing work on genuspreserving simplifications. We propose a simplification framework in which genusreducing and genuspreserving simplifications alternate to yield much better multiresolution hierarchies than would have been possible by using either one of them. In our approach we first identify the holes and the concavities by extending the concept of # hulls to polygonal meshes under the L1 distance metric and then generate valid triangulations to fill them. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation  Display algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling  Curve, surface, solid, and object represent...
New Lower Bounds for Hopcroft's Problem
, 1996
"... We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst cas ..."
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Cited by 33 (6 self)
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We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...
Surface deformation models for nonrigid 3–d shape recovery. to appear
 IEEE Transactions on Pattern Analysis and Machine Intelligence
"... Abstract—Threedimensional detection and shape recovery of a nonrigid surface from video sequences require deformation models to effectively take advantage of potentially noisy image data. Here, we introduce an approach to creating such models for deformable 3D surfaces. We exploit the fact that the ..."
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Cited by 29 (5 self)
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Abstract—Threedimensional detection and shape recovery of a nonrigid surface from video sequences require deformation models to effectively take advantage of potentially noisy image data. Here, we introduce an approach to creating such models for deformable 3D surfaces. We exploit the fact that the shape of an inextensible triangulated mesh can be parameterized in terms of a small subset of the angles between its facets. We use this set of angles to create a representative set of potential shapes, which we feed to a simple dimensionality reduction technique to produce lowdimensional 3D deformation models. We show that these models can be used to accurately model a wide range of deforming 3D surfaces from video sequences acquired under realistic conditions. Index Terms—3D shape recovery, deformation model, nonrigid surfaces. 1
Topology Simplification for Polygonal Virtual Environments
 IEEE Transactions on Visualization and Computer Graphics
, 1998
"... We present a topology simplifying approach that can be used for genus reductions, removal of protuberances, and repair of cracks in polygonal models in a unified framework. Our work is complementary to the existing work on geometry simplification of polygonal datasets and we demonstrate that using ..."
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Cited by 25 (1 self)
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We present a topology simplifying approach that can be used for genus reductions, removal of protuberances, and repair of cracks in polygonal models in a unified framework. Our work is complementary to the existing work on geometry simplification of polygonal datasets and we demonstrate that using topology and geometry simplifications together yields superior multiresolution hierarchies than is possible by using either of them alone. Our approach can also address the important issue of repair of cracks in polygonal models as well as for rapid identification and removal of protuberances based on internal accessibility in polygonal models. Our approach is based on identifying holes and cracks by extending the concept of #shapes to polygonal meshes under the L1 distance metric. We then generate valid triangulations to fill them using the intuitive notion of sweeping a L1 cube over the identified regions. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture...
Robust Geometric Computing in Motion
, 2000
"... In this paper we discuss the gap between the theory and practice of geometric algorithms. We then describe effors to settle this gap and facilitate the successful implementation of geometric algorithms in general and of algorithms for geometric arrangements and motion planning in particular. ..."
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Cited by 24 (2 self)
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In this paper we discuss the gap between the theory and practice of geometric algorithms. We then describe effors to settle this gap and facilitate the successful implementation of geometric algorithms in general and of algorithms for geometric arrangements and motion planning in particular.
Randomized Parallel Algorithms For Trapezoidal Diagrams
, 1992
"... We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal O(A + n log n) expected work and optimal O(logn) time, where A is the number of inters ..."
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Cited by 23 (0 self)
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We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal O(A + n log n) expected work and optimal O(logn) time, where A is the number of intersecting pairs of segments. If the segments form a simple chain, we give an algorithm requiring optimal O(n) expected work and O(logn log log n log n) expected time a , and a simpler algorithm requiring O(n log n) expected work. The serial algorithm corresponding to the latter is among the simplest known algorithms requiring O(n log n) expected operations. For a set of segments forming K chains, we give an algorithm requiring O(A + n log n + K log n) expected work and O(logn log log n log n) expected time. The parallel time bounds require the assumption that enough processors are available, with processor allocations every log n steps. Keywords: randomized, parallel, trapez...