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DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 418 (116 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
The Exact Computation Paradigm
, 1994
"... We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 94 (10 self)
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We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
Approximating Shortest Paths on a Convex Polytope in Three Dimensions
 J. Assoc. Comput. Mach
, 1997
"... Given a convex polytope P with n faces in IR 3 , points s; t 2 @P , and a parameter 0 ! " 1, we present an algorithm that constructs a path on @P from s to t whose length is at most (1+ ")d P (s; t), where dP (s; t) is the length of the shortest path between s and t on @P . The algorithm runs ..."
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Cited by 35 (12 self)
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Given a convex polytope P with n faces in IR 3 , points s; t 2 @P , and a parameter 0 ! " 1, we present an algorithm that constructs a path on @P from s to t whose length is at most (1+ ")d P (s; t), where dP (s; t) is the length of the shortest path between s and t on @P . The algorithm runs in O(n log 1=" + 1=" 3 ) time, and is relatively simple to implement. The running time is O(n+1=" 3 ) if we only want the approximate shortest path distance and not the path itself. We also present an extension of the algorithm that computes approximate shortest path distances from a given source point on @P to all vertices of P . Work by the first and the fourth authors has been supported by National Science Foundation Grant CCR9301259, by an Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by matching funds from Xerox Corporation. Work by the first three authors has been supported by a grant from the U.S.Israeli Binational Science ...
Approximate convex decomposition of polyhedra
 In Proc. of ACM Symposium on Solid and Physical Modeling
, 2005
"... Decomposition is a technique commonly used to partition complex models into simpler components. While decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of compo ..."
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Cited by 29 (2 self)
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Decomposition is a technique commonly used to partition complex models into simpler components. While decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this paper, we explore an alternative partitioning strategy that decomposes a given model into “approximately convex ” pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficiently. Indeed, for many applications, an approximate convex decomposition (ACD) can more accurately represent the important structural features of the model by providing a mechanism for ignoring less significant features, such as surface texture. We describe a technique for computing ACDs of threedimensional polyhedral solids and surfaces of arbitrary genus. We provide results illustrating that our approach results in high quality decompositions with very few components and applications showing that comparable or better results can be obtained using ACD decompositions in place of exact convex decompositions (ECD) that are several orders of magnitude larger. 1 ECD Figure 1: The approximate convex decompositions (ACD) of the Armadillo and the David models consist of a small number of nearly convex components that characterize the important features of the models better than the exact convex decompositions (ECD) that have orders of magnitude more components. The Armadillo (500K edges, 12.1MB) has a solid ACD with 98 components (14.2MB) that was computed in 232 seconds while the solid “ECD ” has more than 726,240 components (20+ GB) and could not be completed because disk space was exhausted after nearly 4 hours of computation. The David (750K edges, 18MB) has a surface ACD with 66 components (18.1MB) while the surface ECD has 85,132 components (20.1MB). 1
Exact Computational Geometry and Tolerancing Metrology
, 1994
"... We describe the relevance of Computational Geometry to tolerancing metrology. We outline the basic issues and define the class of zone problems that is central in this area. In the context of the exact computation paradigm, these problems are prime candidates for "exact solution" since we show that ..."
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Cited by 25 (6 self)
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We describe the relevance of Computational Geometry to tolerancing metrology. We outline the basic issues and define the class of zone problems that is central in this area. In the context of the exact computation paradigm, these problems are prime candidates for "exact solution" since we show that they have boundeddepth. Metrologists in this field have mounted a quest for a reference software which will impose some certainty in a confusing market of metrology software. The use of exact computation in the reference software will solve many intractable difficulties associated with current approaches. In short, here is a practical area in which CG and exact computation can have a real impact. 1 Introduction Researchers in Computational Geometry (CG) have always been convinced that their subject is relevant to a variety of application areas. But CG'ers have often assumed that the application areas would come to CG to find answers to their questions. To what extent is this valid? I will d...
Approximating Shortest Paths on a Nonconvex Polyhedron
 In Proc. 38th Annu. IEEE Sympos. Found. Comput. Sci
, 1997
"... We present an approximation algorithm that, given a simple, possibly nonconvex polyhedron P with n vertices in R 3 , and two points s and t on its surface @P , constructs a path on @P between s and t whose length is at most 7(1 + ")ae, where ae is the length of the shortest path between s and t ..."
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Cited by 21 (3 self)
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We present an approximation algorithm that, given a simple, possibly nonconvex polyhedron P with n vertices in R 3 , and two points s and t on its surface @P , constructs a path on @P between s and t whose length is at most 7(1 + ")ae, where ae is the length of the shortest path between s and t on @P , and " ? 0 is an arbitararily small positive constant. The algorithm runs in O(n 5=3 log 5=3 n) time. We also present a slightly faster algorithm that runs in O(n 8=5 log 8=5 n) time and returns a path whose length is at most 15(1 + ")ae. Work on this paper has been supported by National Science Foundation Grant CCR9301259, by an Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, by matching funds from Xerox Corporation, and by a grant from the U.S.Israeli Binational Science Foundation. y Department of Computer Science, Box 90129, Duke University, krv@cs.duke.edu z Department of Computer Science, Box 90129, Duke University, pa...
Applications of Computational Geometry to Geographic Information Systems
"... Contents 1 Introduction 2 2 Map Data Modeling 4 2.1 TwoDimensional Spatial Entities and Relationships . . . . . . . . . . . . . . . . . . . . . 4 2.2 Raster and Vector Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Subdivisions as Cell Complexes . . . . . . . ..."
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Cited by 21 (1 self)
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Contents 1 Introduction 2 2 Map Data Modeling 4 2.1 TwoDimensional Spatial Entities and Relationships . . . . . . . . . . . . . . . . . . . . . 4 2.2 Raster and Vector Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Subdivisions as Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Topological Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Multiresolution Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Map data processing 8 3.1 Spatial Queries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Map Overlay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Geometric Problems in Map Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Map Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
d_1Optimal Motion for a Rod (Extended Abstract)
"... ) Tetsuo Asano 1 , David Kirkpatrick 2 , and Chee K. Yap 3 1 Osaka ElectroCommunication University, Japan, 2 University of British Columbia, Canada, 3 Courant Institute, New York University, USA December 5, 1994 Abstract We study optimal motion for a rod in the plane amidst polygonal ..."
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Cited by 1 (0 self)
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) Tetsuo Asano 1 , David Kirkpatrick 2 , and Chee K. Yap 3 1 Osaka ElectroCommunication University, Japan, 2 University of British Columbia, Canada, 3 Courant Institute, New York University, USA December 5, 1994 Abstract We study optimal motion for a rod in the plane amidst polygonal obstacles. The optimality criterion is based on minimizing the orbit length of a fixed but arbitrary point (called the focus) on the rod. Our surprising result is that this problem is NPhard if when focus is in the relative interior of the rod whereas it is solvable in polynomial time if the focus is an endpoint of the rod. Other results include a local characterization of d1optimal motion and an approximation algorithm. 1 Introduction Although feasibility motion planning is very well studied, little is known about optimal motion planning except for the case where the robot body is a disc. In this paper we are interested in studying optimal motion for a rod (a directed line segment). Of c...
Two Approximative Algorithms for Calculating MinimumLength Polygons in 3D Space
"... We consider simple cubecurves in the orthogonal 3D grid. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimumlength polygonal curve (MLP) fully contained and complete in the tube ..."
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We consider simple cubecurves in the orthogonal 3D grid. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimumlength polygonal curve (MLP) fully contained and complete in the tube of the curve. So far no provable general algorithm is known for the approximative calculation of such an MLP. This paper presents two approximative algorithms for computing the MLP of a general simple cubecurve in O(n 4) time, where n is the total number of critical edges of the given simple cubecurve.
RangeSensorBased Navigation in . . .
 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH 2001; 20; 6
, 2001
"... This paper is concerned with the problem of sensorbased navigation in three dimensions. The robot, which is modeled as a “bug” or a “point robot, ” has no a priori knowledge of the environment. It must rather use its sensors to perceive the environment and plan a collisionfree path to various targ ..."
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This paper is concerned with the problem of sensorbased navigation in three dimensions. The robot, which is modeled as a “bug” or a “point robot, ” has no a priori knowledge of the environment. It must rather use its sensors to perceive the environment and plan a collisionfree path to various targets. The robot is further required to navigate in the most reactive way possible, retaining the smallest amount of information required for global convergence to the target. We assume a threedimensional polyhedral environment and present two basic results for sensorbased navigation in this environment. First we establish sufficient conditions for rangesensorbased exploration of the entire surface of a general polyhedron and present a strategy for performing this task. Then we characterize the locally shortest path from the current robot location to the target and present a method for estimating this path in time that is linear with the problem