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Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 81 (17 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
A Perturbation Scheme for Spherical Arrangements with Application to Molecular Modeling
, 1997
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Arrangements
, 1997
"... INTRODUCTION Given a finite collection S of geometric objects such as hyperplanes or spheres in R d , the arrangement A(S) is the decomposition of R d into connected open cells of dimensions 0; 1; : : :; d induced by S. Besides being interesting in their own right, arrangements of hyperplanes ..."
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Cited by 37 (14 self)
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INTRODUCTION Given a finite collection S of geometric objects such as hyperplanes or spheres in R d , the arrangement A(S) is the decomposition of R d into connected open cells of dimensions 0; 1; : : :; d induced by S. Besides being interesting in their own right, arrangements of hyperplanes have served as a unifying structure for many problems in discrete and computational geometry. With the recent advances in the study of arrangements of curved (algebraic) surfaces, arrangements have emerged as the underlying structure of geometric problems in a variety of `physical world' application domains such as robot motion planning and computer vision. This chapter is devoted to arrangements of hyperplanes and of curved surfaces in lowdimensional Euclidean space, with an emphasis on combinatorics and algorithms. In the first section we in
Dense Point Sets Have Sparse Delaunay Triangulations
"... Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms ..."
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Cited by 29 (2 self)
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Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since threedimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worstcase running time \Omega (n2). However, this behavior is almost never observed in practice except for highlycontrived inputs. For all practical purposes, threedimensional Delaunay triangulations appear to have linear complexity. This frustrating
Analysis of a bounding box heuristic for object intersection
 Journal of the ACM
, 1999
"... Abstract. Bounding boxes are commonly used in computer graphics and other fields to improve the performance of algorithms that should process only the intersecting objects. A boundingboxbased heuristic avoids unnecessary intersection processing by eliminating the pairs whose bounding boxes are dis ..."
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Cited by 29 (4 self)
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Abstract. Bounding boxes are commonly used in computer graphics and other fields to improve the performance of algorithms that should process only the intersecting objects. A boundingboxbased heuristic avoids unnecessary intersection processing by eliminating the pairs whose bounding boxes are disjoint. Empirical evidence suggests that the heuristic works well in many practical applications, although its worstcase performance can be bad for certain pathological inputs. What is a pathological input, however, is not well understood, and consequently there is no guarantee that the heuristic will always work well in a specific application. In this paper, we analyze the performance of bounding box heuristic in terms of two natural shape parameters, aspect ratio and scale factor. These parameters can be used to realistically measure the degree to which the objects are pathologically shaped. We derive tight worstcase bounds on the performance for bounding box heuristic. One of the significant contributions of our paper is that we only require that objects be well shaped on average. Somewhat surprisingly, the bounds are significantly different from the case when all objects are well shaped.
Robust Geometric Computing in Motion
 INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH
, 2002
"... Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning complexity measures and (more crucially) the handling of robustness issues, namely is ..."
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Cited by 25 (2 self)
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Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning complexity measures and (more crucially) the handling of robustness issues, namely issues related to arithmetic precision and degenerate input. The paper starts with a discussion of the gap between the theory and practice of geometric algorithms, together with a brief review of existing solutions to some of the problems that this dichotomy brings about. We then turn to an overview of the CGAL project and library. The CGAL project is a joint effort by a number of research groups in Europe and Israel to produce a robust software library of geometric algorithms and data structures. The library is now available for use with significant functionality. We describe the main goals and results of the project. The central part of the paper is devoted to arrangements (i.e., space subdivisions induced by geometric objects) and motion planning. We concentrate on the maps and arrangements part of the CGAL library. Then we describe two packages developed on top of CGAL for constructing robust geometric primitives for motion algorithms.
Efficient Maintenance and SelfCollision Testing for Kinematic Chains
, 2002
"... The kinematic chain is a ubiquitous and extensively studied representation in robotics as well as a useful model for studying the motion of biological macromolecules. Both fields stand to benefit from algorithms for efficient maintenance and collision detection in such chains. This paper introduces ..."
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Cited by 20 (3 self)
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The kinematic chain is a ubiquitous and extensively studied representation in robotics as well as a useful model for studying the motion of biological macromolecules. Both fields stand to benefit from algorithms for efficient maintenance and collision detection in such chains. This paper introduces a novel hierarchical representation of a kinematic chain allowing for efficient incremental updates and relative position calculation. A hierarchy of oriented bounding boxes is superimposed on top of this representation, enabling high performance collision detection, selfcollision testing, and distance computation. This representation has immediate applications in the field of molecular biology, for speeding up molecular simulations and studies of folding paths of proteins. It could be instrumental in path planning applications for robots with many degrees of freedom, also known as hyperredundant robots. A comparison of the performance of the algorithm with the current state of the art in collision detection is presented for a number of benchmarks.
A Method for Biomolecular Structural Recognition and Docking Allowing Conformational Flexibility
, 1997
"... In this work we present an algorithm developed to handle biomolecular structural recognition problems, as part of an interdisciplinary research endeavor of the Computer Vision and Molecular Biology fields. A key problem is rational drug design and in biomolecular structural recognition is the genera ..."
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Cited by 18 (1 self)
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In this work we present an algorithm developed to handle biomolecular structural recognition problems, as part of an interdisciplinary research endeavor of the Computer Vision and Molecular Biology fields. A key problem is rational drug design and in biomolecular structural recognition is the generation of binding modes between two molecules, also known as molecular docking. Geometrical fitness is a necessary condition for molecular interaction. Hence, docking a ligand (e.g., a drug molecule or a protein molecule), to a protein receptor (e.g., enzyme), involves recognition of molecular surfaces. Conformational transitions by `hingebending' involves rotational movements of relatively rigid parts with respect to each other. The generation of docked binding modes between two associating molecules depends on their three dimensional structures (3D) and their conformational flexibility. In comparison to the particular case of rigidbody docking, the computational difficulty grows considera...
Motion Planning for a Convex Polygon in a Polygonal Environment
 Geom
, 1997
"... We study the motionplanning problem for a convex mgon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3dimensional space of all free placements of P in Q) in time that is nearquadratic in mn, which i ..."
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Cited by 17 (8 self)
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We study the motionplanning problem for a convex mgon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3dimensional space of all free placements of P in Q) in time that is nearquadratic in mn, which is nearly optimal in the worst case. The algorithm is also conceptually relatively simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also nearquadratic in mn. In addition, we describe an algorithm that preprocesses the computed free configuration space so that `reachability' queries can be answered in polylogarithmic time. All three authors have been supported by a grant from the U.S.Israeli Binational Science Foundation. Pankaj Agarwal has also been supported by a National Science Foundation Gr...
Motion Planning in Environments with Low Obstacle Density
, 1997
"... We present a simple and efficient paradigm for computing the exact solution of the motion planning problem in environments with a low obstacle density. Such environments frequently occur in practical instances of the motion planning problem. The complexity of the free space for such environments is ..."
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Cited by 16 (7 self)
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We present a simple and efficient paradigm for computing the exact solution of the motion planning problem in environments with a low obstacle density. Such environments frequently occur in practical instances of the motion planning problem. The complexity of the free space for such environments is known to be linear in the number of obstacles. Our paradigm is a new cell decomposition approach to motion planning and exploits properties that follow from the low density of the obstacles in the robot's workspace. These properties allow us to decompose the workspace, subject to some constraints, rather than to decompose the higherdimensional free configuration space directly. A sequence of uniform steps transforms the workspace decomposition into a free space decomposition of asymptotically the same size. The approach leads to nearlyoptimal O(n log n) motion planning algorithms for freeflying robots with any fixed number of degrees of freedom in workspaces with low obstacle density.