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31
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...
Algorithmic Motion Planning
, 1997
"... INTRODUCTION Motion planning is a fundamental problem in robotics. It comes in a variety of forms, but the simplest version is as follows. We are given a robot system B, which may consist of several rigid objects attached to each other through various joints, hinges, and links, or moving independen ..."
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Cited by 47 (6 self)
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INTRODUCTION Motion planning is a fundamental problem in robotics. It comes in a variety of forms, but the simplest version is as follows. We are given a robot system B, which may consist of several rigid objects attached to each other through various joints, hinges, and links, or moving independently, and a twodimensional or threedimensional environment V cluttered with obstacles. We assume that the shape and location of the obstacles and the shape of B are known to the planning system. Given an initial placement Z 1 and a nal placement Z 2 of B, we wish to determine whether there exists a collisionavoiding motion of B from Z 1 to Z 2 , and, if so, to plan such a motion. In this simpli ed and purely geometric setup, we ignore issues such as incomplete information, nonholonomic constraints, control issues related to inaccuracies in sensing and motion, nonstationary obstacles, optimality of the planned motion, and so on. Since the early 1980's, motion planning has been an intensiv
New Results on Quantifier Elimination Over Real Closed Fields and Applications to Constraint Databases
 Journal of the ACM
, 1999
"... In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 28, 22] the combinatorial part of the complexity (the part depen ..."
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Cited by 42 (5 self)
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In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 28, 22] the combinatorial part of the complexity (the part depending on the number of polynomials in the input) of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity (the part depending on the degrees of the input polynomials) can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantifier elimination problem. Using the uniform quantifier elimination algorithm, we give a...
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
Arrangements and their Applications in Robotics: Recent Developments
, 1995
"... this paper addresses and survey previous work on these problems. We state the basic new results in Section 3. We exemplify the usefulness of these results by applying them to problems involving robot motion planning (Section 4) and visibility and aspect graphs (Section 5). Section 6 deals with new r ..."
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Cited by 23 (9 self)
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this paper addresses and survey previous work on these problems. We state the basic new results in Section 3. We exemplify the usefulness of these results by applying them to problems involving robot motion planning (Section 4) and visibility and aspect graphs (Section 5). Section 6 deals with new results on Minkowski sums of convex polyhedra in three dimensions, which have applications in robot motion planning and in other related areas. The paper concludes in Section 7, with further applications of the new results and with some open problems.
Hybrid motion planning: Coordinating two discs moving among polygonal obstacles in the plane
 In Workshop on the Algorithmic Foundations of Robotics
, 2002
"... The basic motionplanning problem is to plan a collisionfree motion for objects moving among obstacles between free initial and goal positions, or to determine that no such motion exists. The basic problem as well as numerous variants of it have been intensively studied over the past two decades yi ..."
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Cited by 21 (6 self)
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The basic motionplanning problem is to plan a collisionfree motion for objects moving among obstacles between free initial and goal positions, or to determine that no such motion exists. The basic problem as well as numerous variants of it have been intensively studied over the past two decades yielding a wealth of results and techniques, both theoretical and practical. In this paper, we propose a novel approach to motion planning, hybrid motion planning, in which we integrate complete solutions along with Probabilistic Roadmap (PRM) techniques in order to combine their strengths and offset their weaknesses. We incorporate robust tools, that have not been available before, in order to implement the complete solutions. We exemplify our approach in the case of two discs moving among polygonal obstacles in the plane. The planner we present easily solves problems where a narrow passage in the workspace can be arbitrarily small. Our planner is also capable of providing correct nontrivial “no ” answers, namely it can, for some queries, detect the situation where no solution exists. We envision our planner not as a total solution but rather as a new tool that cooperates with existing planners. We demonstrate the advantages and shortcomings of our planner with experimental results. 1
On the Combinatorial and Topological Complexity of a Single Cell
, 1998
"... The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of n geometric objects in R k of constant description complexity is an important problem in computational geometry which has attracted much attention over the past decad ..."
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Cited by 18 (7 self)
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The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of n geometric objects in R k of constant description complexity is an important problem in computational geometry which has attracted much attention over the past decade. It has been conjectured that the combinatorial complexity of a single cell is bounded by a function much closer to O(n k 1 ) rather than O(n k ) which is the bound for the combinatorial complexity of the whole arrangement. Till now, this was known to be true only for k 3 and only for some special cases in higher dimensions. A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. However, till now no better bounds were known if we restricted attention to a single connected component of a basic semialgebraic set. In this paper, we show how these two problems...
Different bounds on the different Betti numbers of semialgebraic sets
 Proceedings of the ACM Symposium on Computational Geometry
, 2001
"... A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no signif ..."
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Cited by 18 (8 self)
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A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove...
An Improved Algorithm for Quantifier Elimination Over Real Closed Fields
 IEEE FOCS
, 1997
"... In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 25, 20] the combinatorial part of the complexity of this new alg ..."
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Cited by 17 (0 self)
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In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 25, 20] the combinatorial part of the complexity of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantifier elimination problem and apply it to solve a problem arising in the field of constraint databases. No algorithm with reasonable complexity bound was known for this latter problem till now. We also point out i...