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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 78 (22 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...
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 28 (13 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
A New Technique for Analyzing Substructures in Arrangements of Piecewise Linear Surfaces
, 1996
"... . We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewiselinear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of t ..."
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Cited by 26 (3 self)
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. We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewiselinear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of the lower envelope, of a single cell, or of a zone in arrangements of simplices in higher dimensions, and (b) to obtain improved bounds on the complexity of the vertical decomposition of a single cell in an arrangement of triangles in 3space, and of several other substructures in such an arrangement (the entire arrangement, all nonconvex cells, and any collection of cells). The latter results also lead to improved algorithms for computing substructures in arrangements of triangles and for translational motion planning in three dimensions. 1. Introduction The study of arrangements of curves or surfaces is an important area of research in computational and combinatorial geometry, because many...
Incidences of nottoodegenerate hyperplanes
 IN PROC. 21ST
, 2005
"... We present a multidimensional generalization of the SzemerédiTrotter Theorem, and give asharp bound on the number of incidences of points and nottoodegenerate hyperplanes in three or higherdimensional Euclidean spaces. We call a hyperplane nottoodegenerate if at most aconstant portion of its ..."
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Cited by 7 (3 self)
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We present a multidimensional generalization of the SzemerédiTrotter Theorem, and give asharp bound on the number of incidences of points and nottoodegenerate hyperplanes in three or higherdimensional Euclidean spaces. We call a hyperplane nottoodegenerate if at most aconstant portion of its incident points lie in a lower dimensional affine subspace.
On Cell Complexities in Hyperplane Arrangements
, 2000
"... We derive improved bounds on the complexity of many cells in arrangements of hyperplanes in higher dimensions, and use these bounds to obtain a very simple proof of a bound, due to [2], on the sum of squares of cell complexities in such an arrangement. 1 Complexity of Many Cells The main result of ..."
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Cited by 4 (0 self)
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We derive improved bounds on the complexity of many cells in arrangements of hyperplanes in higher dimensions, and use these bounds to obtain a very simple proof of a bound, due to [2], on the sum of squares of cell complexities in such an arrangement. 1 Complexity of Many Cells The main result of the paper, which improves upon previous bounds given in [2], is: Theorem 1.1 The complexity of m distinct cells in an arrangement of n hyperplanes in d dimensions, for d # 4, is O(m 1/2 n d/2 log (#d/2#2)/2 n) with the implied constant of proportionality depending on d. Proof: The proof proceeds by induction on d. The base case d = 4 depends on a sharper bound that is known for d = 3 and will be cited below. Let H be a collection of n hyperplanes in dspace. We will assume that the planes are in general position, meaning that any k planes meet in a d  kflat, if k = 1, . . . , d, and not at all if k > d. It is not di#cult to see that worstcase cell complexity can always be ach...
Polytopes in Arrangements
, 1999
"... Consider an arrangement of n hyperplanes in R d . Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells ..."
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Cited by 2 (1 self)
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Consider an arrangement of n hyperplanes in R d . Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the past. However, no nontrivial bound was known for the general case where the polytopes may have overlapping interiors, for d > 2. We analyze families of polytopes that do not share vertices. In R 3 we show an O(k 1=3 n 2 ) bound on the number of faces of k such polytopes. We also discuss worstcase lower bounds and higherdimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertexdisjoint polytopes in R d is k 1=2 n d=2 ) which is a factor of p n away from the best known upper bound in the range n d 2 ...
Incidence theorems for pseudoflats
"... We prove PachSharir type incidence theorems for a class of curves in R n and surfaces in R 3, which we call pseudoflats. In particular, our results apply to a wide class of generic irreducible real algebraic sets of bounded degree. 1 ..."
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Cited by 1 (0 self)
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We prove PachSharir type incidence theorems for a class of curves in R n and surfaces in R 3, which we call pseudoflats. In particular, our results apply to a wide class of generic irreducible real algebraic sets of bounded degree. 1
On distinct distances in homogeneous sets in the Euclidean space
, 2008
"... A homogeneous set of n points in the ddimensional Euclidean space determines at least Ω(n 2d/(d2 +1) /log c(d) n) distinct distances for a constant c(d)> 0. In threespace, we slightly improve our general bound and show that a homogeneous set of n points determines at least Ω(n.6071) distinct dista ..."
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A homogeneous set of n points in the ddimensional Euclidean space determines at least Ω(n 2d/(d2 +1) /log c(d) n) distinct distances for a constant c(d)> 0. In threespace, we slightly improve our general bound and show that a homogeneous set of n points determines at least Ω(n.6071) distinct distances. 1