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Lines in Space: Combinatorics and Algorithms
, 1996
"... Questions about lines in space arise frequently as subproblems in threedimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements of n lines in threedimensional space. Our main results include: 1. A tight �(n2) b ..."
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Cited by 24 (4 self)
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Questions about lines in space arise frequently as subproblems in threedimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements of n lines in threedimensional space. Our main results include: 1. A tight �(n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to the n given lines. 2. A similar bound of �(n3) for the complexity of the set of all lines passing above the n given lines. 3. A preprocessing procedure using O(n2+ε) time and storage, for anyε>0, that builds a structure supporting O(log n)time queries for testing if a line lies above all the given lines. 4. An algorithm that tests the “towering property ” in O(n4/3+ε) time, for any ε>0: do n given red lines lie all above n given blue lines? The tools used to obtain these and other results include Plücker coordinates for lines in space and εnets for various geometric range spaces.
Algebraic methods in discrete analogs of the Kakeya problem
, 2008
"... Abstract. We prove the joints conjecture, showing that for any N lines in R 3, there are at most O(N 3 2) points at which 3 lines intersect noncoplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line inters ..."
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Cited by 14 (0 self)
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Abstract. We prove the joints conjecture, showing that for any N lines in R 3, there are at most O(N 3 2) points at which 3 lines intersect noncoplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line intersects a set P of points in at least N points then the cardinality of the set of points is Ω(N 3). Both our proofs are adaptations of Dvir’s argument for the finite field Kakeya problem. 1.
Online Point Location in Planar Arrangements and Its Applications
 GEOM
, 2002
"... Recently, HarPeled [HP99b] presented a new randomized technique for online construction of the zone of a curve in a planar arrangement of arcs. In this paper, we present several applications of this technique, which yield improved solutions to a variety of problems. These applications include: ( ..."
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Cited by 10 (6 self)
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Recently, HarPeled [HP99b] presented a new randomized technique for online construction of the zone of a curve in a planar arrangement of arcs. In this paper, we present several applications of this technique, which yield improved solutions to a variety of problems. These applications include: (i) an efficient mechanism for performing online point location queries in an arrangement of arcs; (ii) an efficient algorithm for computing an approximation to the minimumweight Steinertree of a set of points, where the weight is the number of intersections between the tree edges and a given collection of arcs; (iii) a subquadratic algorithm for cutting a set of pseudoparabolas into pseudosegments; (iv) an algorithm for cutting a set of line segments (`rods') in 3space so as to eliminate all cycles in the vertical depth order; and (v) a nearoptimal algorithm for reporting all bichromatic intersections between a set R of red arcs and a set B of blue arcs, where the unions of the arcs in each set are both connected.
An improved bound for joints in arrangements of lines in space
, 2003
"... Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three noncoplanar lines meet. We show that the number of joints of L is O(n ..."
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Cited by 9 (3 self)
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Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three noncoplanar lines meet. We show that the number of joints of L is O(n
PointLine Incidences in Space
, 2002
"... Given a set L of n lines in R , joints are points in R that are incident to at least three noncoplanar lines in L. We show that there are at most O(n ) incidences between L and the set of its joints. ..."
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Cited by 5 (4 self)
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Given a set L of n lines in R , joints are points in R that are incident to at least three noncoplanar lines in L. We show that there are at most O(n ) incidences between L and the set of its joints.
On a question of Bourgain about geometric incidences
 Combinat. Probab. Comput
"... Given a set of s points and a set of n 2 lines in threedimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, then we have s = Ω(n 11/4). This is the first nontrivial answer to a question recently posed by Jean Bourgain. 1 ..."
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Cited by 4 (0 self)
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Given a set of s points and a set of n 2 lines in threedimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, then we have s = Ω(n 11/4). This is the first nontrivial answer to a question recently posed by Jean Bourgain. 1
On Lines and Joints
, 2009
"... Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplifica ..."
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Cited by 4 (0 self)
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Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the followup simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented.
Collinearities in Kinetic Point Sets
, 2011
"... Let P be a set of n points in the plane, each point moving along a given trajectory. A kcollinearity is a pair (L, t) of a line L and a time t such that L contains at least k points at time t, L is spanned by the points at time t (i.e., the points along L are not all coincident), and not all of the ..."
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Let P be a set of n points in the plane, each point moving along a given trajectory. A kcollinearity is a pair (L, t) of a line L and a time t such that L contains at least k points at time t, L is spanned by the points at time t (i.e., the points along L are not all coincident), and not all of the points are collinear at all times. We show that, if the points move with constant velocity, then the number of 3collinearities is at most 2 ()
On Distinct Distances and Incidences: Elekes’s Transformation and the New Algebraic Developments ∗
, 2010
"... We first present a transformation that Gyuri Elekes has devised, about a decade ago, from the celebrated problem of Erdős of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabo ..."
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We first present a transformation that Gyuri Elekes has devised, about a decade ago, from the celebrated problem of Erdős of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. Elekes has offered conjectures involving the new setup, which, if correct, would imply that the number of distinct distances in an selement point set in the plane is always Ω(s/log s). Unfortunately, these conjectures are still not fully resolved. We then review the recent progress made on the transformed incidence problem, based on a new algebraic approach, originally introduced by Guth and Katz. Full details of the results reviewed