## Point-Line Incidences in Space (2002)

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Citations: | 5 - 4 self |

### BibTeX

@MISC{Sharir02point-lineincidences,

author = {Micha Sharir and Emo Welzl},

title = {Point-Line Incidences in Space},

year = {2002}

}

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### Abstract

Given a set L of n lines in R , joints are points in R that are incident to at least three non-coplanar lines in L. We show that there are at most O(n ) incidences between L and the set of its joints.

### Citations

688 |
Algorithms in Combinatorial Geometry
- Edelsbrunner
- 1987
(Show Context)
Citation Context ...und is worst case tight: Choose P as the set of vertices of an M × M grid of points in the plane, for M = ⌊ √ m⌋, and L as a set of n distinct lines that maximize the number of incidences with P; see =-=[5]-=-.If, moreover, all points in P are joints of L then I(P,L) ≤ 6n, (8) since then every point in P must be incident to at least one line not in σ ∪ σ ⊥ , and each such line is incident to at most two p... |

416 | Davenport-Schinzel Sequences and Their Geometric Applications
- Sharir, Agarwal
- 1995
(Show Context)
Citation Context ...cture of reguli in 3-space (see, e.g., [13]), results from extremal graph theory for forbidden complete bipartite subgraphs (see, e.g., [9]), partition schemes from computational geometry (see, e.g., =-=[4, 12]-=-), and methods reminiscent of those developed in [3, 11] for the analysis of joints in line arrangements in space. 2 Elekes was actually interested in a similar problem formulated in the complex space... |

154 | Modern graph theory. Graduate Texts - Bollobas - 1998 |

143 | Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput - Clarkson, Edelsbrunner, et al. - 1990 |

107 | Crossing numbers and hard Erdős problems in discrete geometry
- Székely
- 1997
(Show Context)
Citation Context ...and no new ones are created. In this plane, we can apply the worst-case tight bound of O(m 2/3 n 2/3 +m+n) for point-line incidences in R 2 —a classical result of Szemerédi and Trotter [15] (see also =-=[4, 14]-=-). Hence, the same bound holds for point-line incidences in R 3 . Moreover, since we can choose all points and lines in a common plane in R 3 to begin with, the tight lower bound construction in the p... |

91 | Efficient binary space partitions for hidden-surface removal and solid modeling. Discrete Comput
- Paterson, Yao
- 1990
(Show Context)
Citation Context ...| 1.643 ) for the number of joints was shown, thereby improving on the previous bound of O(|L| 1.75 ) in [3]. An easy construction shows that the number of joints in a set of n lines can be Ω(n 3/2 ) =-=[3, 10]-=-.) We let cp = cp(L), the plane-cover number 1 of p, denote the minimum number of planes that contain all lines in Lp. Note that p is a joint iff cp ≥ 2. For P ⊆ R 3 ∑ a finite set ∑ of points, we set... |

67 |
Extremal problems in discrete geometry, Combinatorica 3
- Szemeredi, Trotter
- 1983
(Show Context)
Citation Context ...are preserved, and no new ones are created. In this plane, we can apply the worst-case tight bound of O(m 2/3 n 2/3 +m+n) for point-line incidences in R 2 —a classical result of Szemerédi and Trotter =-=[15]-=- (see also [4, 14]). Hence, the same bound holds for point-line incidences in R 3 . Moreover, since we can choose all points and lines in a common plane in R 3 to begin with, the tight lower bound con... |

66 |
Analytical Geometry of Three Dimensions
- Sommerville
- 1947
(Show Context)
Citation Context ...oes not apply to Ic, for which we have no lower bounds other than the one implied by (5). Tools. The analysis exploits the Szemerédi-Trotter bound [15], the structure of reguli in 3-space (see, e.g., =-=[13]-=-), results from extremal graph theory for forbidden complete bipartite subgraphs (see, e.g., [9]), partition schemes from computational geometry (see, e.g., [4, 12]), and methods reminiscent of those ... |

36 |
On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry
- Beck
- 1983
(Show Context)
Citation Context ...he proof of the theorem. ✷ Immediate implications. It is interesting to note that one can ‘distill’ from the preceding proof the following result, reminiscent of Beck’s theorem for lines in the plane =-=[2]-=-.� � 3.3. L be a set of n lines incident to ℓ, which meet ℓ at n/2 Let ℓ be a fixed line in 3-space, and let distinct points, such that each of these points is incident to two lines of L, and such th... |

28 |
Efficient partition trees, Discrete Comput
- Matouˇsek
- 1992
(Show Context)
Citation Context ...cles of C; |C| = |P | = m. Project the points of P on some generic plane h, so that no two points are projected to the same point. Apply to the projected set ˜ P the partitioning theorem of Matouˇsek =-=[7]-=-, which, for a given parameter r ≤ m (that will be specified shortly), yields a partitioning of ˜ P into q = O(r) subsets, call them ˜ P1, ˜ P2, . . . , ˜ Pq, each consisting of at least two points an... |

21 | Cutting circles into pseudo-segments and improved bounds on incidences, Discrete Comput. Geom - Aronov, Sharir |

18 | Counting and cutting cycles of lines and rods in space
- Chazelle, Edelsbrunner, et al.
- 1990
(Show Context)
Citation Context ...joints of L. (Joints in line arrangements have been investigated in [11], where a bound of O(|L| 1.643 ) for the number of joints was shown, thereby improving on the previous bound of O(|L| 1.75 ) in =-=[3]-=-. An easy construction shows that the number of joints in a set of n lines can be Ω(n 3/2 ) [3, 10].) We let cp = cp(L), the plane-cover number 1 of p, denote the minimum number of planes that contain... |

17 | Efficient partition trees, Discrete Comput - Matoušek - 1992 |

10 |
On joints in arrangements of lines in space and related problems
- Sharir
- 1994
(Show Context)
Citation Context ... at least three non-coplanar lines (i.e., not all lines in Lp are contained in a single plane), and we let JL denote the set of all joints of L. (Joints in line arrangements have been investigated in =-=[11]-=-, where a bound of O(|L| 1.643 ) for the number of joints was shown, thereby improving on the previous bound of O(|L| 1.75 ) in [3]. An easy construction shows that the number of joints in a set of n ... |

9 |
Constructing cuttings in theory and practice
- Har-Peled
(Show Context)
Citation Context ...|R| curves in the plane. We wish to bound the number of incidences between these points and curves. We choose some parameter r, to be determined below, and construct a (1/r)-cutting of the plane (see =-=[4, 6]-=-) into O(r 2 ) pseudo-trapezoidal cells, each crossed by at most |R|/r projected ‘reguli-curves’. (Such a cutting exists, since the projections of the curves representing the reguli are algebraic of c... |

1 | Lenses in arrangements of pseudocircles and their applications - Nevo, Pach, et al. - 2002 |

1 |
Emde Boas, Another NP-complete covering problem
- van
- 1982
(Show Context)
Citation Context ...p, and k ∈ N, deciding whether cp(L) ≥ k is NP-complete: It is (linear time) equivalent to deciding whether k lines can cover a given planar set of n points, which has been shown to be NP-complete in =-=[16]-=-.Upper Bounds. Our main result is I(JL, L) = O(n 5/3 ) (1) (see Theorem 3.1). We use this bound to derive Ic(P, L) = O(m 4/7 n 5/7 + m + n) (2) (see Theorem 4.1). Note that cp = ⌈|Lp|/2⌉ if no three ... |

1 |
Cutting circles into pseudosegments and new bounds on incidences, Discrete Comput. Geom
- Aronov, Sharir
(Show Context)
Citation Context ... where circles are tangent to each other at the given points, and where the goal is to bound the number of such tangencies. This is handled using recent tools developed for arrangements of circles in =-=[1, 8]-=-. 2. PREREQUISITES We recall some of the tools we need for our proofs. On the way, we show that many lines in a common plane or in a common regulus are counter-productive to having many incidences bet... |