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Incidences Between Points and Circles in Three and Higher Dimensions
 

Summary: Incidences Between Points and Circles in Three and Higher
Dimensions 
Boris Aronov y Vladlen Koltun z Micha Sharir x
June 24, 2002
Abstract
We show that the number of incidences between m distinct points and n distinct circles in
R d , for any d  3, is O(m 6=11 n 9=11 (m 3 =n)+m 2=3 n 2=3 +m+n), where (n) = (log n) O( 2 (n)) ,
and where (n) is the inverse Ackermann function. The bound coincides with the recent
bound of Aronov and Sharir [5], as slightly improved by Agarwal et al. [1], for the planar case.
We also show that the number of incidences between m points and n arbitrary convex plane
curves, no two in a common plane, is O(m 4=7 n 17=21 +m 2=3 n 2=3 +m+ n), in any dimension
d  3. Our results improve the upper bound on the number of congruent copies of a xed
tetrahedron in a set of n points in 4-space, and the lower bound for the number of distinct
distances in a set of n points in 3-space.
1 Introduction
In the main result of this paper, we obtain an improved upper bound for the number of incidences
between m points and n arbitrary circles in three dimensions. 1 The study of the number of
incidences between points in the plane and curves of various types has an extensive history, and
a variety of nontrivial upper (and, more rarely, lower) bounds have been obtained:
 For lines and pseudolines, the maximum number of incidences between m points and n

  

Source: Aronov, Boris - Department of Computer Science and Engineering, Polytechnic Institute of New York University
Sharir, Micha - School of Computer Science, Tel Aviv University

 

Collections: Computer Technologies and Information Sciences