Incidences between points and lines in three dimensions

We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${\mathbb R}^3$, so that no plane contains more than $s$ lines, is $$O\left(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m + n\right)$$ (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between $m$ and $n$). This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step in their solution of Erd{\H o}s's distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth~\cite{Gu14}. The present paper presents a different and simpler derivation, with better bounds than those in \cite{Gu14}, and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions

On lines, joints, and incidences in three dimensions

AbstractWe extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R3 and m of their joints (points incident to at least three non-coplanar lines) is Θ(m1/3n) for m⩾n, and Θ(m2/3n2/3+m+n) for m⩽n. (ii) In particular, the number of such incidences cannot exceed O(n3/2). (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O(n) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O(n3/2), established by Guth and Katz, on the number of joints in a set of n lines in R3. We also present some further extensions of these bounds, and give a trivial proof of Bourgain's conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9]

Incidences with Curves in ℝ^d

We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in R^d is I(P, C) = O (mdk-d+1/^k +ε_n^(dk)−d−+1/dk-d + ∑^(d-1)/_(j=2) m/jk−j+1+/k ε_n d(j−1)(k−1)/(d−1)(jk−j+1) q_j /(d−j)(k−1) (d−1)(jk−j+1) +m + n), where the constant of proportionality depends on k, ε and d, for any ε > 0, provided that no j-dimensional surface of degree c_j (k, d, ε), a constant parameter depending on k, d, j, and ε, contains more than q_j input curves, and that the q_j ’s satisfy certain mild conditions. This bound generalizes a recent result of Sharir and Solomon [20] concerning point-line incidences in four dimensions (where d = 4 and k = 2), and partly generalizes a recent result of Guth [8] (as well as the earlier bound of Guth and Katz [10]) in three dimensions (Guth’s three-dimensional bound has a better dependency on q). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl [7], in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi [4] and by Hablicsek and Scherr [11] concerning rich lines in high-dimensional spaces

Output Sensitive Algorithms for Approximate Incidences and Their Applications

An epsilon-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most epsilon from each other. Given a set of points and a set of objects, computing the approximate incidences between them is a major step in many database and web-based applications in computer vision and graphics, including robust model fitting, approximate point pattern matching, and estimating the fundamental matrix in epipolar (stereo) geometry. In a typical approximate incidence problem of this sort, we are given a set P of m points in two or three dimensions, a set S of n objects (lines, circles, planes, spheres), and an error parameter epsilon>0, and our goal is to report all pairs (p,s) in P times S that lie at distance at most epsilon from one another. We present efficient output-sensitive approximation algorithms for quite a few cases, including points and lines or circles in the plane, and points and planes, spheres, lines, or circles in three dimensions. Several of these cases arise in the applications mentioned above. Our algorithms report all pairs at distance 1. Our algorithms are based on simple primal and dual grid decompositions and are easy to implement. We note though that (a) the use of duality, which leads to significant improvements in the overhead cost of the algorithms, appears to be novel for this kind of problems; (b) the correct choice of duality in some of these problems is fairly intricate and requires some care; and (c) the correctness and performance analysis of the algorithms (especially in the more advanced versions) is fairly non-trivial. We analyze our algorithms and prove guaranteed upper bounds on their running time and on the "distortion" parameter alpha. We also briefly describe the motivating applications, and show how they can effectively exploit our solutions. The superior theoretical bounds on the performance of our algorithms, and their simplicity, make them indeed ideal tools for these applications. In a series of preliminary experimentations (not included in this abstract), we substantiate this feeling, and show that our algorithms lead in practice to significant improved performance of the aforementioned applications