Collinearities in Kinetic Point Sets

TL;DR

This paper investigates the maximum number of collinearities among moving points in the plane, establishing tight bounds for the number of 3-collinearities and general k-collinearities under constant velocity motion.

Contribution

It provides tight bounds on the number of k-collinearities among moving points, including the maximum count for 3-collinearities and constructions with no three points ever collinear.

Findings

Maximum 3-collinearities is 2 * C(n,3), tight bound.

Number of k-collinearities is O(n^3) for constant k, tight asymptotic bound.

Existence of point sets with no three points ever collinear.

Abstract

Let $P$ be a set of $n$ points in the plane, each point moving along a given trajectory. A {\em $k$-collinearity} is a pair $(L,t)$ of a line $L$ and a time $t$ such that $L$ contains at least $k$ points at time $t$, the points along $L$ do not all coincide, and not all of them are collinear at all times. We show that, if the points move with constant velocity, then the number of 3-collinearities is at most $2\binom{n}{3}$, and this bound is tight. There are $n$ points having $\Omega(n^3/k^4 + n^2/k^2)$ distinct $k$-collinearities. Thus, the number of $k$-collinearities among $n$ points, for constant $k$, is $O(n^3)$, and this bound is asymptotically tight. In addition, there are $n$ points, moving in pairwise distinct directions with different speeds, such that no three points are ever collinear.