Collinearities in Kinetic Point Sets

Transcription

1 Collinearities in Kinetic Point Sets Ben Lund 1 George Purdy 1 Justin Smith 1 Csaba Tóth 2 1 Department of Computer Science 2 Department of Mathematics University of Cincinnati University of Calgary

2 k-collinearities A k-collinearity is a line L and a time t such that L contains at least k points at time t. We exclude two degenerate cases: 1. k coincident points, and 2. k always collinear points.

3 Preliminaries What is the largest number of k-collinearities admitted by any set of n kinetic points? Unless stated otherwise, collinearities are counted over the time interval (, ), and the points move in straight lines at constant speeds.

4 Trajectories The motion of a point in the (x, y)-plane...

5 Trajectories... can be represented by its trajectory in (x, y, t)-space.

6 Collisions Definition: A k-collision is a location P and a time t such that k kinetic points are at P at time t. Two kinetic points in R d collide if and only if their trajectories intersect in R d+1.

7 Collisions Theorem(Szemerédi and Trotter): 1 The number of points that lie on at least k of a set of n lines is O(n 2 /k + n/k). Corollary: The number of k-collisions among n kinetic points in R 1 is O(n 2 /k + n/k). 1 E. Szemerédi and W.T. Trotter, Jr. Extremal problems in discrete geometry. Combinatorica, (-4):81-92, 198.

8 Collisions A joint is a point incident to three noncoplanar lines. Theorem(Guth and Katz): 2 There are O(n /2 ) joints in a set of n lines. Corollary: There are O(n /2 ) -collisions among n kinetic points in R 2 if no three points are always collinear. 2 L. Guth and N.H. Katz. Algebraic methods in discrete analogs of the Kakeya problem. Advances in Mathematics, 225(5): , 2010.

9 Results A set of n kinetic points in R 2 admits no more than 2 ( ) n -collinearities. This maximum is attained by a point set with no collisions and no three points always collinear. For any fixed k, the O(n ) upper bound is tight.

10 Upper Bound on -Collinearities, Proof Idea Choose two arbitrary kinetic points, p and q.

11 Upper Bound on -Collinearities, Proof Idea Denote by S p,q the set of all point-time pairs that form a -collinearity with p and q.

12 Upper Bound on -Collinearities, Proof Idea The trajectory of a third point r will intersect S p,q at each -collinearity determined by p, q, and r.

13 Upper bound is tight for -collinearities We construct a set of n kinetic points that determine 2 ( ) n -collinearities. At time 0, the points lie on a circle centered at ( 1, 1). Each point moves with speed 1. The path of each point will carry it through the origin. p 1 p 2

14 Upper bound is tight for -collinearities t << 0

15 Upper bound is tight for -collinearities t = 0

16 Upper bound is tight for -collinearities t >> 0

17 Upper bound is asymptotically tight for k-collinearities We construct n kinetic points determining Ω(n /k 4 + n 2 /k 2 ) k-collinearities.

18 Upper bound is asymptotically tight for k-collinearities The points lie on, and move along, two parallel lines A and B. At time t = 0, k/2 points lie at each of n/k evenly spaced locations on each line. A t = 0, n = 42, k = 6 B

19 Upper bound is asymptotically tight for k-collinearities A set of k/2 coincident points on line A forms a k-collinearity with each set on line B. This gives a total of n 2 /k 2 k-collinearities. A t = 0, n = 42, k = 6 B

20 Upper bound is asymptotically tight for k-collinearities The points move with speeds {0, 1,..., k/2 1}. Each set of k points that are coincident at time 0 includes a point with each speed. t = 1, n = 42, k = 6 1 A B

21 Upper bound is asymptotically tight for k-collinearities At time t, the number of k/2-collisions on each of A and B is n/k (k/2 1)t. These collisions produce (n/k (k/2 1)t) 2 k-collinearities. t = 1, n = 42, k = 6 1 A B

22 Upper bound is asymptotically tight for k-collinearities After (n/k)/(k/2 1) units of time, there will be no further k/2-collisions. The number of k-collinearities over t = [0, ) is at least (n/k)/(k/2 1) t=0 (n/k (k/2 1)t) 2 2n/k 2 (k/2 1) 2 t 2 t=0 2n/k 2 (k/2 1) 2 = Ω(n /k 4 ). t=0 t 2

23 Open problems Open problem 1:What is the maximum number of k-collinearities among n kinetic points in the plane? Is our lower bound Ω(n /k 4 + n 2 /k 2 ) tight? Open problem 2: What is the maximum number of k-collinearities among n kinetic points in the plane if no three points are always collinear and no two points collide? Open problem : A d-collinearity in R d is called full-dimensional if not all points involved in the collinearity are in a hyperplane at all times. What is the maximum number of full-dimensional d-collinearities among n kinetic points in R d?