%0 Journal Article
%T Counting closed billiard paths
%J AUT Journal of Mathematics and Computing
%I Amirkabir University of Technology
%Z 2783-2449
%A Rahmati, Zahed
%A Farahzad, Sina
%A Rahmati, Ali
%D 2020
%\ 09/01/2020
%V 1
%N 2
%P 171-177
%! Counting closed billiard paths
%K Billiard Paths
%K Maximum Path Length
%K computational geometry
%R 10.22060/ajmc.2020.17320.1026
%X Given a pool table enclosing a set of axis-aligned rectangles, with a total of n edges, this paper studies $\it{closed~billiard~paths}$. A closed billiard path is formed by following the ball shooting from a starting point into some direction, such that it doesn’t touch any corner of a rectangle, doesn’t visit any point on the table twice, and stops exactly at the starting position. The $\it{signature}$ of a billiard path is the sequence of the labels of edges in the order that are touched by the path, while repeated edge reflections like $abab$ are replaced by $ab$. We prove that the length of a signature is at most $4.5n−9$, and we show that there exists an arrangement of rectangles where the length of the signature is $1.25n+2$. We also prove that the number of distinct signatures for fixed shooting direction ($45^{\circ}$) is at most $1.5n−6$.
%U https://ajmc.aut.ac.ir/article_3821_7338945819e8a369d3c32dde65cdfafb.pdf