@article {
author = {Rahmati, Zahed and Farahzad, Sina and Rahmati, Ali},
title = {Counting closed billiard paths},
journal = {AUT Journal of Mathematics and Computing},
volume = {1},
number = {2},
pages = {171-177},
year = {2020},
publisher = {Amirkabir University of Technology},
issn = {2783-2449},
eissn = {2783-2287},
doi = {10.22060/ajmc.2020.17320.1026},
abstract = {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 $1.5n−6$.},
keywords = {Billiard Paths,Maximum Path Length,computational geometry},
url = {https://ajmc.aut.ac.ir/article_3821.html},
eprint = {https://ajmc.aut.ac.ir/article_3821_7338945819e8a369d3c32dde65cdfafb.pdf}
}