@article {
author = {Farahzad, Sina and Rahmati, Ali and Rahmati, Zahed},
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 = {},
eissn = {},
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 (ang{45}) 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_c20d01188914b161943269bccd3f81c1.pdf}
}