TY - JOUR
ID - 3821
TI - Counting closed billiard paths
JO - AUT Journal of Mathematics and Computing
JA - AJMC
LA - en
SN - 2783-2449
AU - Rahmati, Zahed
AU - Farahzad, Sina
AU - Rahmati, Ali
AD - Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
AD - Malek-Ashtar University of Technology, Tehran, Iran
Y1 - 2020
PY - 2020
VL - 1
IS - 2
SP - 171
EP - 177
KW - Billiard Paths
KW - Maximum Path Length
KW - computational geometry
DO - 10.22060/ajmc.2020.17320.1026
N2 - 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 $1.5n−6$.
UR - https://ajmc.aut.ac.ir/article_3821.html
L1 - https://ajmc.aut.ac.ir/article_3821_7338945819e8a369d3c32dde65cdfafb.pdf
ER -