TY - JOUR
ID - 3821
TI - Counting closed billiard paths
JO - AUT Journal of Mathematics and Computing
JA - AJMC
LA - en
SN -
AU - Farahzad, Sina
AU - Rahmati, Ali
AU - Rahmati, Zahed
AD - Department of Mathematics and Computer Science, Amirkabir University of Technology
AD - Malek-Ashtar University of Technology, Tehran, Iran
AD - Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)
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 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 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◦ ) is at most 1.5n − 6.
UR - https://ajmc.aut.ac.ir/article_3821.html
L1 - https://ajmc.aut.ac.ir/article_3821_c20d01188914b161943269bccd3f81c1.pdf
ER -