Cutting sequence and Sturmian sequence in billiard

TL;DR

This paper investigates billiard ball trajectories using cutting sequences for rational slopes and Sturmian sequences for irrational slopes, providing a framework to predict ball directions based on sequence transformations.

Contribution

It introduces a method to relate billiard trajectories to cutting and Sturmian sequences, offering new insights into trajectory analysis for different slopes.

Findings

Transformation between trajectory slope and cutting sequence established

Differentiation between cutting and Sturmian sequences for irrational slopes

Sequence-based approach aids in predicting billiard ball directions

Abstract

The winning rule of billiards is to drive the billiard ball on the table into the designated holes. We try to study the trajectory of the billiard ball, so that we can predict the direction of the ball. For rational slopes, we got cutting sequence by setting up the square torus. We simplified cutting sequence using shearing and flipping and we obtain the transformation between trajectory slope and cutting sequence. For irrational slopes, we look at some properties of Sturmian sequence, which help us distinguish between cutting sequence and Sturmian sequence. In conclusion, in the case of different slopes, we use different sequences to do research.