1 Cutting sequence and Sturmian sequence in billiard

2025-04-28 0 0 534.05KB 29 页 10玖币

侵权投诉

Cutting sequence and Sturmian sequence in

billiard

台球中的切割序列和 Sturmian 序列

Zhiyu.Liu18

May 4, 2022

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.

Key word: cutting sequence, Sturmian sequence, continued fractions

摘要

台球的获胜规则是把台球桌上的球打入指定的洞中。我们试图研究台球的运

动轨迹，这样我们就可以预测球的运动方向。对于有理数斜率，通过建立方形环

面得到切割顺序。我们利用剪切和翻转简化了切割顺序，得到了轨迹斜率与切割

顺序之间的变换规则。对于无理数斜率，我们研究了 Sturmian 序列的一些性质，

这有助于我们区分切割序列和 Sturmian 序列。综上所述，在不同斜率的情况下，

我们使用不同的序列来进行研究。


 
Contents 
Introduction ............................................................................................. 4 
A special billiard: square .................................................................... 4 
Unfolding the square table ................................................................. 6 
1 Unfolding a trajectory into a straight line .............................. 6 
2 the square torus .................................................................................. 8 
Cutting sequences .................................................................................... 9 
shearing the square torus ................................................................... 12 
continued fractions and cutting sequences................................... 14 
Every shear can be expressed by basic shears ............................ 16 
Sturmian sequence. Basic properties ............................................. 18 
Continued fractions and Sturmian sequences.............................. 24 
Compare with cutting sequence and Sturmian sequences ... 25 
conclusion ............................................................................................... 26 
 
 
 
 
 
 

1. Introduction

Billiards are one of the most popular ball games in the world in recent years. In this

sport, we hit the ball on the table with a long club in order to make the ball fall into the

hole. According to Newton's first law of motion, in the absence of any external force,

the ball will always move in a straight line. However, the ball does not usually move in

a straight line on a square table and will change direction once it hits the edge of the

table. What we're interested in is the billiard ball's trajectory. We can plot the trajectory

of the ball step by step, using the angle of reflection is equal to the angle of incidence.

Actually, drawing the full trajectory is tedious. Therefore, we try to figure out how to

identify if the trajectory is periodic or non-periodic, and find the period. Actually,

billiard tables are rectangular. To facilitate my research, we study with a square which

is a special quadrilateral. We learned lots of useful methods and tools to express the

trajectory in an easier way. We find when the slope of the trajectory is rational, the

trajectory is periodic. First of all, we generated the torus table by square table to get a

sequence named cutting sequence, and we used this periodic sequence to simplify

complicated trajectories. Then we had a series of studies on cutting sequence, such as

shear and flip. This led transform between trajectory slope and cutting sequence

available. We only need to know one of them to derive the other. Moreover, the

trajectory with irrational slope cannot be ignored. We get Sturmian sequence, and we

learn its properties and compare it with cutting sequence we learned early.

2. A special billiard: square

First of all, Consider the most special case of a polygonal table: a square, because

squares are centrosymmetric. To visualize the trajectory of the ball, let us assume that

the ball is a little point, and there is no friction so that it runs forever. More importantly,

when balls hit the edge of the table, the angle of reflection is equal to the angle of

incidence. (We assume the ball never hits a vertex of square.) If we hit the ball vertically

or horizontally, it will bounce between two points on parallel edges again and again.

We say that this trajectory is periodic, because periodic trajectories mean that the ball

repeats its path all the time. Once the ball starts moving, it goes back to the starting

point after two collisions, so this trajectory has a period of 2. If we change the Angle of

the strike, we can also draw the trajectory of the periods which are more than 2.

Figure 1: Trajectories with period 2 and 4

Can we find any trajectories with period 3? The answer is no. We hit a ball at an

Angle of , then we can get 1=180-2, 3=90-, 4=180-23=2. Finally, we

get 1+4=180, and we know that the angles of a triangle add up to 180 degrees so

there is no triangular in the trajectory. Similarly, we cannot find any trajectories with

odd period.

Figure 2

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

1CuttingsequenceandSturmiansequenceinbilliard台球中的切割序列和Sturmian序列Zhiyu.Liu18May4,20222AbstractThewinningruleofbilliardsistodrivethebilliardballonthetableintothedesignatedholes.Wetrytostudythetrajectoryofthebilliardball,sothatwecanpredictthedirectionoftheball.Forrationalslopes,wegotcuttingsequencebyse...

展开>> 收起<<

1 Cutting sequence and Sturmian sequence in billiard.pdf

共29页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

1 Cutting sequence and Sturmian sequence in billiard

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: