The trajectory of a ball on a billiard table can be plotted if the translational velocity, turning angle velocity, and tilt angle of rotation axis are known.
2011.1　　3cplayerkk　　　　　　　　　kobe　　K.Kobayashi


Content : 　・Almost all movements of a cue ball in three cushion ・How to strike the ball without throw in pocket billiards・Analysis of masse shot and its application.

(2015.4.2 Text, app, and samples of the Masse analysis are now available.)
(2014.11 Confirmation app is added. 2014.11.6 Analysis of slipping part is revised.）
(2013.6 The text of Analysis is now available.)


No one has never wanted to know the movement of the cue ball as to how it slips or rolls on the table to collide balls and cushions. But most of the billiard lovers are playing without a clue on it. I had been one of them.

1. The Movement of Cue Ball in Three Cushion
I had an opportunity to study the rolling of the ball a few years ago. I learnt that the studies at the time generally have analyzed the case in which the rotation axis is horizontal. I have devoted myself to the analysis since then, obtaining before long a general formula of apparent rolling resistance of a ball rolling with the rotation axis in a arbitrary tilt angle. This formula matches well with the actual movement of the ball. Then, I carried on the analysis to study the ball immediately after the strike with a cue, collision of balls, and then collision with cushion. The fundamental part of my analysis was substantially completed at the end of 2009.

The analysis was time-consuming and required patience without much precedents to refer. But the preparation of the confirmation program took much longer.

The confirmation program, which is a simulator per se, produces results that match well with the reality, thereby being a tool sufficient to analyze the quasi movement of a ball. For example, the program makes it easy to repeatedly examine the course of balls of a certain arrangement with the collision with identical ball overlap and speed and different strike points. The program also makes it possible to examine difficult ball arrangement under any conditions. It sometimes leads to a solution, which is out of imagination usually.
The part of the movement was posted on YouTube early 2010. The tag is Billiard 3C Simulator etc.

The analysis are itemized as follows.

・Behavior of the cue ball struck with the cue : Turning angle velocity, tilt angle of the rotation axis, etc. of the cue ball collided at a arbitrary strike point.
　　　　　　(Details of Behavior 1　　2011-3-28 rev1)　

・Changes in behavior of the cue ball before the collision with the object ball : 　Changes in the translational velocity, angle velocity, and rotation axis tilt angle of the cue ball in the state of slipping or rolling before collision with the object ball.

・Behavior of the cue ball immediately after the collision with the object ball : 　The speed, moving angle, angle speed, rotation axis tilt angle, etc. immediately after collision with a arbitrary rightward or leftward ball overlap.
　　　　　　(Details of Behavior 2　　2011-4-18 rev1)　

・Change in Behavior of balls before collision with cushion : 　Changes in the speed, angle speed of balls in the state of slipping or rolling before collision with cushion.

・Behavior of balls immediately after rebounding from cushion : 　Changes in elements immediately after the collision with cushion with a arbitrary angle of incident, speed, angle speed, and rotation axis tilt angle.
　　　　　　(Details of Behavior 3　　2011-4-18 rev1)　
　Changes in Behavior of balls rebounded from cushion : 　It is relatively easy to deal with such balls because the cushion is where the coordinate axis is located.

The above analysis includes various conditions of the billiard table as a matter of course.


The simulator can be completed only when it can describe all information regarding any moment from the strike with the cue to the seizure of the balls. The behavior of the balls rebounding from cushion is especially complicated, and it can be predicted only through analysis. Even the high-speed photography cannot facilitate the analysis because the pattern on the surface of the balls does not make a simple rotation.

The analysis introduced here is based on three cushion but the result of the analysis can be applied to pocket billiards commonly. I would like to mention the essential points regarding the pocket billiard and masses in the future. (2015-4-2 completed)


There are many themes for which I have spent a lot of time in the analysis, for some of which not much progress has been made for time being. I write up my analysis, starting those including the formula thereof.
　　　　　　(By-Talks : including formula 2014 rev4)　

As it was completed a while ago, I would like to gradually introduce the text of the analysis section by section to the extent that I have written, for the reference of those who may be interested in.
Chapter 4 has been revised to add some more on the knowledge at the time of preparing the program.
　　　　　　　(It's being reconsidered)


2. How to strike a ball without throw in pocket billiards

I have spent a lot of time to contemplate how to deal with the analysis of pocket billiards. There are many points different from three cushions. What is fundamental is how to move the object ball as intended. At the same time, the destination of the cue ball should be under control. But the movements of them are influenced with the other balls in many cases. These points are largely different from three cushion. The analysis of subsequent collision is not something an amateur can handle. Thus, I examined throw, which is the difficulty of pocket billiards.

Throw is a phenomenon in which the object ball travels in a direction forwardly shifted from the direction that the object ball is supposed to travel geometrically when the player try to enter the object ball into a pocket. What is difficult in this problem is the magnitude of the shift is not so reproducible. The instability thereof is caused because the surface of the balls are covered with moisture or oil in different ways always. That is, the friction coefficient on the surface of the balls varies, destabilizing the friction applied on the object ball due to the collision with the cue ball, so that the angle of destination varies even if the cue ball collides the object ball with the same overlapping.

Thus, I theoretically worked out the rotation condition of the cue ball in which the friction force (horizontal component), which causes the throw, is zero, and applied it to the practical analysis.
I successfully obtained two relational expression after struggles. Based on the expression, I could describe how to strike the cue ball without throw.
　　　　　　(How to strike a ball without throw)　　　　（2014-10-11）

　　　　　　(How to strike a ball without throw)　(english)


3. Confirmation Application 　　

I introduced the application I prepared for the confirmation and evaluation of the results of analysis, as a reference for the explanation.
　A Change in the tilt angle of the rotation axis of a ball struck with the cue.　(2014-3-15)
　　　　　　Download : motion of the axis zip　　　　　　　

　A change in the direction of rolling in slipping　　(2014-11-19)
　　　　　　Download ： KOBA-8 modifi.3　　　

　　　　　　(Samples of a setting of KOBA-8 modifi.3)　
(It does not have a function to judge whether a value inputted in the input field is appropriate or not. If the value is not appropriate, it causes freeze-up of the program. :　2015-3 Additional comment)


Supplemental Remarks.　　(15-4-2)

In three cushion, there is a case a player strikes the cue ball to collide against two object balls after travelling around on the table (that is, after rebounding more than 3 times). When actually playing it, one may feel that it is somehow easy to hit an object ball in distance. I examined it, considering that it may be difficult if the object ball is aimed by extending the rolling distance straight. The result was clear. The first object ball can be apparently 3 times bigger in size beyond the three cushion.
　　　　　　(Convergence of translational direction via cushion)　　(14-10-11)


・Analysis of masse completed.　　(2015-4-2)

When I started the analysis of billiard, I thought it would be nice if I could do it up to the analysis of masse. At the time, I had no clue for the analysis. I had kicked it off, working on it to some extent in parallel with what I have already started. But there was sometimes a problem with smoothness of the curved line after a shot or collision.. I have already posted on YouTube the extent of the analysis for which I could obtained smoothness at the time.

In 2014, I thought of a way to smoothen the conventional slipping process and rewrote the program. After that, the curve will not be twisted and the program is dramatically shortened. This paved the way to the analysis of masse. I carried on the analysis, eliminating discontinuity which had been observed sometimes.

More than 10 years have passed since the start, it has be so hard for a person over 77 years old to go through the details of a program and revise it. It was the final theme I thought I would do it if I could. Now I am filled with a sense of satisfaction.

The simulator of masse can be a reference for high-level players. High-level players have an ability to sense where to strike, which direction to swing, how much to swing, and how strong to strike, once he sees the ball arrangement and finds the course to aim. I hope this tool can help you to feel which element is most sensitive in a particular case.

For players at intermediate level or less, I would like them to use the simulator to find out how the course is varied by changing any one of the setting gradually. Masse is forbidden in pool halls in principle. So, if you have a chance to try it in a game, you should ask yourself whether you can do it, and then ask for the permission. I am sure you will become unpopular if you are keen to try masse without a skill. You must not just practice masse in a pool hall. If you do, you will be kicked out for sure.

The only and most important point that should be kept in mind at masse is you should hold the cue softly, strike down with a good snap of wrist securing a rest, and stop applying a force onto the cue at the moment the tip of the cue collides against the cue ball in order to stop the cue. You have to do it perfectly, otherwise you will damage the cloth of the table.

The text of the masse analysis is presented below, but I have to admit that the explanation is not enough for the length of arms necessary for torque calculation. It requires a cubic diagram essentially, but I omitted it intentionally because a complicate drawing would cause misunderstanding undesirably.
　　　　　　(Text of the analysis of masse　　　2015-4-2)　

　　　　　　　　　　　　　　　　　　　　　　　mail : 3cplayerkk@ares.eonet.ne.jp