對於人們打撞球時如何運用不同的擊球方式，使得球會有推桿、定桿和拉桿等不同運動行為的產生，是本論文的研究動機。文中透過基本的力學概念，分析撞球過程中球所受力的狀況，其中包括球桿擊球、球與球的碰撞以及球在球檯上的運動等三個部份的探討，更修正了滾動摩擦力(rolling resistance)的架構，並且運用於分析方法當中。本論文在分析方法中考慮了球桌布給球的摩擦力，也考慮在真實當中球桌布有微小凹陷的現象。最後推導出推桿、定桿和拉桿三種基本桿法的數學模式，接著以此模式進行反向推導，求出擊球時所需要的擊球條件，並透過3D Studio Max 8 軟體建構出虛擬撞球場景，進行三種桿法的模擬，並且實際訪問專家的意見作為驗證。

由模擬結果可知定桿和拉桿能同時準確控制母球和子球最終的停止位置，而推桿為了使母球能夠往前推移足夠的距離，將無法準確控制子球停止的位置，並且可以發現擊球條件通常並非唯一。

The objective of this thesis is to explore how human beings manipulate their dexterous operation skills to produce follow shot, stop shot, and draw shot while playing billiard. According to basic mechanics, the forces exerted on a billiard ball are analyzed in three different conditions. They include the cue stick hitting the cue ball, collision
between the cue ball and the object ball, and the interaction between the ball and the surface. Mathematical models for those three fundamental techniques: the follow shot, the stop shot, and the draw shot, are developed by incorporating the rolling resistance. The friction between
the surface and the ball, and the dent on the surface due to the mass of the ball are considered. Furthermore, reverse derivations are also implemented to solve initial hitting conditions between the cue stick and the cue ball for certain final states for the cue ball and the object ball. Computer simulations in virtual reality are conducted using the 3D Studio Max 8 software tool. Opinions from billiard experts are also collected for the purpose of verification.

Based on simulation results, reversed derivation can accurately predict final states of the cue ball and the object ball for both stop shot and draw shot. However, for the case of follow shots, final position of the object ball cannot be fully determined. Besides, no unique solution exists for the hitting condition in all three types of shot.

目錄 .............................................................................................................i
圖目錄 ....................................................................................................... iv
表目錄 ...................................................................................................... vii
摘要 ......................................................................................................... viii
Abstract ..................................................................................................... ix
第一章 緒論 ............................................................................................. 1
1.1 研究動機與目的 ........................................................................... 1
1.2 撞球基本桿法的簡介 ................................................................... 2
1.2.1 推桿 ..................................................................................... 2
1.2.2 定桿 ..................................................................................... 3
1.2.3 拉桿 ..................................................................................... 4
1.2.4 下塞 ..................................................................................... 4
1.3 文獻回顧 ........................................................................................ 6
1.3.1 球桿衝擊母球 ..................................................................... 6
1.3.2 球與球的碰撞 ..................................................................... 7
1.3.3 球與球檯邊的碰撞 ............................................................. 7
1.3.4 單顆球在球檯上運動狀態的轉變 ...................................... 8
1.4 研究方法與步驟 ............................................................................ 8
1.5 本文架構 ........................................................................................ 9
第二章 建立基本桿法的數學模型 ...................................................... 11
2.1 球桿衝擊母球 ............................................................................. 11
2.2 球在球檯上的運動 ..................................................................... 15
2.3 球與球的碰撞 ............................................................................. 28
第三章 擊球條件的反算 ...................................................................... 30
3.1 子球最終停止位置 ..................................................................... 30
3.2 母球最終停止位置 ..................................................................... 33
3.3 母球與子球初始相對距離......................................................... 42
3.4 擊球條件 ..................................................................................... 52
第四章 三種桿法模擬與專家回饋的意見 .......................................... 54
4.1 場景建立 ..................................................................................... 54
4.2 單一顆球受球桿衝擊後滾動至停止的模擬 ............................ 56
4.2.1 推桿 ................................................................................... 57
4.2.2 定桿 ................................................................................... 58
4.2.3 拉桿 ................................................................................... 59
4.3 母球受衝擊後與子球碰撞的模擬 ............................................ 61
4.3.1 推桿 ................................................................................... 61
4.3.2 定桿 ................................................................................... 62
4.3.3 拉桿 ................................................................................... 63
4.4 三種桿法控制母球與子球最終停止位置的模擬 .................... 65
4.4.1 推桿 ................................................................................... 65
4.4.2 定桿 ................................................................................... 68
4.4.3 拉桿 ................................................................................... 70
4.5 專家回饋的意見 ......................................................................... 72
第五章 結論與建議 ................................................................................ 73
參考文獻 ................................................................................................... 76

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