從利沙球圖形到擺線間的幾何轉換

Abstract

本論文是研究利沙球圖形與擺線的幾何轉換。利沙球圖形與擺線皆在幾何曲線中扮演非常 重要角色。然而過去的研究中並沒有對於這兩群重要的幾何曲線有任何相關性的連結。我的 研究首先利用在物理系統中常見的簡諧運動為基礎，進而解得其古典軌跡落在利沙球圖形 上。進一步透過群論中SU(2)矩陣的巧妙轉換，發展出一系列介於利沙球圖形與擺線之間有 趣的幾何曲線。透過SU(2)轉換的概念不僅引導出有趣的幾何圖像，其中所對應的物理意義 也值得我們深入探討。

This thesis is the research of the geometric transformstion between Lissajous and trochoidal curves. Lissajous and trochoidal curves are important in geometric curves. However there is not any connection between Lissajous and trochoidal curves in early researches. Firstly, we start from the simple harmonic motions, and the solution is found as Lissajous parametric curves. Furthermore, by means of the transformation, the matrix SU(2) in group theory , a series of curves between Lissajous and trochoids are demonstrated. They have not been discussed until now. Through the concept of SU(2), we obtained the intriguing geometric curves. Importantly, the physics of the transformation is worthy to discussed further in the future.