幾何中的海倫－擺線趣談

Helen of Geometry-The Interesting Cycloid

劉豐榮(Feng-Jung Liu)

《立德學報》 3卷1期 (2005/12) Pp. 123-135

https://doi.org/10.7080/JLU.200512.0123

摘要

在數學發展史上從未有任何曲線像「擺線」一樣引起那麼多位數學家的注目與研究。它的發展史可說是十七、十八世紀西方數學發展史的縮影，同時也深深影響著其後數學的發展，特別是「微分方程」及「變分法」。不僅如此，在物理上它同時也具許多有趣的性質。它是這樣美麗又引起那麼多的爭議，因此有人把它比喻成古希臘時期引起「特洛伊」(Troy)戰爭的美女海倫(Helen)。在這篇文章中我們將討論擺線的發展歷史、其數學及物理性質，同時也介紹關於擺線的一些趣聞，祈能引起讀者對於數學探索的樂趣。
本文計分為三段，第一段將介紹擺線的定義及其數學方程式。第二段我們將以微積分的手法導出其弧長及面積等數學性質。第三段則介紹它的相關物理性質包括：「等時性」、「最速降性」。

關鍵字

擺線 ； 時性 ； 最速降性

並列摘要

In the history of development of mathematics, there has been no such curve as ”cycloid” arousing so many mathematicians' attention and interests in studying it. The history of development of cycloid can be considered as an epitome of that of western mathematics in the 17th and 18th centuries. Meanwhile, it had been deeply affecting the later development of mathematics, especially ”differential equations” and ”calculus of variations.” In fact cycloid has many interesting natures in physics. It is so beautiful, though argumentative. Thus, people like to take the beautiful goddess Helen of Troy as the metaphor of cycloid. This study intends to discuss the history of development of cycloid, and its mathematical and physical natures. Besides, in this study some interesting episodes about cycloid shall be introduced, hoping to arouse readers' interests in mathematical exploration.
This paper is divided into 3 parts. The first part introduces the definition of cycloid and its mathematical equation. The second part uses calculus to induce its mathematical features, like arc length and area. The third part introduces its related physical natures, including tautochrone, brachistochrone.

並列關鍵字

cycloid ； tautochrone ； brachistochrone