摆线

一条由滚动的圆所生成的摆线

数学中,摆线(Cycloid)的定义为圆在一条直线上滚动时,圆边界上一定点所形成的轨迹。它是一般旋轮线的一种,亦称圆滚线

摆线也是最速降线问题等时降落问题的解。

历史

摆线的研究最初开始于库萨的尼古拉,之后马兰·梅森也有针对摆线的研究。1599年伽利略为摆线命名。1634年吉勒斯·德·罗贝瓦勒英语Gilles de Roberval指出摆线一拱的區域面积是滾動圆的面积的三倍。1658年克里斯多佛·雷恩也向人们指出摆线的长度是滾動圆的直径的四倍。在这一时期,伴随着许多发现,也出现了众多有关发现权的争议,甚至抹杀他人工作的现象,而因此摆线也被人们称作“几何学中的海伦”(The Helen of Geometers)。[1]

方程式

由半径为2的圆所生成的摆线

过原点半径为r的摆线参数方程为

在这里实参数t是在弧度制下,圆滚动的角度;摆线的第一道拱由参数t在(0, 2π)区间内的点组成。对每一个给出的t,圆心的坐标为(rt, r)。

通过替换解出t可以求的笛卡尔坐标方程

也可寫成

摆线也满足下面的微分方程

面积

一条由半径为r的圆所生成的拱形面积可以由下面的参数方程界定:

微分,

于是可以求得

弧长

弧形的长度可以由下面的式子计算出:

其它相关联的曲线

一些曲线同摆线紧密相关。当我们弱化定点只能固定在圆边界上时,我们得到了短擺線(curtate cycloid)和長擺線(prolate cycloid),兩者合稱為次擺線(trochoid),前面的情形是定点在圆的内部,后者则是在圆外。次摆线则是上述三种曲线的统称。更进一步,如果我们让圆也沿着一个圆滚动而不是直线的话,我们会得到外摆线(epicycloid,沿着圆的外部运动,定点在圆的边缘),内摆线(hypocycloid,沿着圆内部滚动,定点在圆的边缘)以及外旋轮线(epitrochoid)和内旋轮线(hypotrochoid,定点可以在圆内的任一点包括边界。)

应用

Cycloidal arches at the Kimbell Art Museum

在建筑物的设计方面,摆线曾被路易·卡恩用来设计德克萨斯州沃思堡的建筑金贝尔艺术博物馆英语Kimbell Art Museum。 它也曾被用于设计新罕布什尔州汉诺威的霍普金斯中心。

参考

  1. ^ 卡乔里, 弗洛里安. 数学史. 纽约: 切尔西. 1999: 177. ISBN 978-0821821022. 
  • Wells D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. 1991: 445–47. ISBN 0-14-011813-6. 

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