# 摆线

## 方程式

${\displaystyle x=r(t-\sin t)\,}$
${\displaystyle y=r(1-\cos t)\,}$

${\displaystyle x=r\cos ^{-1}\left(1-{\frac {y}{r}}\right)-{\sqrt {y(2r-y)}}}$

${\displaystyle \left({\frac {dy}{dx}}\right)^{2}={\frac {2r-y}{y}}.}$

## 面积

${\displaystyle x=r(t-\sin t),\,}$
${\displaystyle y=r(1-\cos t),\,}$
${\displaystyle 0\leq t\leq 2\pi .\,}$

${\displaystyle {\frac {dx}{dt}}=r(1-\cos t),}$

{\displaystyle {\begin{aligned}A&=\int _{x=0}^{x=2\pi r}y\,dx=\int _{t=0}^{t=2\pi }r^{2}(1-\cos t)^{2}\,dt\\&=\left.r^{2}\left({\frac {3}{2}}t-2\sin t+{\frac {1}{2}}\cos t\sin t\right)\right|_{t=0}^{t=2\pi }\\&=3\pi r^{2}.\end{aligned}}}

## 弧长

{\displaystyle {\begin{aligned}S&=\int _{t=0}^{t=2\pi }{\sqrt {\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dx}{dt}}\right)^{2}}}\,dt\\&=\int _{t=0}^{t=2\pi }2r\sin \left({\frac {t}{2}}\right)\,dt\\&=8r.\end{aligned}}}

## 应用

Cycloidal arches at the Kimbell Art Museum

## 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.