Cycloid Animation [gnuplot]

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Formulation

Parametric equations

$$\begin{align*} \v{OC} & =\begin{bmatrix} r\theta\\ r\ \end{bmatrix} \\ \v{OP} &=\begin{bmatrix} r(\theta-\sin\theta)\\ r(1-\cos\theta) \end{bmatrix} \end{align*}$$

Arc length $L$

$$\begin{equation*} \frac{\diff x}{\diff \theta} = r(1-\cos\theta)\ ,\ \ \frac{\diff y}{\diff \theta} = r\sin\theta \end{equation*}$$ $$\begin{align*} L &= \int_0^{2\pi}\sqrt{\left(\frac{\diff x}{\diff \theta}\right)^2+\left(\frac{\diff y}{\diff \theta}\right)^2}\,\diff \theta\\ &= \int_0^{2\pi}\sqrt{\left[r(1-\cos\theta)\right]^2+\left[r\sin\theta\right]^2}\,\diff \theta\\ &= \int_0^{2\pi}\sqrt{2r^2(1-\cos\theta)}\,\diff \theta\\ &= \int_0^{2\pi}\sqrt{2r^2\cdot 2\sin^2\frac{\theta}{2}}\,\diff \theta\\ &= \int_0^{2\pi}2r\left|\sin\frac{\theta}{2}\right|\,\diff \theta\\ &= 2r\left[-2\cos\frac{\theta}{2}\right]_0^{2\pi} = 8r \end{align*}$$

Area $S$

$$\begin{align*} S &= \int_{x=0}^{x=2\pi r} y \,\diff x\\ &= \int_0^{2\pi} r(1-\cos\theta) \cdot r(1-\cos\theta)\,\diff \theta\\ &= r^2 \int_0^{2\pi} \left(1-2\cos\theta+\frac{1+\cos 2\theta}{2}\right) \diff \theta\ \ \left(\because \cos^2\theta=\frac{1+\cos 2\theta}{2}\right)\\ &= r^2\left[\frac{3}{2}\theta-2\sin\theta+\frac{1}{4}\sin 2\theta\right]_0^{2\pi} = 3\pi r^2 \end{align*}$$

Simulation [gnuplot]

Source code (PLT file)

Output (PNG file → GIF file)

Another example

External links

▶︎ Cycloid - Wikipedia