Chapter 10.6, Problem 102E

To determine

Theparametric equations for the epicycloid.

Answer to Problem 102E

The parametric equations for the epicycloid are $x=3\cos t-\cos 3t$ and $y=3\sin t-\sin 3t$ .

Explanation of Solution

Given information:

The curve traced by point $P$ is shown in figure (1).

  

Figure (1)

Calculation:

Using the above figure (1) the point $P$ moves around the circle with radius $2$ . The parametric equation for the circle are.

  $\begin{array}{l}x=\cos \theta \\ y=\sin \theta \end{array}$

The circle of radius $1$ rotates around the circle with radius $2$ . the point $P$ moves through an angle.

  $\begin{array}{l}x=\cos \left(\theta +t\right)\\ y=\sin \left(\theta +t\right)\end{array}$

The circle moves without slipping, therefore the arc length $s$ must be equal to $s\text{'}$ .

  $\begin{array}{c}s=2t\\ s\text{'}=\theta \\ s=s\text{'}\\ 2t=\theta \end{array}$

The equation will be.

  $\begin{array}{c}x=\cos \left(2t+t\right)\\ =\cos \left(3t\right)\\ y=\sin \left(2t+t\right)\\ =\sin \left(3t\right)\end{array}$

The center $C$ of circle radius $1$ travels around the other circle with radius.

  $\begin{array}{l}=2+1\\ =3\end{array}$

  $\begin{array}{c}x=3\cos t\\ y=3\sin t\end{array}$

Let $O$ is the origin.

  $OP=OC-CP$

Calculate the parametric equation for epicycloid.

  $\begin{array}{c}x=3\cos t-\cos 3t\\ y=3\sin t-\sin 3t\end{array}$

Therefore, the parametric equations for the epicycloid are $x=3\cos t-\cos 3t$ and $y=3\sin t-\sin 3t$ .