Question

An epicycloid is another variation of a cycloid in which the point tracing the path is on the circumference of a wheel, but the wheel is rolling without slipping on the outside of another wheel, instead of along a horizontal track. If the radius of the rolling wheel is k and the radius of the fixed wheel is r, find parametric equations for the epicycloid.

Short answer

As a response, the parametric equations

$$\begin{gathered} x=(r+k)\cos \theta -k\cos \left(\frac{r}{k}+1\right)\theta \\ y=(r+k)\sin \theta -r\sin \left(\frac{r}{k}+1\right)\theta \end{gathered}$$

Step-by-step solution

Step: 1 Given information

Consider an epicycloid in which the path is traced around the circumference of a wheel, but the wheel is not slipping on the outside of another wheel.

Step: 2: Calculation

The angles $\theta ,\varphi$are related to each other.

Let $A{A}^{\text{'}}$d denote the arc length.

The arc length is equal to $r\theta .$

$A{A}^{\text{'}}=r\theta$while $A^{\text{'}}p=k(\theta +\varphi )$

We can conclude that$\varphi =\left(\frac{r}{k}+1\right)\theta$

Step: 3: Further Calculation

The coordinates of $p$with respect to origin are.

$$\begin{gathered} x=(r+k)\cos \theta -k\cos \left(\frac{r}{k}+1\right)\theta \\ y=(r+k)\sin \theta -r\sin \left(\frac{r}{k}+1\right)\theta \end{gathered}$$

The minus sign in the second component of the $x$-coordinate equation comes from the fact that the rolling circle's rotation and its center's motion are in the same direction.