Question

Look up the curves called hypocycloid and epicycloid. Write a report on what you find. Be sure to draw comparisons with the cycloid.

Short answer

An epicycloid is a plane curve created by tracing a chosen point on the edge of a circle of radius r rolling on the outside of a circle of radius R.

Step-by-step solution

Step 1. Given Information

Hypocycloid: A curve traced by a point on the circumference of a circle rolling internally on the circumference of a fixed circle.

Epicycloid:The curve described by a point on the circumference of a circle as this circle rolls around the outside of another fixed circle,

Step 2. Figure of Hypocycloid and Epicycloid

Step 3. Comparison of cycloid with Epicycloid and Hypocycloid

A cycloid is a curve generated by a point on the circumference of the circle as the circle rolls along a straight line with out slipping.

The parametric equations for Epicycloid and Hypocycloid are

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

respectively.

Where $r$is the radius of rolling circle and $R$is the radius of fixed circle, $\theta$is the angle between radius vector of the point of contact of the circles and the $x$-axis.