Chapter 8.4, Problem 64E

Epicycloid If the circle $C$ of Exercise $63$ rolls on the outside of the larger circle, the curve traced out by $P$ is called an epicycloid. Find parametric equations for the epicycloid.

Hypocycloid A circle $C$ of radius $b$ rolls on the inside of a larger circle of radius $a$ centered at the origin. Let $P$ be a fixed point on the smaller circle, with the initial position at the point $\left(a,0\right)$ as shown in the figure. The curve traced out by $P$ is called a hypocycloid.

(a) Show that parametric equations of hypocycloid are

$x=\left(a-b\right)\cos \theta +b\cos \left(\frac{a-b}{b}\theta \right)$

$y=\left(a-b\right)\sin \theta -b\sin \left(\frac{a-b}{b}\theta \right)$

(b) If $a=4b$, the hypocycloid is called an asteroid. Show that in this case parametric equations can be reduced to

$x=a{\cos }^3\theta$ $y=a{\sin }^3\theta$

Sketch the curve. Eliminate the parameter to obtain an equation for the asteroid in rectangular coordinates.