Chapter 10.6, Problem 72PE

A cycloid is a curve created by rolling a circle along a line. We can also create families of curves by rolling a circle around another circle. If a circle of radius $r$ rolls around the interior of a larger circle of radius $R,$ a fixed point on the smaller circle traces out a curve called a hypocycloid. For Exercises 71-72, use the following parametric equations for a hypocycloid.

$x=\left(R-r\right)\cos t+r\cos \left[\left(\frac{R-r}{r}\right)t\right]$ and $y=\left(R-r\right)\sin t-r\sin \left[\left(\frac{R-r}{r}\right)t\right]$

a. Write parametric equations for a hypocycloid with $R=4$ and $r=1$ . The curve defined by these equations is called an astroid (not to be confused with asteroid).

b. Graph the exterior circle given by $x=4\cos t$ and $y=4\sin t$ and the astroid from part (a). Use $0\le t\le 2\pi$ and a viewing window of $\left[-6.4,6.4,1\right]$ by $\left[-4,4,1\right]$ .

c. For what values of $t$ on the interval $\left[0,2\pi \right)$ do the astroid curve and circle intersect?