Chapter 10.6, Problem 67PE

Suppose that an imaginary string of unlimited length is attached to a point on a circle and pulled taut so that it is tangent to the circle. Keeping the string pulled tight, wind (or unwind) the string around the circle. The path traced out by the end of the string as it moves is called an involute of the circle (shown in blue).

Given a circle of radius $r$ with parametric equations $x=r\cos \theta ,$ $y=r\sin \theta ,$ show that the parametric equations of the involute of the circle are $x=r\left(\cos \theta +\theta \sin \theta \right)$ and $y=r\left(\sin \theta -\theta \cos \theta \right)$ .