Chapter 12, Problem 5PS

Cycloid Consider one arch of the cycloid

$r(\theta )=(\theta -\sin \theta )i+(1-\cos \theta )j,0\le \theta \le 2\pi$ as shown in the figure. Let $s(\theta )$ be the arc length from the highest point on the arch to the point $(x(\theta ),y(\theta ))$, and let $\rho (\theta )=1/K$ be the radius of curvature at the point $(x(\theta ),y(\theta ))$. Show that $s$ and $\rho$ are related by the equation $s^2+{\rho }^2=16$. (This equation is called a natural equation for the curve.)