Chapter 12, Problem 6PS

Cardioid Consider the cardioid

$r=1-\cos \theta ,0\le \theta \le 2\pi$

as shown in the figure. Let $s(\theta )$ be the arc length from the point $(2,\pi )$ on the cardioid to the point $(r,\theta )$, and let $p(\theta )=1/K$ be the radius of curvature at the point $(r,\theta )$. Show that s and $\rho$ are related by the equation $s^2+9{\rho }^2=16$. (This equation is called a natural equation for the curve.)