Chapter 13, Problem 30PE

To determine

Show that the radius of curvature of a twice-differentiable plane curve $r\left(t\right)=f\left(t\right)i+g\left(t\right)j$ is given by the formula $\rho =\frac{{\dot{x}}^2+{\dot{y}}^2}{\sqrt{{\ddot{x}}^2+{\ddot{y}}^2-{\ddot{s}}^2}}$, where $\ddot{s}=\frac{d}{dt}\sqrt{{\dot{x}}^2+{\dot{y}}^2}$.