Say I have some polar curve $r=f(\theta)$ in the Cartesian plane (smooth and twice-differential) and I've found a formula for the curvature, $k$. Does a clever trick exist (that isn't merely finding the maxima and minima of $k$) that allows us to find the principle curvature of our curve?
What about the mean curvature?
Since the curvature $k=d\theta/dl$, then average curvature is $k_a=\Delta\theta/\Delta l$. So you break you curve into segments at the points where $k=0$ or $k=\infty$. For each segment you calculate $|\Delta\theta|$ — angle the tangent of the curve rotated. You sum all $|\Delta \theta|$ and divide by total length of the curve.
Example. Let's calculate the average curvature for $y=\sin x$ from $0$ to $2\pi$. Curvature is zero for $x=\pi$. So we consider two segments. At first segment the tangent rotated from $\theta_0=\pi/4$ to $\theta_\pi=-\pi/4$, so $|\Delta\theta_1|=\pi/2$. It's the same for the second segment $|\Delta\theta_2|=\pi/2$. The length of the curve is $$l=\int_0^{2\pi}\sqrt{1+\cos^2x}dx=4\sqrt{2}E(1/2)\approx7.64$$ So mean curvature is $k_a=2(\pi/2)/l\approx0.41$