Mean curvature of a curve

Question

I'd like to know if there exist some meaning for the mean value of the curvature function in a parametric function $r(t)$. This is a different topic of mean curvature in surfaces as in this topic. For example, given the curve $r(t)=(\sin t + \sin 2t, \cos t + \cos 2t)$, the curvature is given by $\kappa=-\frac{6\cos t + 9}{4 \cos t + 5}$. Does it make any sense to compute the mean value of $\kappa$?

Answer 1

The (signed) curvature $\kappa$ is the derivative of any function $\theta$ such that the unit tangent at $t$ is given by $(\cos(\theta(t)), \sin(\theta(t)))$. Thus, the mean curvature is $$\frac{1}{b - a} \int_a^b \kappa(t) \, dt = \frac{\theta(b) - \theta(a)}{b - a},$$ which can be thought of as the average angular speed of the tangent line from $a$ to $b$.

Answer 2

The mean can always be used to understand a varying number, so it might be useful in understanding the average curve. Also, I think that integrating the curvature over the curve over a section tells how many radians the curve has turned over that section, so integrating over the whole curve should give you $2\pi$ times the number of total loops and the average should be some notion of inverse average radius of the curve.

In general, I think the range of the curvature and a graph of the function are much more informative.