Differential geometry quantity related to the energy of a curve

Question

I have come across the quantity $$W=\frac{f'}{1+f'^2}$$

where $f=f(x)$ and prime represents the derivative with respect to $x$. I know $1+f'^2$ is sometimes referred to as the energy density of the curve $(x,f(x))$, but I cannot seem to come up with a good physical or geometrical interpretation of the quantity $W$.

Is there an obvious interpretation of this quantity?

Answer 1

In terms of the graph $\gamma(x) = (x,f(x))$, all I can come up with is that $W = v^{-1} \sin \theta$ where $v = |\dot \gamma|$ is the velocity of the curve and $\theta$ is the heading angle of the curve (measured from horizontal).