While trying to characterize the complexity of a closed differentiable curve (for a path planning application), I've been using a notion which is similar in spirit to the winding number of a curve. For now I'll call it the *unwinding number*, but my question is whether it has a proper name or if there is another concept in topology which seems closely related to it.

In plain english, the *unwinding number* of a curve $\gamma$ with respect to a point $p$, is the number of times that the outer-pointing normal of $\gamma$ points towards $p$.

Trying to be formal: Consider a closed differentiable curve $\gamma: [0,1]->\mathbb{R}^2$ and let $\gamma'': [0,1] \to SO(2)$ define its outer-pointing normal. That is, $\gamma''(t)$ is the outer-pointing normal at the point $\gamma(t)$ in the curve $\gamma$. Given a point $p \in \mathbb{R}^2$ the *unwinding* number of $\gamma$ is defined as the number of times that $\gamma''(t) = (p-\gamma(t))/||p-\gamma(t)||$ where $t \in [0,1)$.

I am sure there are lots of degenerate cases where this definition doesn't make sense, but I am mostly interested in curves that model obstacles in the real world, and not so much in the corner cases. However, if possible, I would like to avoid reinventing the wheel, since I am sure smarter people than me have thought about similar things before.