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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.

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    $\begingroup$ Do you count +1 when the extension of the normal sweeps across p in one direction and -1 when it sweeps across p in the other direction, or do you count +1 for both? $\endgroup$
    – Yoav Kallus
    Commented Jul 30, 2013 at 17:13
  • $\begingroup$ @Yoav: It counts as +1 in both cases. $\endgroup$
    – rick
    Commented Jul 30, 2013 at 17:30
  • $\begingroup$ WHat are the properties of this? $\endgroup$
    – Igor Rivin
    Commented Jul 30, 2013 at 20:00
  • $\begingroup$ @IgorRivin, I don't have a succinct answer for this. I know some properties of this unwinding number, but only with respect to a very concrete algorithm I've been looking at. However, I don't know of any natural/geometric properties for this definition. My hope was that someone familiar with geometry/topology would see a resemblance between this definition and some other more natural/well-known definition in the literature. $\endgroup$
    – rick
    Commented Jul 30, 2013 at 21:25

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For your invariant to be well-defined your curve needs to be an immersion, and disjoint from the point $p$. There are only two independent invariants of such curves (up to 1-parameter families of such curves), the winding number of the curve about the point $p$, and the Whitney-Graustein number, which is the winding number of the curve's unit normal (or tangent) vector.

So your number, whatever it is, should be expressible in terms of these two numbers. Let's call the winding number about the point $p$, $W(\gamma,p)$, and the Whitney-Graustein number $WG(\gamma)$. I believe your quantity then has to be

$$WG(\gamma) - W(\gamma,p)$$

So that's what your invariant has to be if you want it to be independent of small $C^1$-perturbations of the curve. This means we're interpreting the "number of times" you refer to in a signed sense. If you want it to be a literal, unsigned count, then your number is a very unstable quantity, but it is bounded below by the absolute value of $WG(\gamma)-W(\gamma,p)$.

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