Let $c: I \rightarrow \mathbb{R}^2$ be a closed curve which is parameterized by arc length. Further, let $P$ be the period of $c$ and $Q \not\in \mathtt{trace}(c)$.

The winding number of $c$ w.r.t. $Q$ is defined by $$ W(c, Q) := \frac{1}{2\pi} (\vartheta(P) - \vartheta(0)) $$

where $\vartheta$ is a lift of $(c-Q)/\lvert c-Q \rvert$.

Now, what I want to show is that $$ W(c, Q) = -\frac{1}{2\pi} \int_0^P \frac{\langle c(t) - Q, N(t) \rangle}{\lvert c(t) - Q \rvert^2} dt. $$

Attempt: Let $\kappa$ be the curvature of $(c-Q)/\lvert c-Q \rvert$. Then we would have to show that $\kappa$ is the integrand multiplied by $-1$. This would prove the statement, since $\kappa = \vartheta'$.


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