Alternate Representation for the Winding Number

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 $$\kappa = \vartheta'$$.