A Universe of Sorts

All curves are parametrized by arc length to avoid weird artefacts by time parametrization.
So $r(s)$ is a function from length of the curve to $\mathbb R^3$ .
The (unit?) tangent to a curve is given by $T(s) \equiv dr/ds = r'(s)$ .
The curvature is given by $\kappa(s) \equiv |dr^2/ds^2|$ .
The unit normal is given by $\hat N(s) r''(s) / \kappa(s)$ .
We wish to consider the total curvature, given by $\int_0^L \kappa(s) ds$ where $L$ is the total length of a closed curve on the plane.
TODO: how to prove that this will be a multiple of $2 \pi$ ?

§ Integrating Curvature in 1D [TODO ]