§ Integrating Curvature in 1D [TODO ]

• 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 $2 \pi$?