I am trying to understand a theorem that involves curvature in the complex plane. The author of the theorem makes the statement that a smooth function $K : T \rightarrow R$ is the curvature of a smooth, simple, and closed curve $\gamma_\kappa$ parametrized by arc length if and only if three specific conditions are satisfied. Here $T$ is the unit circle in the complex plane. The first of these three conditions is that $\int_{0}^{2\pi} K ds = 2\pi$. The author makes the comment that this condition expresses that $\gamma_\kappa$ has a well-determined tangent at $s = 0$.
I am trying to understand the significance of this statement, but I can't seem to figure it out. I know that this condition shows that $\gamma_\kappa$ is in arc length parametrization, but I'm not sure what the well-determined tangent means and why it is significant.