Let $\alpha: I \rightarrow \mathbb{R}^3$ a parametrized smooth curve such that $\Vert \alpha'(t) \Vert = 1$. We define the curvature at the point $\alpha(t)$ as $k(t)=\Vert \alpha''(t) \Vert$.
My question is what happens if our curve intersects itself. In this case $k(t)$ could take two different values. In some texts they define the curvature at the instant $t$ instead of at the point $\alpha(t)$.
I wanted to know what the most accurate definition is.
It's the second one. If $\alpha(t_1)=\alpha(t_2)$, with $t_1\ne t_2$, it may well happen that $\kappa(t_1)\ne\kappa(t_2)$. So, the curvature is the curvature at a certain moment, not at a certain point of the curve.