Normal curvature

Let $\vec{\tau}$ be a unit tangent vector at point $x$ of a regular surface $S$. Then the normal curvature of $S$ in the direction $\vec{\tau}$ is the curvature of a normal section of $S$ at $x$ (a normal section is a cut plane containing the normal):

$$\kappa(\vec{\tau}) = \mathbb{II}(\vec{\tau}, \vec{\tau})$$

In the case when $\vec\tau$ is not a unit vector:

$$\kappa(\vec\tau_x) =\frac{\mathbb{II}(\vec\tau_x)}{\mathbb{I}(\vec\tau_x)}$$