Normal Curvature

Let $C$ be a curve on the surface $S$ . At any point $p$ on $C$ (and hence on $S$ ), let $\theta$ denote the angle between the normal $\mathbf{n}$ to $C$ and the normal $\mathbf{N}$ to $S$ . If $k$ is the curvature of $C$ at $p$ , then the quantity $k_n = k \cos \theta$ is called the normal curvature of $C \subset S$ at $p$ .

We can express $k_n$ differently, using the tangent $\mathbf{t}$ to the curve. $k_n = k \mathbf{n}\cdot\mathbf{N} = \mathbf{t}'\cdot \mathbf{N} = -\mathbf{t}\cdot\mathbf{N}' = -\langle \mathbf{t} \vert d\mathbf{N}_p \vert \mathbf{t} \rangle$ , which is just the second fundamental form evaluated at point $p$ in the tangent direction.

The extreme values of the normal curvature at any point of the surface are equal to the principal curvatures $\kappa_1$ and $\kappa_2$ . The tangent directions to the curves with the extreme normal curvature are the eigenvectors of the shape operator and are called the principal directions. The unit vectors along the principal directions are denoted by $e_1$ and $e_2$ , and form an orthonormal basis for $T_p(S)$ .