Last week I considered again *principal curvature (pc)* and *principal curvature directions (pcd)* of a, for the sake of simplicity, 2-manifold embedded in 3-space. In this simple case, the pc and pcd of at a point are the eigenvalues and eigenvectors of the shape operator. The magnitude of the pc's corresponds to the minimal and maximal normal curvature at the point. My question, however, is:

What does the principal curvatures direction

magnituderepresents?

In the textbooks I looked up in (Kühnel and do Camro) I couldn't find a reference to the principal curvature direction's magnitude. Is there something known about this? Is it something basic (maybe even from linear algebra)?

Edit 1: A somewhat more general, but related, question is:

What is the geometrical meaning of an eigenvector's magnitude?