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:



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> What does the principal curvatures direction **magnitude** represents?
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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:



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> What is the geometrical meaning of an eigenvector's magnitude?
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Principal curvatures and curvature directions
Unless there's an additional constraint on the defintion of the principal curvature directions, being just defined as eigenvectors means their magnitude is arbitrary and so meaningless.


