Formal Definition
Let M be a surface in Euclidean space with second fundamental form . Fix a point p∈M, and an orthonormal basis X1, X2 of tangent vectors at p. Then the principal curvatures are the eigenvalues of the symmetric matrix
If X1 and X2 are selected so that the matrix is a diagonal matrix, then they are called the principal directions. If the surface is oriented, then one often requires that the pair (X1, X2) to be positively oriented with respect to the given orientation.
Without reference to a particular orthonormal basis, the principal curvatures are the eigenvalues of the shape operator, and the principal directions are its eigenvectors.
Read more about this topic: Principal Curvature
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