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
Famous quotes containing the words formal and/or definition:
“This is no argument against teaching manners to the young. On the contrary, it is a fine old tradition that ought to be resurrected from its current mothballs and put to work...In fact, children are much more comfortable when they know the guide rules for handling the social amenities. It’s no more fun for a child to be introduced to a strange adult and have no idea what to say or do than it is for a grownup to go to a formal dinner and have no idea what fork to use.”
—Leontine Young (20th century)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)