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Principal curvature

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The normal curvature of a surface in a principal direction, i.e. in a direction in which it assumes an extremal value. The principal curvatures and k _ {2} are the roots of the quadratic equation

\tag{* } \left | \begin{array}{ll} L - kE &M - kF \\ M - kF &N - kG \\ \end{array} \right | = 0,

where E , F and G are the coefficients of the first fundamental form, while L , M and N are the coefficients of the second fundamental form of the surface, computed at the given point.

The half-sum of the principal curvatures k _ {1} and k _ {2} of the surface gives the mean curvature, while their product is equal to the Gaussian curvature of the surface. Equation (*) may be written as

k ^ {2} - 2Hk + K = 0,

where H is the mean, and K is the Gaussian curvature of the surface at the given point.

The principal curvatures k _ {1} and k _ {2} are connected with the normal curvature \widetilde{k} , taken in an arbitrary direction, by means of Euler's formula:

\widetilde{k} = k _ {1} \cos ^ {2} \phi + k _ {2} \sin ^ {2} \phi ,

where \phi is the angle formed by the selected direction with the principal direction for k _ {1} .

Comments

In the case of an m - dimensional submanifold M of Euclidean n - space E ^ {n} principal curvatures and principal directions are defined as follows.

Let \xi be a unit normal to M at p \in M . The Weingarten mapping (shape operator) A _ \xi of M at p in direction \xi is given by the tangential part of - \overline \nabla \; _ {\overline \xi \; } , where \overline \nabla \; is the covariant differential in E ^ {n} and \overline \xi \; is a local extension of \xi to a unit normal vector field. A _ \xi does not depend on the chosen extension of \xi . The principal curvatures of M at p in direction \xi are given by the eigen values of A _ \xi , the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of A _ \xi define the higher mean curvatures of M , which include as extremal cases the mean curvature as the trace of A _ \xi and the Lipschitz–Killing curvature as its determinant.

References

[a1] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a2] B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)
[a3] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a4] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a5] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145
[a6] H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60
[a7] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)
[a8] B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
[a9] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5
How to Cite This Entry:
Principal curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_curvature&oldid=49535
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article