Principal curvature
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
are the roots of the quadratic equation
![]() | (*) |
where ,
and
are the coefficients of the first fundamental form, while
,
and
are the coefficients of the second fundamental form of the surface, computed at the given point.
The half-sum of the principal curvatures and
of the surface gives the mean curvature, while their product is equal to the Gaussian curvature of the surface. Equation (*) may be written as
![]() |
where is the mean, and
is the Gaussian curvature of the surface at the given point.
The principal curvatures and
are connected with the normal curvature
, taken in an arbitrary direction, by means of Euler's formula:
![]() |
where is the angle formed by the selected direction with the principal direction for
.
Comments
In the case of an -dimensional submanifold
of Euclidean
-space
principal curvatures and principal directions are defined as follows.
Let be a unit normal to
at
. The Weingarten mapping (shape operator)
of
at
in direction
is given by the tangential part of
, where
is the covariant differential in
and
is a local extension of
to a unit normal vector field.
does not depend on the chosen extension of
. The principal curvatures of
at
in direction
are given by the eigen values of
, the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of
define the higher mean curvatures of
, which include as extremal cases the mean curvature as the trace of
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 |
Principal curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_curvature&oldid=12102