Difference between revisions of "Principal curvature"
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− | + | The [[Normal curvature|normal curvature]] of a surface in a principal direction, i.e. in a direction in which it assumes an extremal value. The principal curvatures $ k _ {1} $ | |
+ | 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|first fundamental form]], while $ L $, | ||
+ | $ M $ | ||
+ | and $ N $ | ||
+ | are the coefficients of the [[Second fundamental form|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|mean curvature]], while their product is equal to the [[Gaussian curvature|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==== | ====Comments==== | ||
− | In the case of an | + | 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 | + | 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|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==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish pp. 1–5</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish pp. 1–5</TD></TR></table> |
Latest revision as of 14:54, 7 June 2020
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 $ k _ {1} $
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 |
Principal curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_curvature&oldid=12102