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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746602.png" /> are the roots of the quadratic equation
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746606.png" /> are the coefficients of the [[First fundamental form|first fundamental form]], while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746609.png" /> are the coefficients of the [[Second fundamental form|second fundamental form]] of the surface, computed at the given point.
+
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
  
The half-sum of the principal curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466011.png" /> 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
+
$$ \tag{* }
 +
\left |
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466012.png" /></td> </tr></table>
+
\begin{array}{ll}
 +
L - kE  &M - kF  \\
 +
M - kF  &N - kG  \\
 +
\end{array}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466013.png" /> is the mean, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466014.png" /> is the Gaussian curvature of the surface at the given point.
+
\right |  = 0,
 +
$$
  
The principal curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466016.png" /> are connected with the normal curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466017.png" />, taken in an arbitrary direction, by means of Euler's formula:
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466018.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466019.png" /> is the angle formed by the selected direction with the principal direction for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466020.png" />.
+
$$
 +
k  ^ {2} - 2Hk + K  = 0,
 +
$$
  
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466021.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466022.png" /> of Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466023.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466024.png" /> principal curvatures and principal directions are defined as follows.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466025.png" /> be a unit normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466026.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466027.png" />. The Weingarten mapping (shape operator) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466030.png" /> in direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466031.png" /> is given by the tangential part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466033.png" /> is the [[Covariant differential|covariant differential]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466035.png" /> is a local extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466036.png" /> to a unit normal vector field. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466037.png" /> does not depend on the chosen extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466038.png" />. The principal curvatures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466039.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466040.png" /> in direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466041.png" /> are given by the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466042.png" />, the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466043.png" /> define the higher mean curvatures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466044.png" />, which include as extremal cases the mean curvature as the trace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466045.png" /> and the Lipschitz–Killing curvature as its determinant.
+
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
How to Cite This Entry:
Principal curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_curvature&oldid=12102
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article