Namespaces
Variants
Actions

Difference between revisions of "Geodesic curvature"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
''at a point of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440701.png" /> on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440702.png" />''
+
<!--
 +
g0440701.png
 +
$#A+1 = 30 n = 0
 +
$#C+1 = 30 : ~/encyclopedia/old_files/data/G044/G.0404070 Geodesic curvature
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The rate of rotation of the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440703.png" /> around the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440704.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440705.png" />, i.e. the projection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440706.png" /> of the vector of the angular rate of rotation of the tangent moving along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440707.png" />. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440709.png" /> are regular and oriented, and that the velocity is taken relative to the arc length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407010.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407011.png" />. The geodesic curvature can be defined as the curvature of the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407012.png" /> on the plane tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407013.png" /> at the point under consideration. The geodesic curvature is
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/g/g044/g044070/g04407014.png" /></td> </tr></table>
+
''at a point of a curve  $  \gamma = \mathbf r ( t) $
 +
on a surface  $  F $''
  
where a prime denotes differentiation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407015.png" />.
+
The rate of rotation of the tangent to $  \gamma $
 +
around the normal  $  \mathbf n $
 +
to  $  F $,
 +
i.e. the projection on  $  \mathbf n $
 +
of the vector of the angular rate of rotation of the tangent moving along  $  \gamma $.  
 +
It is assumed that  $  \gamma $
 +
and  $  F $
 +
are regular and oriented, and that the velocity is taken relative to the arc length  $  s $
 +
along  $  \gamma $.  
 +
The geodesic curvature can be defined as the curvature of the projection of  $  \gamma $
 +
on the plane tangent to  $  F $
 +
at the point under consideration. The geodesic curvature is
  
The geodesic curvature forms part of the function expressing the variation of the length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407016.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407017.png" /> is varied on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407018.png" />. If the ends are fixed:
+
$$
 +
k _ {g= \
  
<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/g/g044/g044070/g04407019.png" /></td> </tr></table>
+
\frac{( \mathbf r  ^  \prime  , \mathbf r  ^ {\prime\prime} , \mathbf n ) }{| \mathbf r  ^  \prime  |  ^ {3} }
 +
,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407020.png" /> is the vector of variation of the curve. Curves for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407021.png" /> are geodesic lines (cf. [[Geodesic line|Geodesic line]]).
+
where a prime denotes differentiation with respect to  $  t $.
  
The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407022.png" /> is called the total geodesic curvature, or the rotation, of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407023.png" />. The connection between the rotation of a closed contour and the total curvature of the included region on the surface is given by the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]].
+
The geodesic curvature forms part of the function expressing the variation of the length  $  L( \gamma ) $
 +
as  $  \gamma $
 +
is varied on $  F $.
 +
If the ends are fixed:
  
The geodesic curvature forms a part of the [[Interior geometry|interior geometry]] of the surface, and can be expressed in terms of the metric tensor and the derivatives of the intrinsic surface coordinates with respect to its parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407024.png" />. If the geometry of a Riemannian space is studied without considering the latter to be immersed in Euclidean space, then the geodesic curvature is the only curvature which can be defined for a curve and the word "geodesic" is omitted. In considering curves on a submanifold of a Riemannian space, the curvature of a curve may be defined in the external space and in the submanifold — just like the spatial curvature and the geodesic curvature of a curve on a surface.
+
$$
 +
\delta L  = - \int\limits k _ {g} ( \mathbf v , \delta \mathbf r )  ds,\ \
 +
\textrm{ where } \
 +
\mathbf v = \left [ \mathbf n ,  
 +
\frac{d \mathbf r }{ds }
 +
\right ]
 +
$$
  
The concept of the geodesic curvature may be introduced for a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407025.png" /> on a general convex surface. If the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407026.png" /> has a length and each one of its arcs has a certain rotation, the right (left) geodesic curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407027.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407028.png" /> is the limit of the ratio of the right (left) rotation of the arc to its length, under the condition that the arc is contracted towards the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407029.png" />.
+
and  $  \delta \mathbf r $
 +
is the vector of variation of the curve. Curves for which  $  k _ {g} \equiv 0 $
 +
are geodesic lines (cf. [[Geodesic line|Geodesic line]]).
  
Two concepts of geodesic curvature are defined in a Finsler space. These differ in the manner in which the length of the vector replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407030.png" /> is determined. On geodesics these geodesic curvatures are zero.
+
The integral  $  \int k _ {g}  ds $
 +
is called the total geodesic curvature, or the rotation, of the curve  $  \gamma $.  
 +
The connection between the rotation of a closed contour and the total curvature of the included region on the surface is given by the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]].
  
 +
The geodesic curvature forms a part of the [[Interior geometry|interior geometry]] of the surface, and can be expressed in terms of the metric tensor and the derivatives of the intrinsic surface coordinates with respect to its parameter  $  t $.
 +
If the geometry of a Riemannian space is studied without considering the latter to be immersed in Euclidean space, then the geodesic curvature is the only curvature which can be defined for a curve and the word  "geodesic"  is omitted. In considering curves on a submanifold of a Riemannian space, the curvature of a curve may be defined in the external space and in the submanifold — just like the spatial curvature and the geodesic curvature of a curve on a surface.
  
 +
The concept of the geodesic curvature may be introduced for a curve  $  \gamma $
 +
on a general convex surface. If the curve  $  \gamma $
 +
has a length and each one of its arcs has a certain rotation, the right (left) geodesic curvature of  $  \gamma $
 +
at a point  $  x $
 +
is the limit of the ratio of the right (left) rotation of the arc to its length, under the condition that the arc is contracted towards the point  $  x $.
 +
 +
Two concepts of geodesic curvature are defined in a Finsler space. These differ in the manner in which the length of the vector replacing  $  \mathbf v $
 +
is determined. On geodesics these geodesic curvatures are zero.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a2]</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">[a3]</TD> <TD valign="top">  B. O'Neill,  "Elementary differential geometry" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a2]</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">[a3]</TD> <TD valign="top">  B. O'Neill,  "Elementary differential geometry" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR></table>

Latest revision as of 19:41, 5 June 2020


at a point of a curve $ \gamma = \mathbf r ( t) $ on a surface $ F $

The rate of rotation of the tangent to $ \gamma $ around the normal $ \mathbf n $ to $ F $, i.e. the projection on $ \mathbf n $ of the vector of the angular rate of rotation of the tangent moving along $ \gamma $. It is assumed that $ \gamma $ and $ F $ are regular and oriented, and that the velocity is taken relative to the arc length $ s $ along $ \gamma $. The geodesic curvature can be defined as the curvature of the projection of $ \gamma $ on the plane tangent to $ F $ at the point under consideration. The geodesic curvature is

$$ k _ {g} = \ \frac{( \mathbf r ^ \prime , \mathbf r ^ {\prime\prime} , \mathbf n ) }{| \mathbf r ^ \prime | ^ {3} } , $$

where a prime denotes differentiation with respect to $ t $.

The geodesic curvature forms part of the function expressing the variation of the length $ L( \gamma ) $ as $ \gamma $ is varied on $ F $. If the ends are fixed:

$$ \delta L = - \int\limits k _ {g} ( \mathbf v , \delta \mathbf r ) ds,\ \ \textrm{ where } \ \mathbf v = \left [ \mathbf n , \frac{d \mathbf r }{ds } \right ] $$

and $ \delta \mathbf r $ is the vector of variation of the curve. Curves for which $ k _ {g} \equiv 0 $ are geodesic lines (cf. Geodesic line).

The integral $ \int k _ {g} ds $ is called the total geodesic curvature, or the rotation, of the curve $ \gamma $. The connection between the rotation of a closed contour and the total curvature of the included region on the surface is given by the Gauss–Bonnet theorem.

The geodesic curvature forms a part of the interior geometry of the surface, and can be expressed in terms of the metric tensor and the derivatives of the intrinsic surface coordinates with respect to its parameter $ t $. If the geometry of a Riemannian space is studied without considering the latter to be immersed in Euclidean space, then the geodesic curvature is the only curvature which can be defined for a curve and the word "geodesic" is omitted. In considering curves on a submanifold of a Riemannian space, the curvature of a curve may be defined in the external space and in the submanifold — just like the spatial curvature and the geodesic curvature of a curve on a surface.

The concept of the geodesic curvature may be introduced for a curve $ \gamma $ on a general convex surface. If the curve $ \gamma $ has a length and each one of its arcs has a certain rotation, the right (left) geodesic curvature of $ \gamma $ at a point $ x $ is the limit of the ratio of the right (left) rotation of the arc to its length, under the condition that the arc is contracted towards the point $ x $.

Two concepts of geodesic curvature are defined in a Finsler space. These differ in the manner in which the length of the vector replacing $ \mathbf v $ is determined. On geodesics these geodesic curvatures are zero.

Comments

References

[a1] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a2] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a3] B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
[a4] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a5] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
How to Cite This Entry:
Geodesic curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_curvature&oldid=17534
This article was adapted from an original article by Yu.S. Slobodyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article