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''of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g0441702.png" /> on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g0441703.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g0441704.png" />''
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The rate of rotation of the tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g0441705.png" /> around the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g0441706.png" />. The rate is measured with respect to the arc length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g0441707.png" /> during the movement of the tangent lines along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g0441708.png" />. The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g0441709.png" /> and the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417010.png" /> are supposed to be regular and oriented. The geodesic torsion on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417011.png" /> is determined by the points and the direction of the curve and equals the torsion of the geodesic line in that direction. The geodesic torsion is given by
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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/g/g044/g044170/g04417012.png" /></td> </tr></table>
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''of a curve  $  \gamma $
 +
on a surface  $  F $
 +
in  $  E  ^ {3} $''
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417013.png" /> is the radius vector of the curve; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417014.png" /> is the unit normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417015.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417016.png" /> is the ordinary torsion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417017.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417018.png" /> is the angle between the osculating plane of the curve and the tangent plane to the surface; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417020.png" /> are the principal curvatures of the surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417021.png" /> is the angle between the curve and the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044170/g04417022.png" />.
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The rate of rotation of the tangent plane to  $  F $
 +
around the tangent to $  \gamma $.  
 +
The rate is measured with respect to the arc length  $  s $
 +
during the movement of the tangent lines along  $  \gamma $.
 +
The curve $  \gamma $
 +
and the surface  $  F $
 +
are supposed to be regular and oriented. The geodesic torsion on  $  F $
 +
is determined by the points and the direction of the curve and equals the torsion of the geodesic line in that direction. The geodesic torsion is given by
  
 +
$$
 +
\tau _ {g}  =  \left (
  
 +
\frac{d \mathbf r }{ds }
 +
 +
\mathbf n
 +
\frac{d \mathbf n }{ds }
 +
\right )  = \
 +
\tau +
 +
\frac{d \phi }{ds }
 +
  =  ( k _ {2} - k _ {1} )  \sin  \alpha  \cos  \alpha .
 +
$$
 +
 +
Here  $  \mathbf r $
 +
is the radius vector of the curve;  $  \mathbf n $
 +
is the unit normal to  $  F $;
 +
$  \tau $
 +
is the ordinary torsion of  $  \gamma $;
 +
and  $  \phi $
 +
is the angle between the osculating plane of the curve and the tangent plane to the surface;  $  k _ {1} $
 +
and  $  k _ {2} $
 +
are the principal curvatures of the surface and  $  \alpha $
 +
is the angle between the curve and the direction of  $  k _ {1} $.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  pp. 395  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 153; 261</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''3''' , Publish or Perish  pp. 1–5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  pp. 395  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 153; 261</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''3''' , Publish or Perish  pp. 1–5</TD></TR></table>

Latest revision as of 19:41, 5 June 2020


of a curve $ \gamma $ on a surface $ F $ in $ E ^ {3} $

The rate of rotation of the tangent plane to $ F $ around the tangent to $ \gamma $. The rate is measured with respect to the arc length $ s $ during the movement of the tangent lines along $ \gamma $. The curve $ \gamma $ and the surface $ F $ are supposed to be regular and oriented. The geodesic torsion on $ F $ is determined by the points and the direction of the curve and equals the torsion of the geodesic line in that direction. The geodesic torsion is given by

$$ \tau _ {g} = \left ( \frac{d \mathbf r }{ds } \mathbf n \frac{d \mathbf n }{ds } \right ) = \ \tau + \frac{d \phi }{ds } = ( k _ {2} - k _ {1} ) \sin \alpha \cos \alpha . $$

Here $ \mathbf r $ is the radius vector of the curve; $ \mathbf n $ is the unit normal to $ F $; $ \tau $ is the ordinary torsion of $ \gamma $; and $ \phi $ is the angle between the osculating plane of the curve and the tangent plane to the surface; $ k _ {1} $ and $ k _ {2} $ are the principal curvatures of the surface and $ \alpha $ is the angle between the curve and the direction of $ k _ {1} $.

Comments

References

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) pp. 395 (Translated from French)
[a2] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 153; 261
[a3] M. Spivak, "A comprehensive introduction to differential geometry" , 3 , Publish or Perish pp. 1–5
How to Cite This Entry:
Geodesic torsion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_torsion&oldid=18383
This article was adapted from an original article by Yu.S. Slobodyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article