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Difference between revisions of "Geodesic torsion"

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are the principal curvatures of the surface and  $  \alpha $
 
are the principal curvatures of the surface and  $  \alpha $
 
is the angle between the curve and the direction of  $  k _ {1} $.
 
is the angle between the curve and the direction of  $  k _ {1} $.
 
====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>
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<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>
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Latest revision as of 18:47, 11 April 2023


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} $.

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=53779
This article was adapted from an original article by Yu.S. Slobodyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article