# Geodesic torsion

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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=47086
This article was adapted from an original article by Yu.S. Slobodyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article