Geodesic torsion

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