Geodesic torsion

of a curve on a surface in

The rate of rotation of the tangent plane to around the tangent to . The rate is measured with respect to the arc length during the movement of the tangent lines along . The curve and the surface are supposed to be regular and oriented. The geodesic torsion on 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

Here is the radius vector of the curve; is the unit normal to ; is the ordinary torsion of ; and is the angle between the osculating plane of the curve and the tangent plane to the surface; and are the principal curvatures of the surface and is the angle between the curve and the direction of .

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