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
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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
.
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 |
Geodesic torsion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_torsion&oldid=47086