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Torsion of a geodesic. The torsion of a geodesic passing through point P of a surface in the direction of the unit tangent vector T is given by

where is the unit surface normal at point P and is the position vector at point P, or, equivalently, by

where E, F, G, L, M, N are the first and second fundamental coefficients.

Derivation. We begin with the Frenet-Serret formula

1) dn/ds = τ_{g}B - κT

which relates the geodesic torsion τ_{g} on the curve at point P to the quantities κ, T, n and B where
T, n and B are the unit tangent, principal normal and binormal vectors to the curve at point P.
We now take the dot product of both sides of 1) with the vector B

(dn/ds)∙B = (τ_{g}B - κT)∙B

(dn/ds)∙B = τ_{g}B∙B - κT∙B

Now B∙B = 1 and T∙B = 0 so

τ_{g} = (dn/ds)∙B

Now a geodesic C on a surface S has the properties that at each point of C the principal normal coincides with the normal to S. Hence n = N where N is the unit surface normal at P. Letting n = N we have

2) τ_{g} = (dN/ds)∙B

Now B = T n = T N. Substituting B = T N into 2) we have

3) τ_{g} = (dN/ds)∙(T
N)

Now T = where is the position vector of point P on the curve. Substituting into 3) we have

Now
=
du +
dv and dN = N_{u}du + N_{v}dv) so

5) dN
N = (
du +
dv )
(N_{u}du + N_{v}dv)∙N

= (N _{u}
N)du^{2} + [N _{u}
N + N _{v}
N ]dudv + (N _{v}
N)dv^{2}

Now setting N = / in 5) and evaluating all products , 4) becomes

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