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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 ole1.gif is the unit surface normal at point P and ole2.gif 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 = τgB - κT

which relates the geodesic torsion τg 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 = (τgB - κT)∙B


(dn/ds)∙B = τgBB - κTB

Now BB = 1 and TB = 0 so


τg = (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 = ole4.gif N where N is the unit surface normal at P. Letting n = N we have


2)        τg = (dN/ds)∙B

Now B = T ole5.gif n = T ole6.gif N. Substituting B = T ole7.gif N into 2) we have


3)        τg = (dN/ds)∙(T ole8.gif N)

Now T = ole9.gif where ole10.gif is the position vector of point P on the curve. Substituting into 3) we have


Now ole12.gif = ole13.gif du + ole14.gif dv and dN = Nudu + Nvdv) so

5)        dN ole15.gif ole16.gif N = ( ole17.gif du + ole18.gif dv ) ole19.gif (Nudu + Nvdv)∙N

                        = (N u ole20.gif N)du2 + [N u ole21.gif N + N v ole22.gif N ]dudv + (N v ole23.gif N)dv2

Now setting N = ole24.gif / ole25.gif in 5) and evaluating all products , 4) becomes


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