Page 154 - Intro to Tensor Calculus

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where κ (g) is called the geodesic curvature of the curve. In a similar manner it can be shown that δu α
δs
α
α
is a surface vector orthogonal to t . Let δu α = αt where α is a scalar constant to be determined. By
δs
α β
differentiating the relation a αβ t u = 0 intrinsically and simplifying we find that α = −κ (g) and therefore
δu α α
= −κ (g) t . (1.5.113)
δs
The equations (1.5.112) and (1.5.113) are sometimes referred to as the Frenet-Serret formula for a curve
relative to a surface.
Taking the intrinsic derivative of equation (1.5.111), with respect to the parameter s, we find that

δT i i δt α i du β α
= x α + x α,β t . (1.5.114)
δs δs ds
Treating the curve as a space curve we use the Frenet formulas (1.5.13). If we treat the curve as a surface
curve, then we use the Frenet formulas (1.5.112) and (1.5.113). In this way the equation (1.5.114) can be
writteninthe form
β α
α
i
i
κN = x κ (g) u + x i t t . (1.5.115)
α α,β
By using the results from equation (1.5.71) in equation (1.5.115) we obtain
i α β
i
i
κN = κ (g) u + b αβ n t t (1.5.116)
i
α
where u is the space vector counterpart of the surface vector u . Let θ denote the angle between the surface
i
i
i
normal n and the principal normal N , then wehavethatcos θ = n i N . Hence, by multiplying the equation
(1.5.116) by n i we obtain
α β
κ cos θ = b αβ t t . (1.5.117)
α
Consequently, for all curves on the surface with the same tangent vector t ,the quantity κ cos θ will remain
constant. This result is known as Meusnier’s theorem. Note also that κ cos θ = κ (n) is the normal component
of the curvature and κ sin θ = κ (g) is the geodesic component of the curvature. Therefore, we write the
equation (1.5.117) as
α β
κ (n) = b αβ t t (1.5.118)
α
which represents the normal curvature of the surface in the direction t . The equation (1.5.118) can also be
writteninthe form
α
du du β B
κ (n) = b αβ = (1.5.119)
ds ds A
which is a ratio of quadratic forms.
The surface directions for which κ (n) has a maximum or minimum value is determined from the equation
(1.5.119) which is written as
α β
(b αβ − κ (n) a αβ )λ λ =0. (1.5.120)
The direction giving a maximum or minimum value to κ (n) must then satisfy

β
(b αβ − κ (n) a αβ )λ =0 (1.5.121)