Page 68 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 57

Further differentiation gives
2
2 1
2
1
2 2
d r d u d u ∂a 1 du

= a 1 + a 2 +
ds 2 ds 2 ds 2 ∂u 1 ds
1 2 2 2
∂a 1 ∂a 2 du du ∂a 2 du
+ + +
∂u 2 ∂u 1 ds ds ∂u 2 ds
2 β
d u ∂a β du β du γ
= a β 2 + γ ,
ds ∂u ds ds

and hence, with d/ds denoted by ( ) ,
β
2 β
∂a β d u du du γ

β γ
r · a α = a α · a β u β + a α · u u = a αβ +[βγ, α] , (3.77)
∂u γ ds 2 ds ds
using eqns (3.17) and (3.49).
Comparison with eqn (3.70) now shows that for a geodesic

r · a α = 0, α = 1, 2, (3.78)

which implies that the direction of r is orthogonal to the tangent plane to the surface

at any point. Recalling the significance of r from Chapter 2, we deduce that geodesics
have the property that the principal normal is always in the same direction as the normal
to the surface.
3.5 Surface Frenet equations and geodesic curvature
In Chapter 2 we looked at the curvature and torsion of a space-curve and derived the
1
Serret-Frenet equations. Now we discuss a general curve on a surface on which u and
2
u are surface co-ordinates. The curve is given by
α α
u = u (s)
in terms of arc-length as parameter, and the unit surface contravariant tangent vector
α
λ at any point satisfies
α β
a αβ λ λ = 1 (3.79)
by eqn (3.36). We can apply intrinsic differentiation to this equation, bearing in mind
that the a αβ s are effectively constant with respect to intrinsic differentiation, because
of eqns (2.33) and (3.60).
We obtain
δλ β δλ α β α δλ β
α
a αβ λ + a αβ λ = 2a αβ λ = 0, (3.80)
δs δs δs
β
using the symmetry of a αβ . Thus by eqn (3.39) δλ /δs are the contravariant compo-
α
nents of a surface vector which is orthogonal to λ .
α
α
Now let a contravariant unit vector parallel to δλ /δs be ν . We choose the sense
α
of ν such that
α β
ε αβ λ ν = 1, (3.81)