Page 69 - Basic Structured Grid Generation

58 Basic Structured Grid Generation

or, equivalently,
α
ν = ε βα λ β , (3.82)
in terms of the associated covariant components λ β , where lowering the index is
represented here by
λ α = a αβ λ β

rather than eqn (1.53). To verify eqn (3.82), note that, using eqn (3.45),
α β
β γ
α
γ
γ α
ε αβ λ ν = ε αβ λ (ε γβ λ γ ) = δ λ λ γ = λ λ γ = a βγ λ λ = 1 (3.83)
α
and
α β α γβ γβ
a αβ λ ν = a αβ λ (ε λ γ ) = ε λ β λ γ = 0,
due to the symmetric and skew-symmetric natures of λ β λ γ and ε γβ , respectively.
Similarly we can verify that
α
λ =−ε βα ν β . (3.84)
We now write α
δλ α
= κ g ν , (3.85)
δs
where the scalar magnitude κ g is called the geodesic curvature of the curve at the point
in question.
Because of eqns (2.33) and (3.62), ε αβ also acts as a constant for the purposes of
intrinsic differentiation. Hence eqn (3.82) yields
δν α βα δλ β βα
= ε = κ g ε ν β ,
δs δs
using the associated covariant components of the surface vector quantities δλ/δs, ν in
eqn (3.85). So, from eqn (3.84), we obtain
δν α α
=−κ g λ . (3.86)
δs
Equations (3.85) and (3.86) constitute the surface-Frenet equations for curves on
surfaces.
α
For an actual geodesic curve, eqn (3.72) shows that δλ /δs = 0. This means that
the geodesic curvature of a geodesic is zero.
The surface-Frenet equations may be expressed, using the surface form of eqn (2.32),
as
dλ α α β du γ dλ α α β γ α
+ βγ λ = + βγ λ λ = κ g ν ,
ds ds ds
dν α α β du γ dν α α β γ α
+ βγ ν = + βγ ν λ =−κ g λ . (3.87)
ds ds ds
An explicit formula for κ g may be obtained by multiplying the first of these equations
ν
through by ε να λ , implying summation over α and ν, and making use of eqn (3.83).