Page 66 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 55

Hence, with γ in place of α, eqn (3.67) becomes

1 ∂a αβ α β d 1 β
˙ u ˙u − a γβ ˙u = 0, γ = 1, 2. (3.68)
2f ∂u γ dt f
A simpler form of these equations results if we express the parametric form of the
solution curve in terms of arc-length s rather than t. Along a solution curve we now
have f = 1, a constant, by eqn (3.36). Thus eqn (3.68) becomes

α
d du β 1 ∂a αβ du du β
a γβ − = 0, (3.69)
ds ds 2 ∂u γ ds ds
that is,
α
α
2 β
d u ∂a γβ du du β 1 ∂a αβ du du β
a γβ 2 + α − γ = 0,
ds ∂u ds ds 2 ∂u ds ds
which may be expressed as
2 β
β
d u 1 ∂a γβ ∂a γα ∂a βα du du α
a γβ 2 + 2 α + β − γ = 0,
ds ∂u ∂u ∂u ds ds
β
α
exploiting the symmetry of (du /ds)(du /ds) with respect to β and α.Inother words,
using eqn (3.49),
β
2 β
d u du du α
a γβ 2 +[βα, γ ] = 0, γ = 1, 2. (3.70)
ds ds ds
Re-writing α as δ, multiplying through by a αγ (implying summation over γ ), and
using eqn (3.50), now gives
β
2 α
2 α
β
d u αγ du du δ d u α du du δ
+ a [βδ, γ ] = + βδ = 0, α = 1, 2. (3.71)
ds 2 ds ds ds 2 ds ds
Equations (3.70) and (3.71) are second-order differential equations which define
geodesic curves, curves whose length is stationary (rather than necessarily being a
minimum) with respect to small variations given two fixed end-points. If we are given
α
α
starting values of u and du /ds at a point on a surface, we can in principle solve
either of these pairs of equations and obtain a geodesic curve.
By the two-dimensional version of eqn (2.32), we can express eqn (3.71) in the
simple form
δ du α
= 0. (3.72)
δs ds
In practice eqns (3.71) may be quite complicated. For the surface of revolution as
defined by eqn (3.32) they become, using eqn (3.54),
2
d u 2 2 −1 du 2 2 2 dv 2

+ (f + g ) (f f + g g ) − (f + g )ff = 0,
ds 2 ds ds
2
d v du dv
f
+ 2f −1 = 0.
ds 2 ds ds
(3.73)