Page 77 - Basic Structured Grid Generation

66 Basic Structured Grid Generation

the principal directions at P. Surface curves to which the tangents are everywhere in
the principal directions are called lines of curvature.
The case λ max = λ min arises at points where the components b αβ are proportional
to a αβ , in other words there is a constant k such that b αβ = ka αβ for all α, β ranging
over the values 1, 2. In this case the normal curvature at the point must be equal to
k for all normal sections, by eqn (3.101). Such points (for example, all points on the
surface of a sphere) are called umbilics.
Note that from b αβ we can derive the associated mixed second-order surface tensor
α
αβ
αβ
b = a b γβ , raising one index by the application of the surface metric tensor a .
γ
Equations (3.111) and (3.112) can now be written as
2
1 2
1 2
1
1 α
1
α
κ m = b = (b + b ) and κ G = det(b ) = b b − b b . (3.118)
2 α 2 1 2 β 1 2 2 1
α
If we assume that we can choose our surface co-ordinates u such that the surface
co-ordinate curves coincide with the lines of curvature, certain simplifications occur.
Firstly, since the lines of curvature are orthogonal to each other, we have a 12 = F =
0, by eqn (3.41). Secondly, writing the principal curvatures as κ a and κ b (without
specifying which is the maximum or minimum), eqns (3.107) become, taking the unit
√
α
1
1
vector λ in the u -direction, where the principal curvature is κ a and λ = 1/ a 11 ,
2
λ = 0 by eqn (3.40),
1 1
(b 11 − κ a a 11 )√ = 0and (b 21 − κ a a 21 )√ = 0.
a 11 a 11
Hence we also have b 12 = b 21 = M = 0, and the principal curvatures are
b 11 L b 22 N
κ a = = and κ b = = (3.119)
a 11 E a 22 G
similarly.
Example: For the surface of revolution as given by eqn (3.32), we have already seen
that F = M = 0. Hence the co-ordinate curves are also lines of curvature in that case.
From eqns (3.33) and (3.99) we immediately obtain the principal curvatures
1
2 −
3 (f 2 + g ) 2 g
2 −

κ a = (f 2 + g ) 2 (f g − f g ) and κ b = .
f
α
More generally, consider a unit directional vector µ making an angle ψ with the
κ a line of curvature. Then
1 1
1 2
µ = √ cos ψ and µ = √ sin ψ
a 11 a 22
making use of eqns (3.39) and (3.40). The curvature of a normal section in this direction
is, by eqn (3.93), given by
α β b 11 2 b 22 2 2 2
κ = b αβ µ µ = cos ψ + sin ψ = κ a cos ψ + κ b sin ψ, (3.120)
a 11 a 22
and this is sometimes referred to as Euler’s Theorem.