Page 76 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 65

as well as the Gaussian curvature
1 2 det(b αβ ) LN − M 2
κ G = κ max κ min = (b 11 b 22 − b ) = = . (3.112)
12
a det(a αβ ) EG − F 2

Exercise 10. Show that
EN − 2FM + GL
κ m = . (3.113)
2
2(EG − F )
Exercise 11. For the surface z = f (x, y), with covariant metric tensor components
given by eqn (3.23), show that the unit surface normal, the coefficients of the second
fundamental form, and the Gaussian curvature, are given by
1
−
∂f ∂f ∂f ∂f
2 2 2
N = − i − j + k 1 + + , (3.114)
∂x ∂y ∂x ∂y
2
2
2
1 ∂ f 1 ∂ f 1 ∂ f
L = √ , M = √ , N = √ , (3.115)
a ∂x 2 a ∂x∂y a ∂y 2

2 2
∂f ∂f
where a = 1 + + ,and
∂x ∂y
2 −2
2 2 2 2 2
∂ f ∂ f ∂ f ∂f ∂f
κ G = − 1 + + . (3.116)
∂x 2 ∂y 2 ∂x∂y ∂x ∂y
For elliptic points, as defined in the last section, κ max and κ min have the same
sign, and κ G is positive, whereas for hyberbolic points κ max and κ min have oppo-
site signs and κ G is negative. At parabolic points either κ max or κ min is zero, and
so is κ G .
If the surface tangents corresponding to these curvatures are λ α and λ α min ,
max
eqn (3.107) gives
β β
b αβ λ = κ max a αβ λ and
max max
β β
b αβ λ = κ min a αβ λ .
min min
Hence
(κ max − κ min )a αβ λ α λ β = κ max a αβ λ β λ α − κ min a αβ λ α λ β
max min max min max min
= b αβ λ β λ α − b αβ λ α λ β = 0,
max min max min
where the symmetry of both a αβ and b αβ has been used. It follows that, provided
κ max = κ min ,
a αβ λ α λ β = 0. (3.117)
max min
We see from eqn (3.47) that the corresponding surface vectors λ max , λ min must
be orthogonal to each other, except when κ max = κ min , in which case all normal
sections through P have the same curvature. The two orthogonal directions are called