Page 58 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 47

where g ij is the covariant metric tensor of the background curvilinear system; a αβ
is called the covariant metric tensor of the surface. The justification for calling it a
covariant tensor is that it has second-order covariant tensor properties with respect to
1
2
transformations of surface co-ordinates, in the sense that a transformation from (u ,u )
1
2
to a different pair (u , u ) produces a new set of values
γ δ γ δ
∂y i ∂y i ∂u ∂y i ∂u ∂y i ∂u ∂u
a αβ = α = α = α a γδ (3.18)
γ
β
∂u ∂u β ∂u ∂u ∂u ∂u δ ∂u ∂u β
by eqn (3.17). Thus a αβ transforms according to eqn (1.76), except that the indices
γ, δ are summed only over the values 1 and 2.
The first fundamental form is also commonly denoted in terms of u, v as
2
2
2
ds = E(du) + 2F du dv + G(dv) , (3.19)
where clearly
E = a 11 , F = a 12 = a 21 , G = a 22 . (3.20)
We also have the formula

α
ds du du β
= a αβ , (3.21)
dt dt dt
giving arc-length
t

α β
s = a αβ ˙u ˙u dt (3.22)
t 0
measured from some point on the curve with t = t 0 , with derivatives with respect to
t denoted by a dot.
2
1
Exercise 1. For the surface z = f (x, y) parametrized with u = x, u = y, show that
2 2

∂f ∂f ∂f ∂f
a 11 = 1 + , a 12 = , a 22 = 1 + . (3.23)
∂x ∂x ∂y ∂y
2
2
Exercise 2. For the surface r = r(u, v) = (u + v ,u + v, uv), show that
2
2
2
2
a 11 = 1 + 4u + v , a 12 = 2u + 2v + uv, a 22 = 1 + u + 4v .
The quadratic form (3.16) must be positive definite, and this holds if and only if
2
a 11 > 0 and a = a 11 a 22 − (a 12 ) > 0, (3.24)
where a is the second-order determinant of the 2 × 2matrix (a αβ ).
If inequalities (3.24) are both satisfied, then we necessarily also have a 22 > 0. Arc-
1
1
2
length along the u and u co-ordinate curves is given by √ a 11 du and √ a 22 du 2
respectively. The second inequality in eqn (3.24) corresponds to the inequality
2
2
2
2
|a 1 × a 2 | =|a 1 | |a 2 | − (a 1 · a 2 ) > 0,
which is guaranteed if the point P under consideration is non-singular. It follows,
incidentally, that
√
|a 1 × a 2 |= a. (3.25)