Page 57 - Basic Structured Grid Generation

46 Basic Structured Grid Generation

3.2 Intrinsic geometry of surfaces

Intrinsic properties of surfaces are those which can be formulated without reference to
3
the space (in the present case E ) in which they are embedded, in particular without
reference to vectors normal to the surface. These properties depend essentially on a
certain quadratic differential form, the so-called first fundamental form, which contains
all the basic information about the metrical properties of the surface.
Here we assume that a given surface S is parametrized as in eqn (3.3), with param-
α
eters u ,α = 1, 2 (or, where this would make for clarity, u, v). It is again convenient
α
to use superscript indices, as the u will serve as curvilinear co-ordinates for the sur-
face, but the convention now is that Greek indices can take only the values 1 or 2.
Moreover, repeated Greek indices in an expression normally imply summation over
the values 1 and 2, according to the corresponding summation convention. We shall
also retain the background rectangular cartesian co-ordinates y 1 ,y 2 ,y 3 (or x, y, z), and
3
3
2
1
a set of curvilinear co-ordinates x ,x ,x (or ξ, η, ς) for the space E . On S we have
2
1
y i = y i (u ,u ), i = 1, 2, 3.
α
At a point P on S two co-ordinate curves u = constant intersect, and vectors in the
tangent plane at P, tangential to these curves, are
∂r
a α = , α = 1, 2. (3.12)
∂u α
With reference to the background space we have
∂r ∂x i ∂x i
∂r ∂y i ∂y i
a α = = i i = = g i , (3.13)
i
∂y i ∂u α ∂u α ∂x ∂u α ∂u α
where i i and g i are the unit basis vectors and covariant base-vectors of the background
three-dimensional cartesian and curvilinear systems, respectively. Here the index i is
taken to be summed from 1 to 3.
A curve C on the surface S can be given in terms of a parameter t by
α α
u = u (t), α = 1, 2. (3.14)
An infinitesimal line-segment of C may be represented by
∂r α α
dr = du = a α du , (3.15)
∂u α
with α summed over the values 1 and 2. Hence differentials of arc-length ds are
given by
2 α β α β
ds = dr · dr = (a α du ) · (a β du ) = a α · a β du du .
This gives the first fundamental form referred to above:

2
α
β
ds = a αβ du du , (3.16)
with
i j i j
∂y i ∂y i ∂x ∂x ∂x ∂x
a αβ = a α · a β = = g i · g j = g ij , (3.17)
α
α
∂u ∂u β ∂u α ∂u β ∂u ∂u β