Page 71 - Basic Structured Grid Generation

60 Basic Structured Grid Generation

3.6 The second fundamental form

3
The basic geometry of surfaces in E depends on two quadratic differential forms,
the first of which generates the metrical properties considered above. The study of the
shape of a surface S as viewed from the enveloping space gives rise to another quadratic
2
1
β
α
differential form b αβ du du in the differentials of surface co-ordinates u , u .We
consider an arbitrary surface curve C on S passing through a point P at which the
curve has tangent vector t = dr/ds (in the direction in which the arc-length parameter
s is increasing) and principal normal n.Then
dt
= κn = κ, (3.89)
ds
where κ is the curvature of C at P and κ is the curvature vector.
If N is a unit normal to the surface at P, we can decompose κ into

κ = κ N + κ g , (3.90)
where κ N is in the direction of N, κ N (called the normal curvature of C at P) is the
component κ ·N,and κ g is tangential to the surface at P. Since N·t = 0, differentiation
with respect to s as parameter gives
dt dN dN dr dN dr
N · + · t = N · (κ N + κ g ) + · = κ N + · = 0,
ds ds ds ds ds ds
which gives
α
α
dN dr ∂N du α du β du du β b αβ du du β
κ N =− · =− · a β = b αβ = , (3.91)
ds ds ∂u α ds ds ds ds ds 2
where it is convenient to make b αβ explicitly symmetric, with
1 ∂N ∂N
b αβ =− · a β + · a α . (3.92)
2 ∂u α ∂u β
We can also write
α β
κ N = b αβ λ λ , (3.93)
α
where λ represents a unit surface contravariant tangent vector to C.
Note that the sign of κ N does not depend on the orientation of C (the direction
in which s is measured), but it does depend on the direction in which the surface
normal N is taken.
Since N is normal to the surface tangent vectors, differentiating N · a α = 0 with
β
respect to u gives
2
∂N ∂a α ∂ r
· a α =−N · =−N · ,
α
∂u β ∂u β ∂u ∂u β
which we may note is in fact already symmetric in α and β. Thus an alternative
expression for b αβ is
2
∂ r
b αβ = N · . (3.94)
α
∂u ∂u β