Page 78 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 67

3.8 Weingarten, Gauss, and Gauss-Codazzi equations

Here we investigate the spatial derivatives of the surface covariant base vectors a α
and the surface normal N.Since N is a unit vector, we have N · N = 1, and partial
α
differentiation with respect to u gives immediately
∂N
N · = 0,
∂u α
α
which implies that the vectors ∂N/∂u lie in the tangent plane and must be linear
combinations of a 1 , a 2 . We have already seen in eqn (3.91) that a definition of b αβ was

∂N
b αβ =− · a β .
∂u α
Consequently, if we write
∂N
γ
= c a γ
α
∂u α
γ
for some set of coefficients c α , taking scalar products of both sides with a β gives
γ
−b αβ = c a βγ .
α
It follows, using eqn (3.29), that
γ
c =−b αβ a βγ ,
α
and hence
∂N βγ γ
=−b αβ a a γ =−b a γ , α = 1, 2, (3.121)
α
∂u α
which are Weingarten’s equations.
Exercise 12. Show that in an alternative notation Weingarten’s equations can be
written:
∂N MF − LG LF − ME
= a 1 + a 2 ,
∂u EG − F 2 EG − F 2
∂N NF − MG MF − NE
= a 1 + a 2 .
∂v EG − F 2 EG − F 2
We have previously noted that the derivatives of the surface covariant base vectors
with respect to the surface co-ordinates are not themselves surface vectors. However,
β
2
α
β
recalling that ∂a α /∂u = ∂ r/∂u ∂u , it may be seen that eqns (3.50) and (3.94)
β
immediately provide the required projections of ∂a α /∂u in the directions a 1 , a 2 ,and
N. Thus, as may be verified by taking scalar products of both sides of the following
µ
with a and N,
∂a α γ
= a γ + b αβ N, (3.122)
αβ
∂u β
which is Gauss’s formula.