Page 86 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 75

We can establish a connection with the Gauss Formula, eqn (3.122), if we multiply
that equation through by a αβ to get
2
∂ r γ ∂r
αβ αβ αβ
a = a αβ + a b αβ N.
α
∂u ∂u β ∂u γ
Hence, by eqns (3.111) and (3.163),
2
∂ r γ ∂r
αβ
a + ( B u ) = 2κ m N, (3.165)
α
∂u ∂u β ∂u γ
which may be written, in view of (3.164), as the identity
B r = 2κ m N. (3.166)
However, if we use the expression eqn (3.161) for B , we see that we have again
derived the surface metric identity (3.136).