Page 80 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 69

so also must det(b αβ ) be intrinsic. Moreover, by eqn (3.112) we obtain Gauss’s
result
R 1212
κ G = , (3.128)
a
which shows that the Gaussian curvature is also an intrinsic quantity, that is, it depends
only on the tensor components a αβ and their derivatives. Such a quantity is sometimes
referred to as a bending invariant, as it remains unchanged by any deformation of
the surface which involves pure bending without stretching, shrinking, or tearing, thus
preserving distances between points, angles between directions at a point, and the
coefficients of the first fundamental form and their derivatives.
Equation (3.124) may be written in the form

∂b αβ µ µ ∂b αγ µ µ
b
− αγ βµ − βγ αµ = − b γµ − b
b
∂u γ ∂u β αβ βγ αµ
by subtracting the same term from each side, which gives covariant derivative
identities
b αβ,γ = b αγ,β . (3.129)
This contains only two non-trivial results:

b 11,2 = b 12,1 and b 21,2 = b 22,1 . (3.130)
These are the Codazzi equations(or the Mainardi-Codazzi equations).

Exercise 13. Express the Codazzi equations in the form
∂L ∂M 1 2 1 2
− = L 12 + M( 12 − ) − N 11
11
∂v ∂u
∂M ∂N 1 2 1 2
− = L 22 + M( 22 − ) − N . (3.131)
12
12
∂v ∂u
Exercise 14. Deduce from eqns (3.131) and (3.52) that when the co-ordinate curves
coincide with lines of curvature (so that F = M = 0), the Codazzi equations may be
expressed as
∂L 1 ∂E L N 1 ∂E
= + = (κ a + κ b )
∂v 2 ∂v E G 2 ∂v
∂N 1 ∂G L N 1 ∂G
= + = (κ a + κ b ). (3.132)
∂u 2 ∂u E G 2 ∂u

To derive the Gauss equation in the form due to Brioschi, we start with eqns (3.97)
and
2
2
2
2
1 ∂ r ∂r ∂r ∂ r ∂r ∂r ∂ r ∂r ∂r
2
LN − M = , , , , − , ,
a ∂u 2 ∂u ∂v ∂v 2 ∂u ∂v ∂u∂v ∂u ∂v
 

1  r uu · r vv r uu · r u r uu · r v r uv · r uv r uv · r u r uv · r v 

= r u · r vv r u · r u r u · r v − r u · r uv r u · r u r u · r v ,

a  
r v · r vv r v · r u r v · r v r v · r uv r v · r u r v · r v