Page 75 - Basic Structured Grid Generation

64 Basic Structured Grid Generation

α
α
or, in terms of the surface tangent vector λ = du /ds where the normal section cuts
the surface,
α β
(b αβ − κ N a αβ )λ λ = 0, (3.105)
that is,
2

dv dv
(b 11 − κ N a 11 ) + 2(b 12 − κ N a 12 ) + (b 22 − κ N a 22 ) = 0, (3.106)
du du
1
2
writing (u, v) instead of (u ,u ). We may look for stationary values of κ N as the
directional parameter dv/du varies (or as the plane of the normal section at P is rotated
about the normal N). Differentiation of eqn (3.106) with respect to this parameter,
assuming that κ N is stationary, gives

dv
(b 12 − κ N a 12 ) + (b 22 − κ N a 22 ) = 0,
du
and it follows, substituting this result in eqn (3.106), that we also have

dv
(b 11 − κ N a 11 ) + (b 12 − κ N a 12 ) = 0.
du
The last two equations may be summarized in the form
β
(b αβ − κ N a αβ )λ = 0, α = 1, 2. (3.107)
α
This is a set of two homogeneous linear equations in the unknown quantities λ ,
with non-trivial solutions if κ N satisfies the quadratic equation
det(b αβ − κ N a αβ ) = 0, (3.108)
which may be expressed as
2
0 = (b 11 − κ N a 11 )(b 22 − κ N a 22 ) − (b 12 − κ N a 12 )
2 2 2
= (b 11 b 22 − b ) − κ N (a 11 b 22 + a 22 b 11 − 2a 12 b 12 ) + κ (a 11 a 22 − a ).
12
N
12
Using eqns (3.24) and (3.30), we obtain
2
12
11
22
0 = det(b αβ ) − aκ N (a b 22 + a b 11 + 2a b 12 ) + aκ .
N
In other words, the stationary curvatures are the roots of the quadratic equation
1
2 αβ
κ − a b αβ κ N + det(b αβ ) = 0. (3.109)
N
a
The roots must give the maximum and minimum values of κ N :

1 αβ 1 2 1
2
αβ
κ max,min = a b αβ ± a b αβ − (b 11 b 22 − b ). (3.110)
12
2 2 a
These are called the principal curvatures of the surface at P. We can also define the
mean curvature of the surface
1
1 αβ
κ m = (κ max + κ min ) = a b αβ , (3.111)
2 2