Page 72 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 61

α β 1 2
The second fundamental form b αβ du du may be written, putting u = u, u =
v,as
2
2
L(du) + 2M du dv + N(dv) ,
where
2
2
2
∂ r ∂ r ∂ r
b 11 = L = N · , b 12 = b 21 = M = N · , b 22 = N = N · . (3.95)
∂u 2 ∂u∂v ∂v 2
If we define the direction of N in the usual right-handed sense with respect to the
tangent vectors a 1 , a 2 , we can write
a 1 × a 2 1
N = = √ a 1 × a 2 , (3.96)
|a 1 × a 2 | a
by eqn (3.25). Hence eqns (3.95) can be written as scalar triple products
2
2
2
1 ∂ r ∂r ∂r 1 ∂ r ∂r ∂r 1 ∂ r ∂r ∂r
L = √ , , ,M = √ , , ,N = √ , ,
a ∂u 2 ∂u ∂v a ∂u∂v ∂u ∂v a ∂v 2 ∂u ∂v
(3.97)
or as determinants, for example, with cartesian co-ordinates x, y, z,
2
2
2
∂ x ∂ y ∂ z

2 2
∂u ∂u ∂u 2

1 ∂x ∂y ∂z
L = √ , (3.98)
a ∂u ∂u ∂u

∂x ∂y ∂z

∂v ∂v ∂v
and similarly for M and N.
Exercise 9. For the surface of revolution defined by eqn (3.32) show that
1
2 2 −
L = (f (u) + g (u)) 2 [f (u)g (u) − f (u)g (u)], M = 0,
1
2
2

−
N = (f (u) + g (u)) 2 f(u)g (u). (3.99)
The second fundamental form is directly related to the distance, to second order
in small quantities, between points on the surface in a neighbourhood of the surface
point P and the tangent plane at P. The increment in position vector from P with
co-ordinates (u, v) to a neighbouring point Q(u + δu, v + δv) is, by Taylor Series
expansion,
δr = r(u + δu, v + δv) − r(u, v)
2
2
2
1 ∂ r 2 ∂ r ∂ r 2
= a 1 δu + a 2 δv + (δu) + 2 δu∂v + (δv)
2 ∂u 2 ∂u∂v ∂v 2
to second order in δu, δv. The distance between Q and the tangent plane through P
is given by projecting δr in the direction of the unit normal N to the tangent plane,