DESIGN OF SURFACES 209

n

ij

(0.5, 0.5) = ij

46.5 + 46.5

46.5 + 46.5

=^1

2

( + )

22

Now ruu(u,v) = UMuuGMTVT = [(42u– 26) (–42u + 16) 0]

ruv(u,v) = UMuGMvTVT = [0 0 0]

rvv(u,v) = UMGMvvTVT = [0 0 0]

using which

L = ruu(u,v)·n (u,v) =

• 1260 + 1260 – 780
  (–126 + 96 + 30) + (– 126 + 156 )

2

2222

uu

uu u u

M = ruv(u,v)·n (u,v) = 0

N = rvv(u,v)·n (u,v) = 0

It can be seen that LN–M^2 = 0 for which the Gaussian curvature K LN M
GG G

= –

= 0

2

(^112212)
2 at all points
on the surface. Hence the surface is developable. Further, G 11 ,G 12 ,G 22 are given from Eq. (7.18) by
G 11 = ru(u,v)·ru(u,v) = (21u^2 – 26u)^2 + (–21u^2 + 16u + 5)^2
G 12 = ru(u,v)·rv(u,v) = 0
G 22 = rv(u,v)·rv(u,v) = 36
= 36 [(21u^2 – 26u)^2 + (–21u^2 + 16u + 5)^2 ]
H
GN G L GM
GG G
L
G G
= + – 2 uu
2[ – ]

2

(0.5, 0.5) (0.5, 0.5)
2 (0.5, 0.5)
11 22 12
11 22 122 11 11
rn⋅
Hereruu(0.5, 0.5) = (–5i– 5j),n(0.5, 0.5) = (i + j)/√2, for which H = – 0.0294. The equation of the
tangent plane at (u = 0.5, v = 0.5) is given using Eq. (7.17) as
Figure 7.4 Ferguson patch for Example 7.2.
6
4
2
(^06)
4
2
0
–5
0
(a)
6 5 4 3 2 1 0 6 4 2
0 6 4
2
0
z
(b)
x
5
y