174 COMPUTER AIDED ENGINEERING DESIGN
which using Taylor series expansion is
d
u
du d
u
dud
u
+ + +^1 du d
2
() +^1
2
()
22
2
2 2
2
≅ ∂^2
∂
∂
∂
∂
∂∂
∂
∂
∂
∂
⎡
⎣
⎢
⎤
⎦
⎥⋅
rr rr r
n
v
v
v
v
v
v
= + + +^1
2
() +^1
2
u ()
22
2
2 2
2
rn rn⋅⋅∂ r n r n r n^2
∂∂
⎛
⎝⎜
⎞
⎠⎟
⋅ ∂
∂
⋅ ∂
∂
du d ⋅
u
dud
u
v v du d
v
v
v
v
Sincen is perpendicular to the tangent plane, ru·n=rv·n = 0, hence
d
u
dud
u
= +^1 du d
2
() +^1
2
()
(^22)
2
2 2
2
∂ 2
∂∂
⎛
⎝
⎜
⎞
⎠
⎟⋅
∂
∂
⋅ ∂
∂
⋅
r
n r n r n
v
v
v
v
=^1
2
[ ]
du d
du
d
uu u
u
v
v
v
vvv
rnrn
rnrn
⋅⋅
⋅⋅
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥ (6.16)
The matrix D
rnrn
rnrn
uu u
u
⋅⋅
⋅⋅
⎡
⎣
⎢
⎤
⎦
⎥
v
vvv
is called the second fundamental matrix of the surface. Using
G 11 G 22 – G 12 G 21 = (ru×rv)· (ru×rv) in Eq. (6.4), we get
n
rr
rr
rr
| |
11 22 – 122
u
u
u
GG G
×
×
v ×
v
v
(6.17)
From Eq. (6.16)
2 = [ ]
ddu d = ( ) + 2^22 + ( )
du
d
du dud d
uu u
u
v v uu u vv
v
vvv
vvv
rnrn
rnrn
rn rn rn
⋅⋅
⋅⋅
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥ ⋅⋅⋅
or
2 d = L(du)^2 + 2Mdudv + N(dv)^2
where L = ruu·n, M = ruv·n, N = rvv·n (6.18)
S : r = r (u,v)
n
P
R
d
Tangent plane
Figure 6.9 Deviation of R from the tangent plane