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