DIFFERENTIAL GEOMETRY OF SURFACES 175
HereL,M and N are the second fundamental form coefficients. The second fundamental matrix D can
then be expressed as
D =
LM
MN
⎡
⎣
⎢
⎤
⎦
⎥ (6.19)
Using Eqs. (6.17), (6.18) and (6.19), we have
D
rnr n
rnrn
rrrrrr
rrrrrr
=
=^1
( ) ( )
11 22 ( ) ( )
12
2
uu u
u
uu u u u
GG G uu u
⋅⋅
⋅⋅
⎡
⎣
⎢
⎤
⎦
⎥
⋅× ⋅×
⋅× ⋅×
⎡
⎣
⎢
⎤
⎦
⎥
v
vvv
vvv
vvvvv
(6.20)
From Eq. (6.1) we have
ruu=xuui + yuuj+zuuk
ruv=xuvi + yuvj+zuvk
rvv=xvvi + yvvj + zvvk (6.21a)
and rr
ijk
uuuuxyz
xyz
= × v
vvv
we get from Eq. (6.20) (6.21b)
D =^1
22 2 + +
11 12
ABC 21 22
DD
DD
⎡
⎣
⎢
⎤
⎦
⎥
where A = yuzv–yvzu, B = zuxv–zvxu, C = xuyv–xvyu
and D
xyz
xyz
xyz
DD
xyz
xyz
xyz
D
xyz
xyz
xyz
uu uu uu
uuu
uuu
11 = , 12 = 21 = uuu , 22 = uuu^
vvv
vvv
vvv
vvvvvv
vvv
(6.22)
6.4 Classification of Points on a Surface
From Eq. (6.18), we observe that deviation d of a point R on the surface from a point P along the
normaln through P is given by
dLduMdudNd =^1
2
(^22 + 2 vv + ) (6.23)
To realize on which side of the tangent plane R lies, we can determine whether d is positive, negative
or zero. The tangent plane at point P will intersect the surface at all points where d= 0, that is
Ldu M dud Nd du
MMLN
L
(^22) d
2
- 2 + = 0 =
- –
vvv⇒
±
(6.24)