Computer Aided Engineering Design

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)