Computer Aided Engineering Design

DIFFERENTIAL GEOMETRY OF SURFACES 169

6.1.2 Tangent Plane and Normal Vector on a Surface

Referring to Eq. (6.2) for tangents ru(u,v) and rv(u,v) at P(u,v) on the surface, the normal at P is
a vector perpendicular to the plane containing ru(u,v) and rv(u,v). The normal N(u,v) and the unit
normaln(u,v) are given by

N(u,v) = ru(u,v)×rv (u,v), n

rr

rr

(, ) =

(, ) (, )

| ( , ) ( , ) |

u

uu

uu

u

u

v

vv

vv

v

v

×

×

(6.4)

and ru(u 0 ,v 0 )×rv(u 0 ,v 0 )≠ 0

that is, the slopes ru and rv should exist at P and that their cross product should not be zero. If all
points on the surface are regular points, the surface is a regular surface. If at some point P, ru(u 0 ,v 0 )
×rv(u 0 ,v 0 ) = 0 , then P is a singular point where the slopes ru and rv are either undefined (non-unique/
non-existing) or zero or coincident. Condition (Eq. (6.2)) requires that at least one of the Jacobians
(J 1 ,J 2 ,J 3 ) described below is non-zero.

If x

xu

u

x

xu

y

yu

u

y

yu

z

zu

u

z

zu

uuu =

(, )

, =

(, )

,

(, )

, =

(, )

, =

(, )

, =

∂ (, )

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

vv

v

vv

v

vv

vvvv

then rr

ijk

uuuuuuxyz

xyz

( 00 , ) (vvv 00 , ) =

vvv

×

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

= J 1 i + J 2 j + J 3 k

with the Jacobians

J

yz

yz

J

zx

zx

J

xy

xy

uu u u u u

12 3 = , = , =^

vvvvvv

, (6.3)

Figure 6.5 shows some examples of singular points or lines on the surface. In Figure 6.5 (a) and
(b), slope ru is not uniquely defined while in (c), P represents the tip of a cone where the slope again
is non-unique.

Figure 6.5 Singular points and lines on surfaces

rv

ru

ru

rv

ru

ru

P

(a) (b) (c)