Computer Aided Engineering Design

170 COMPUTER AIDED ENGINEERING DESIGN

For a surface in implicit form, that is, f(x,y,z) = 0, the normal N and unit normal n at a point can
be obtained from

Ni jknN

N

= + + , =

| |

∂

∂

∂

∂

∂

∂

f

x

f

y

f

z

(6.5)

The plane containing the tangent vectors ru(u,v) and rv(u,v) at P(u 0 ,v 0 ) = P(x 0 ,y 0 ,z 0 ) on
the surface is called the tangent plane. To determine its equation, we can select any generic point
Q(x,y,z) on the tangent plane, different from P. Since the normal N(u 0 ,v 0 ) and the vector PQ are
perpendicular to each other, their scalar product is zero. With

PQ = (x–x 0 )i + (y–y 0 )j + (z–z 0 )k and PQ·N = 0

we have

PQ (r ( , ) ( , )) = r

–––

= 0

000

⋅×uuuuuu

xx yy zz

xyz

xyz

vvv

vvv

(6.6)

where (xu,yu,zu) and (xv,yv,zv) are defined in Eq. (6.3) and are evaluated at (u 0 ,v 0 ). Following the
expression of the normal in Eq. (6.5), for a surface in the form f (x,y,z) = 0, the tangent plane is given by

(– xx 000 ) + ( – ) + ( – ) = 0

f

x

yy

f

y

zz

f

z

∂

∂

∂

∂

∂

∂

(6.7)

with the derivatives evaluated at (x 0 ,y 0 ,z 0 ).

From the foregoing discussion, we may realize that at a regular point, the normal to the suface is
well-defined.

Figure 6.6 Normal and tangent plane

v

u

N (u 0 ,v 0 )

n rv

Tangent plane

ru

Q(x,y,z)

r(u,v) Z

Y

X

k

j

i

O

p