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