Rendering - Vector Analysis for Computer Graphics - page 230

and

N = N × N A

then

N × D = N A

Furthermore,

N × D = N A (10.14)

But as N is perpendicular to D , we have

sin 90 = N × D = N D N D (10.15)

Equations (10.14) and (10.15) imply that D = A .

N is rotated , as shown in Fig. 10.8(a), where

=

D

=

A

tan

N N

or

⎛
⎝

⎞
⎠

F
v

P
u −

F
u

P
v

tan − 1 =

P
u ×

P
v

So this allows us to develop a perturbation strategy based upon rotations, rather than offsets.
Now let's illustrate the above analysis with an example. To keep the mathematics simple, let's
use a cylinder as the bivariate vector function, as shown in Fig. 10.9.

Y

v

v

Z

P (u , v)

u u

X

Figure 10.9.

For instance, if the radius and the height of the cylinder equal 1 and uv ∈ 02, then

the x component is given by cos u,
the y component is given by v,
the z component is given by − sinu,