The Plane - Vector Analysis for Computer Graphics - page 103

uv ×

R

w

v

u

P

T

S

Figure 4.4.

We start with three points, Rx R y R z R S x S y S z S , and T x T y T z T , as shown
in Fig. 4.4, and create two vectors, u = − RT. The three points are assumed to be

in a counterclockwise sequence viewed from the direction of the surface normal. The vector
product u

= RS and v

× v provides a vector normal to the plane containing the points:

i j k
x u y u z u
x v y v z v

u × v =

= − RP. We now take another point, P xyz, which is on the plane, and form a vector, w

Therefore, the scalar product w 0. This condition can be expressed as a determinant

and converted into the general equation of a plane:

· u × v =

x w y w z w
x u y u z u
x v

w · u × v = = 0 (4.5)

y v z v

Expanding Eq. (4.5), we obtain
x w

+

y w

+

z w

=

y u z u
y v z v

z u x u
z v x v

x u y u
x v y v

0

which becomes

x R

+

y R

+

z R

=

y S − y R z S − z R z S − z R x S − x R x S − x R y S − y R

x − y − z − 0

y T − y R z T − z R z T − z R x T − x R x T − x R y T − y R

This can be arranged in the Cartesian form of the line equation

ax + by + cz = d

where

1 y R z R
1 y S z S
1 y T z T

y S − y R z S − z R

a = or a =

y T − y R z T − z R