Graphics Reference
In-Depth Information
6.16 Two intersecting planes
Hopefully, it is obvious that two planes give rise to a straight line at their intersection. We
can discover the vector representing this line by exploiting the fact that the intersecting line is
perpendicular to the normal vectors associated with the planes. Therefore, the cross product
of the plane's normal vectors reveals the vector, the direction, of which, is determined by the
order of the normal vectors. Figure 6.26 illustrates such a scenario.
Y
n
2
n
1
v
P
T
p
t
Z
X
Figure 6.26.
To derive a parametric line equation for the intersection, we require the coordinates of a point
on the line. It is convenient to select a point whose position vector
t
is perpendicular to
v
.
We begin by defining the two plane equations
a
1
x
+
b
1
y
+
c
1
z
=
d
1
a
2
x
+
b
2
y
+
c
2
z
=
d
2
where
n
1
=
a
1
i
+
b
1
j
+
c
1
k
n
2
=
a
2
i
+
b
2
j
+
c
2
k
Therefore,
n
1
×
n
2
=
v
where
v
is the vector representing the line of intersection.
The line equation is
p
=
t
+
v
where
is a scalar
