Graphics Reference
In-Depth Information
As an example, consider the three points
R
(0,0,1),
S
(1,0,0),
T
(0,1,0). There-
fore
101
100
110
011
110
010
001
101
011
a
=
=1
b
=
=1
c
=
=1
d
=
−
(1
×
0+1
×
0+1
×
1) =
−
1
and the plane equation is
x
+
y
+
z
−
1=0
10.8 Intersecting Planes
When two non-parallel planes intersect they form a straight line at the in-
tersection, which is parallel to both planes. This line can be represented as a
vector, whose direction is revealed by the vector product of the planes' sur-
face normals. However, we require a point on this line to establish a unique
vector equation; a useful point is chosen as
P
0
, whose position vector
p
0
is
perpendicular to the line.
Figure 10.35 shows two planes with normal vectors
n
1
and
n
2
intersecting
to create a line represented by
n
3
, whilst
P
0
(
x
0
,y
o
,z
0
) is a particular point
on
n
3
and
P
(
x, y, z
) is any point on the line.
We start the analysis by defining the surface normals:
n
1
=
a
1
i
+
b
1
j
+
c
1
k
n
2
=
a
2
i
+
b
2
j
+
c
2
k
next we define
p
and
p
0
:
p
=
x
i
+
y
j
+
z
k
p
0
=
x
0
i
+
y
0
j
+
z
0
k
Y
n
2
n
1
P
0
P
n
3
p
0
p
Z
X
Fig. 10.35.
Two intersecting planes create a line of intersection.
