Graphics Reference
In-Depth Information
Now we state the plane equations in vector form:
n
1
·
p
+
d
1
=0
p
+
d
2
=0
The geometric significance of the scalars
d
1
and
d
2
has already been described
above. Let's now define the line of intersection as
n
2
·
p
=
p
0
+
λ
n
3
where
λ
is a scalar.
Because the line of intersection must be orthogonal to
n
1
and
n
2
n
3
=
a
3
i
+
b
3
j
+
c
3
k
=
n
1
×
n
2
Now we introduce
P
0
as this must satisfy both plane equations, therefore
n
1
·
p
0
=
−d
1
(10.82)
n
2
·
p
0
=
−
d
2
(10.83)
and as
P
0
is such that
p
0
is orthogonal to
n
3
n
3
·
p
0
= 0 (10.84)
Equations (10.82)-(10.84) form three simultaneous equations, which reveal
the point
P
0
. These can be represented in matrix form as
⎡
⎤
⎡
⎤
⎡
⎤
−
d
1
−
a
1
b
1
c
1
x
0
y
0
z
0
⎣
⎦
=
⎣
⎦
·
⎣
⎦
d
2
0
a
2
b
2
c
2
a
3
b
3
c
3
or
⎡
⎤
⎡
⎤
⎡
⎤
d
1
d
2
0
a
1
b
1
c
1
x
0
y
0
z
0
⎣
⎦
=
⎣
⎦
·
⎣
⎦
−
a
2
b
2
c
2
a
3
b
3
c
3
therefore
x
0
y
0
z
0
=
−
1
DET
=
=
d
1
b
1
c
1
a
1
d
1
c
1
a
1
b
1
d
1
d
2
b
2
c
2
a
2
d
2
c
2
a
2
b
2
d
2
0
b
3
c
3
a
3
0
c
3
a
3
b
3
0
which enables us to state
d
2
−
d
1
b
1
c
1
b
2
c
2
b
3
c
3
b
3
c
3
x
0
=
DET
d
2
−
d
1
a
3
c
3
a
3
c
3
a
1
c
1
a
2
c
2
y
0
=
DET
d
2
−
d
1
a
1
b
1
a
2
b
2
a
3
b
3
a
3
b
3
z
0
=
DET
