Graphics Reference
In-Depth Information
Y
1
T
l
a
e
b
t
P
p
Z
X
Fig. 10.32.
The plane is defined by the vectors
a
and
b
, and the point
T
(1
,
1
,
1).
Let's illustrate this vector approach with an example.
Figure 10.32 shows a plane containing the vectors
a
=
i
and
b
=
k
,and
the point
T
(1, 1, 1) with its position vector
t
=
i
+
j
+
k
.
By inspection, the plane is parallel to the
xz
-plane and intersects the
y
-axis
at
y
=1.
From (10.79) we can write
p
=
t
+
λ
a
+
ε
b
where
λ
and
ε
are arbitrary scalars.
For example, if
λ
=2and
ε
=1
x
P
=1+2
×
1+1
×
0=3
y
P
=1+2
×
0+1
×
0=1
z
P
=1+2
×
0+1
×
1=2
Therefore, the point (3, 1, 2) is on the plane.
10.7.4 Converting from the Parametric to the General Form
It is possible to convert from the parametric form to the general form of the
plane equation using the following formulae:
λ
=
(
a
·
b
)(
b
·
t
)
−
(
a
·
t
)
b
2
b
)
2
a
2
b
2
−
(
a
·
2
ε
=
(
a
·
b
)(
a
·
t
)
−
(
b
·
t
)
a
b
)
2
a
2
b
2
−
(
a
·
The resulting point
P
(
x
P
,y
P
,z
P
) is perpendicular to the origin.
If vectors
a
and
b
are unit vectors,
λ
and
ε
become
λ
=
(
a
·
b
)(
b
·
t
)
−
a
·
t
(10.80)
(
a
·
b
)
2
1
−
