Graphics Reference
In-Depth Information
10.7.2 General Form of the Plane Equation
The general form of the equation of a plane is expressed as
Ax
+
By
+
Cz
+
D
=0
which means that the Cartesian form is translated into the general form by
making
A
=
a,
B
=
b,
C
=
c,
D
=
−
d
10.7.3 Parametric Form of the Plane Equation
Another method of representing a plane is to employ two vectors and a point
that lie on the plane. Figure 10.31 illustrates a scenario where vectors
a
and
b
,andthepoint
T
(
x
T
,y
T
,z
T
) lie on a plane.
We now identify any other point on the plane
P
(
x
,
y
,
z
) with its associated
position vector
p
.
The point
T
also has its associated position vector
t
.
Using vector addition we can write
c
=
λ
a
+
ε
b
where
λ
and
ε
are two scalars such that
c
locates the point
P
.
We can now write
p
=
t
+
c
(10.79)
therefore
x
P
=
x
T
+
λx
a
+
εx
b
y
P
=
y
T
+
λy
a
+
εy
b
z
P
=
z
T
+
λz
a
+
εz
b
which means that the coordinates of any point on the plane are formed from
the coordinates of the known point on the plane, and a linear mixture of the
components of the two vectors.
Y
P
a
c
p
l
a
b
e
b
t
T
Z
X
Fig. 10.31.
The plane is defined by the vectors
a
and
b
and the point
T
(
x
T
,y
T
,z
T
).
