Graphics Reference
In-Depth Information
Y
P
a
c
b
p
λ
a
ε
b
T
t
Z
X
Figure 4.3.
Let vectors
a
and
b
be parallel to the plane and defined as
a
=
x
a
i
+
y
a
j
+
z
a
k
b
=
x
b
i
+
y
b
j
+
z
b
k
and the point T x
T
y
T
z
T
is on the plane. Therefore,
c
=
a
+
b
where and are scalars. But
p
=
t
+
c
Therefore,
p
=
t
+
a
+
b
More explicitly,
x
P
=
x
T
+
x
a
+
x
b
y
P
=
y
T
+
y
a
+
y
b
z
P
=
z
T
+
z
a
+
z
b
·
=
If
a
and
b
are unit vectors and are mutually perpendicular, i.e.,
a
b
0, and become linear
measurements along the
a
- and
b
-axes relative to T.
4.4 A plane equation from three points
We know that two points are required to define a line, but three points are needed to define a
plane. So let us now consider how the plane equation is derived from three such points.
