Graphics Reference
In-Depth Information
The value of
d
has the following geometric interpretation.
In Figure 10.29 the perpendicular distance from the origin to the plane is
h
=
p
0
cos(
α
)
therefore
n
·
p
0
=
n
p
0
cos(
α
)=
h
n
therefore the plane equation can be also expressed as
ax
+
by
+
cz
=
h
n
(10.78)
Dividing (10.78) by
n
we obtain
a
n
b
n
c
n
x
+
y
+
z
=
h
where
=
a
2
+
b
2
+
c
2
What this means is that when a unit normal vector is used,
h
is the perpen-
dicular distance from the origin to the plane.
Let's investigate this equation with an example.
Figure 10.30 shows a plane represented by the normal vector
n
=
j
+
k
andapointontheplane
P
0
(0
,
1
,
0)
Using (10.77) we have
h
=
n
0
x
+1
y
+1
z
=0
×
0+1
×
1+1
×
0=1
therefore, the plane equation is
y
+
z
=1
If we normalize the equation to create a unit normal vector, we have
y
√
2
+
z
√
2
=
1
√
2
1
√
2
.
where the perpendicular distance from the origin to the plane is
Y
1
P
0
n
O
1
X
Z
Fig. 10.30.
A plane represented by the normal vector
n
and a point
P
0
(0
,
1
,
0).
