Graphics Reference
In-Depth Information
Y
1
v
n
P
(
x
, y,
z
)
T
1
1
Z
X
Fig. 10.44.
P identifies the point where a line intersects a plane.
and
v
=
x
v
i
+
y
v
j
+
z
v
k
therefore, the line and plane will intersect for some
λ
such that
n
·
(
t
+
λ
v
)+
d
=
n
·
t
+
λ
n
·
v
+
d
=0
therefore
λ
=
−
(
n
·
t
+
d
)
v
for the intersection point. The position vector for
P
is
p
=
t
+
λ
v
If
n
n
·
v
= 0 the line and plane are parallel.
Let's test this result with the scenario shown in Figure 10.44.
Given the plane
·
x
+
y
+
z −
1=0
n
=
i
+
j
+
k
and the line
p
=
t
+
λ
v
where
t
=0
and
v
=
i
+
j
then
λ
=
−
(1
×
0+1
×
0+1
×
0
−
1)
=
1
2
1
×
1+1
×
1+1
×
0
therefore, the point of intersection is
P
1
2
,
0
.
2
,
1
