Graphics Reference
In-Depth Information
The third step is to find the point of intersection using
x
a
−
x
b
=
x
s
−
x
t
y
a
−
y
b
=
y
s
−
y
t
z
a
−
z
b
=
z
s
−
z
t
Therefore,
2
+
2
=
2
−
0
−
=
1
−
1
−
2
−
2
=
0
−
2
and
+
=
1
=
+
=
1
Therefore,
1
2
1
2
=
and
=
Substituting and in Eq. (6.14), we get
=
+
+
1
2
2
i
+
−
=
+
3
2
j
+
p
j
2
k
j
2
k
i
k
Double checking gives
1
2
3
2
j
q
=
2
i
+
j
+
−
2
i
+
j
+
2
k
=
i
+
+
k
and the point of intersection is
1 1
2
1
, which is correct.
6.7 A line intersecting a plane
There are two configurations of a line and plane: either they intersect or they are parallel, and
we must be able to detect both possibilities. The plane could be defined either using a plane
equation or using two vectors. As the former is the most probable format, let's proceed with
this.
We define the plane equation using the Cartesian form
ax
+
by
+
cz
=
d
where the normal vector
n
is given by
n
=
a
i
+
b
j
+
c
k