Graphics Reference
In-Depth Information
√
2
2
=
−
q
·
×
·
+
q
a
b
k
i
k
1
√
2
=
d
=
=
×
+
a
b
i
k
which confirms our prediction.
3
6.6 Two intersecting lines in
R
As we have seen in the previous section, two straight lines in
3
can cross one another but not
necessarily intersect. Therefore, in order to calculate a possible intersection, we have to ensure
two things:
R
1. The two lines are not parallel.
2. The two lines touch.
Y
λ
a
ε
b
p
q
T
S
t
s
Z
X
Figure 6.8.
We begin by defining the two lines, as shown in Fig. 6.8:
p
=
t
+
a
and
q
=
s
+
b
where and are scalars:
t
=
x
t
i
+
y
t
j
+
z
t
k
and
s
=
x
s
i
+
y
s
j
+
z
s
k
a
=
x
a
i
+
y
a
j
+
z
a
k
and
b
=
x
b
i
+
y
b
j
+
z
b
k
Step 1:
If
a
×
b
=
0, the lines are parallel and do not intersect.
Step 2:
The distance d between two skew lines is given by
=
t
−
s
·
a
×
b
d
a
×
b
If
t
−
s
·
a
×
b
=
0, the lines do not intersect.
