Graphics Reference
In-Depth Information
6.5 The shortest distance between two skew lines in
R
3
Having seen how to cope with two-dimensional lines, circles and ellipses, now let's explore the
relationship between two lines in R
3
.
We already know that lines in R
2
either intersect or are parallel with one another. However,
in R
3
a third option is possible — one where the lines approach one another then recede,
allowing a shortest distance to be calculated. Such lines are called
skew
lines, and the shortest
distance between them will be on a mutual perpendicular to both lines.
Y
a
T
Q
Q
′
d
q
′
q
b
t
T
′
t
′
Z
X
Figure 6.6.
Figure 6.6 shows two lines with direction vectors
a
and
b
. The shortest distance d between the
lines is the magnitude of the vector
−
TT
, which is perpendicular to both lines. Therefore,
−
OT
=
+
q
a
(6.10)
and
−
OT
=
q
+
b
(6.11)
But
−
TT
is perpendicular to
a
and
b
and parallel to
a
×
b
. Therefore,
−
TT
=
d
·
a
×
b
a
×
b
But
−
OT
=
−
OT
+
−
TT
and
−
OT
=
−
OT
d
a
×
b
+
(6.12)
a
×
b
