Graphics Reference
In-Depth Information
3.7.2 The parametric form of the line equation
Y
Q
λ
v
q
T
t
O
X
Figure 3.17.
With reference to Fig. 3.17, let the straight-line equation be
q
=
t
+
v
(3.31)
The objective is to establish the value of .
Q is nearest to O when
q
is perpendicular to. Therefore,
v
·
q
=
0
(3.32)
Using
v
, take the dot product of Eq. (3.31) and substitute it in Eq. (3.32):
v
·
q
=
v
·
t
+
v
·
v
=
0
Therefore,
=
−
v
·
t
=
−
v
·
t
2
v
·
v
v
The position vector for Q is then given by
v
v
·
t
q
=
t
−
(3.33)
2
v
=
and the distance OQ
.
Eq. (3.33) is greatly simplified if
v
is a unit vector
q
ˆ
v
:
q
=
t
+
v
ˆ
Therefore,
=−ˆ
v
·
t
and
q
=
t
−
v
ˆ
·
t
ˆ
v
(3.34)
Again, let's test Eq. (3.34) with an example.
