Graphics Reference
In-Depth Information
Y
Y
ou
P
d
y
v
o
Q
C
p
c
q
X
Z
Figure 11.2.
The time for the object to reach the “close encounter” at C is t
C
, where
c
=
q
+
t
C
s
O
ˆ
v
O
(11.1)
=
−−→
⊥ˆ
·ˆ
v
O
=
If we let
d
Y
ou
C, then
d
v
O
and
d
0.
⊥ˆ
ˆ
And as
d
v
O
,
y
and
c
have identical projections on
v
O
. Therefore,
v
O
·
ˆ
c
=ˆ
v
O
·
y
(11.2)
ˆ
Multiplying Eq. (11.1) throughout by
v
O
, we obtain
v
O
·
ˆ
c
=ˆ
v
O
·
q
+
t
C
s
O
ˆ
v
O
·ˆ
v
O
(11.3)
Substituting Eq. (11.2) in Eq. (11.3), we get
v
O
·
ˆ
y
=ˆ
v
O
·
q
+
t
C
s
O
ˆ
v
O
·ˆ
v
O
But
ˆ
v
O
·ˆ
v
O
=
1. Therefore,
ˆ
v
O
·
y
=ˆ
v
O
·
q
+
t
C
s
O
and
t
C
=
ˆ
v
O
·
y
−
q
(11.4)
s
O
The shortest distance between you and the object is given by
d
. Now
d
=
c
−
y
Therefore,
d
=
c
−
y
(11.5)
Substituting Eq. (11.1) in Eq. (11.5), we get
d
=
q
+
t
C
s
O
ˆ
v
O
−
y
(11.6)