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
v O ·
y
(11.2)
ˆ
Multiplying Eq. (11.1) throughout by
v O , we obtain
v O ·
ˆ
c
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
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
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)
 
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