Game Development Reference
In-Depth Information
After removing a factor of two, we can write these equations in matrix form as
follows.
(
)
V
2
−⋅
VV
t
Vt
SSV
−⋅
 
1
1
2
1
2
1
1
=
(5.10)
 
(
)
2
SSV
−⋅
VV
  
2
1
1
2
2
2
2
Solving this equation for t and t gives us
1
(
)
t
V
2
−⋅
VV
SSV
−⋅

1
1
1
2
2
1
1
=

2
(
)
t
VV
V
SSV
−⋅

2
1
2
2
2
1
2
2
(
)
V
VV
SSV
−⋅
1
2
1
2
2
1
1
=
.
(5.11)
(
)
2
2
2
(
)
VV
VV
−⋅
VV
V
2
SSV
−⋅

1
2
1
2
1
2
1
2
1
2
Plugging these values of t and t back into the function f gives us the minimum
squared distance between the two lines. Taking a square root gives us the actual
distance that we want. If the direction vectors
V and
V have unit length, then
Equation (5.11) simplifies a bit since
V
2
=
1
and
V
2
2
=
1
.
1
(
)
2
2
2
VV is zero, then the lines are parallel, in which
case the distance between the two lines is equal to the distance between any point
on one of the lines and the other line. This is illustrated in Figure 5.3. In particu-
lar, we can use Equation (5.3) to measure the distance from the point S to the
line
If the quantity
VV
1
2
1
2
()
or the distance from the point S to the line
()
P
t
P
t
.
22
11
()
P
t
11
S
1
()
P
t
22
Figure 5.3. The distance between parallel lines is given by the distance from a point on
one line to the other line.
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