Game Development Reference

In-Depth Information

Q

QS

d

V

S

(

)

proj

V
QS

−

Figure 5.1.
The distance
d
from a point
Q
to the line

SV
is found by calculating the

+

t

length of the perpendicular component of

−

S
with respect to the line.

Q

rem, the squared distance between the point
Q
and the line can be obtained by

subtracting the square of the projection of

−

S
onto the direction
V
from the

Q

square of

−

S
. This gives us

Q

2

(

)

2

[

(

)

]

2

d

=− −

QS

proj

QS

QSV

−

V

(

)

2

−⋅

(

)

2

=− −

QS

V
.

(5.3)

2

V

Simplifying a bit and taking the square root gives us the distance
d
that we

desire:

[

(

)

]

2

QSV

V
−⋅

(

)

2

d

=−−

QS

.

(5.4)

2

5.1.2 Distance Between Two Lines

In two dimensions, two lines are either parallel or they intersect at a single point.

In three dimensions, there are more possibilities. Two lines that are not parallel

and do not intersect are called
skew
. A formula giving the minimum distance be-

tween points on skew lines can be found by using a little calculus.

Suppose that we have two lines, as shown in Figure 5.2, defined by the par-

ametric functions

()

P

t

=+

=+

S

t

V

11

1

11

P

()

t

S

t

,

(5.5)

22

2

22

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