Game Development Reference

In-Depth Information

()

P

t

11

() ()

P

t

−

P

t

11

2 2

()

P

t

22

()

()

Figure 5.2.
The distance between skew lines

P

t

and

P

t

is calculated by finding

11

22

()

( )

the parameters
t
and
t
minimizing

P

t

−

P

t

.

11

2 2

where
t
and
t
range over all real numbers. Then the squared distance between a

point on the line

()

()

P

t

and a point on the line

P

t

can be written as the fol-

11

22

lowing function of the parameters
t
and
t
.

(

)

(

)

(

)

2

(5.6)

f

tt

,

=

P

t

−

P

t

12

1 1

2 2

()

Expanding the square and substituting the definitions of the functions

P

t

and

11

()

P

t

gives us

22

(

)

()

2

()

2

()

()

ftt

,

=

P

t

+

P

t

−

2

P

t

⋅

P

t

12

11

2 2

11

2 2

(

)

(

)

=+

SV

t

2

+ +

S V

t

2

1

1

1

2

2

2

(

)

−

2

SS

⋅

+

t

VS

⋅

+

t

VS

⋅

+

t t

VV

⋅

1

2

1

1

2

2

2

1

1 2

1

2

2

2

2

2

2

2

=+

S

t Vt

+

2

SV

⋅

++

S

t Vt

+

2

S V

⋅

1

1

1

1

1

1

2

2

2

2

2

2

(

)

−

2

SS

⋅

+

t

VS

⋅

+

t

VS

⋅

+

t t

VV
.

⋅

(5.7)

1

2

1

1

2

2

2

1

1 2

1

2

The minimum value attained by the function
f
can be found by setting partial

derivatives with respect to
t
and
t
equal to zero. This provides us with the

equations

∂

f

2

11

=

2

tV

+

2

SV

⋅

−

2

VS

⋅

−

2

t

VV

⋅

=

0

(5.8)

1

1

1

2

2 1

2

∂

1

and

∂

f

2

22

=

2

tV

+

2

SV

⋅

−

2

VS

⋅

−

2

t

VV

⋅

=

0

.

(5.9)

2

2

2

1

1 1

2

∂

2

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