Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 422

FIGURE B.12—CONT'D

The points P 1 and P 2 at which the lines are closest form a line segment perpendicular to both lines
( Figure B.13 ). They can be represented parametrically as points along the lines, and then the parametric
interpolants can be solved for by satisfying the equations that state the requirement for perpendicularity
( Eq. B.30 ) .

ð P 2 P 1 Þ V ¼ 0

ð P 2 P 1 0

B þ tW A þ sV

Þ W ¼

ð ð Þ Þ V ¼ 0

ð 0

t ¼ B V þ A V þ sV V
W V

B þ tW A þ sV ð Þ Þ W ¼

t ¼ B W þ A W þ sV W
W W

(B.30)

B V þ A V þ sV V
W V

¼ B W þ A W þ sV W
W W

ð B V þ A V þ sV V Þ W W ð Þ¼ B W þ A W þ sV W ð Þ W V ð Þ

ð A W B W Þ W V ð ÞþB V A V ð Þ W W ð Þ

s ¼

2

ð V V Þ W W ð ÞV W ð Þ

ð B V A V Þ W V ð ÞþA W B W ð Þ V V ð Þ

t ¼

2

ð W W Þ V V ð ÞV W ð Þ