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Line-Line Distance


The distance between two skew lines with equations

x=x_1+(x_2-x_1)s
(1)
x=x_3+(x_4-x_3)t
(2)

is given by

 D=(|(x_3-x_1)·[(x_2-x_1)x(x_4-x_3)]|)/(|(x_2-x_1)x(x_4-x_3)|)
(3)

(Gellert et al. 1989, p. 538). This can be written in the concise form

 D=(|c·(axb)|)/(|axb|)
(4)

by defining

a=x_2-x_1
(5)
b=x_4-x_3
(6)
c=x_3-x_1.
(7)

See also

Skew Lines, Line-Line Angle, Line-Line Intersection

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References

Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989.Hill, F. S. Jr. "The Pleasures of 'Perp Dot' Products." Ch. II.5 in Graphics Gems IV (Ed. P. S. Heckbert). San Diego: Academic Press, pp. 138-148, 1994.

Referenced on Wolfram|Alpha

Line-Line Distance

Cite this as:

Weisstein, Eric W. "Line-Line Distance." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Line-LineDistance.html

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