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Skew Lines

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SkewLines

Two or more lines which have no intersections but are not parallel, also called agonic lines. Since two lines in the plane must intersect or be parallel, skew lines can exist only in three or more dimensions.

Two lines with equations

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

are skew if

(x_1-x_3)·[(x_2-x_1)x(x_4-x_3)]!=0
(3)

(Gellert et al. 1989, p. 539).

This is equivalent to the statement that the vertices of the lines are not coplanar, i.e.,

|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|!=0.
(4)

Three skew lines always define a one-sheeted hyperboloid, except in the case where they are all parallel to a single plane but not to each other. In this case, they determine a hyperbolic paraboloid (Hilbert and Cohn-Vossen 1999, p. 15).

See also

Coplanar, Director, Gallucci's Theorem, Intersecting Lines, Line-Line Distance, Parallel Lines, Regulus

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References

Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 1, 1979.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.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 15, 1999.

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Skew Lines

Cite this as:

Weisstein, Eric W. "Skew Lines." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SkewLines.html

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