Skew lines

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In geometry, skew lines are two lines in Euclidean space that do not intersect but are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Lines that are coplanar either intersect or are parallel, so skew lines exist only in three or more dimensions.

If each line is defined by two points, then these four points must not be coplanar, so they must be the vertices of a tetrahedron of nonzero volume; conversely, any two pairs of points defining a tetrahedron of nonzero volume also define a pair of skew lines. Therefore, we may test whether two pairs of points (a,b) and (c,d) define skew lines by applying the formula for the volume of a tetrahedron, V = (1/6)·|det(a−b, b−c, c−d)|, and testing whether the result is nonzero.

If four points are chosen at random within a unit cube, they will with probability one define a pair of skew lines, because (after the first three points have been chosen) the fourth point will define a non-skew line if and only if it is coplanar with the first three points, and the plane through the first three points forms a subset of measure zero of the cube. Similarly, a very small perturbation of two parallel or intersecting lines will almost surely turn them into skew lines. In this sense, skew lines are the "usual" case, and parallel or intersecting lines are special cases.

1 Configurations of multiple skew lines
2 Skew lines and ruled surfaces
3 Distance between two skew lines
4 References
5 External links

[edit] Configurations of multiple skew lines

A configuration of skew lines is a set of lines in which all pairs are skew. Two configurations are said to be isotopic if it is possible to continuously transform one configuration into the other, maintaining throughout the transformation the invariant that all pairs of lines remain skew. Any two configurations of two lines are easily seen to be isotopic, and configurations of the same number of lines in dimensions higher than three are always isotopic, but the same is not true for configurations of three or more lines in three dimensions (Viro and Viro 1990). The number of nonisotopic configurations of n lines in R³, starting at n = 1, is

1, 1, 2, 3, 7, 19, 74, ... (sequence A110887 in OEIS).

[edit] Skew lines and ruled surfaces

If one rotates a line L around another line L' skew but not perpendicular to it, the surface of revolution swept out by L is a hyperboloid of one sheet. The copies of L within this surface make it a ruled surface; it also contains another family of lines that are skew to L' at the same distance from it but with the opposite angle. An affine transformation of this ruled surface produces a surface which in general has an elliptical cross-section rather than the circular cross-section produced by rotating L around L'; such surfaces are also called hyperboloids of one sheet, and again are ruled by two families of mutually skew lines. A third type of ruled surface is the hyperbolic paraboloid. Like the hyperboloid of one sheet, the hyperbolic paraboloid has two families of skew lines; in each of the two families the lines are parallel to a common plane although not to each other. Any three skew lines in R³ lie on exactly one ruled surface of one of these types (Hilbert and Cohn-Vossen 1952).

[edit] Distance between two skew lines

We now derive a formula for computing the minimum distance between two skew lines, determined by two pairs of points (v₁,v₂) and (v₃,v₄).

Any two points on these two lines may be written as vectors in the form t(v₂-v₁) - v₁ and s(v₄-v₃) - v₃, where s and t are real-valued parameters. The distance between two such points can be calculated by applying the Pythagorean theorem to the coordinates and regrouping the resulting polynomials in s and t as

$\displaystyle As^2+2Bst+Ct^2+2Ds+2Et+F,$

where

$A = (v_4-v_3) \cdot (v_4-v_3), B=(v_4-v_3) \cdot (v_1-v_2),$

$C = (v_1-v_2) \cdot (v_1-v_2), D=(v_4-v_3) \cdot (v_3-v_1),$

$E=(v_1-v_2) \cdot (v_3-v_1), F=(v_3-v_1) \cdot (v_3-v_1).$

Finding the minimum of this expression, we obtain the minimum distance between two lines as

$d^2 = \frac{ACF+2BDE-AE^2-CD^2-FB^2}{AC-B^2} = \frac{\det R}{\det S}$

where $R=\begin{bmatrix}A&B&D\\B&C&E\\D&E&F\end{bmatrix}$ and $S=\begin{bmatrix}A&B\\B&C\end{bmatrix}$ .

By using the Lagrange identity this may be rewritten in terms of wedge products:

$d = \frac{||(v_4-v_1) \wedge (v_3-v_1) \wedge (v_2-v_1)||}{||(v_4-v_3) \wedge (v_2 -v_1)||}.$

The numerator of this expression is six times the volume of the tetrahedron defined by v₁, v₂, v₃ and v₄.

[edit] References

Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination, 2nd ed., Chelsea, 13–17. ISBN 0-8284-1087-9.

Viro Drobotukhina, Julia; Viro, Oleg (1990). "Configurations of skew lines". Leningrad Math. J. 1 (4): 1027–1050. Revised version in English: arXiv:math.GT/0611374.