Skew lines
In solid geometry, skew lines are two lines that do not intersect and are not parallel. Equivalently, they are lines that are not coplanar. 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.
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[edit] Explanation
If each line in a pair of skew lines 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, a test of whether two pairs of points (a,b) and (c,d) define skew lines is to apply the formula for the volume of a tetrahedron, V = |det(a−b, b−c, c−d)|/6, and testing whether the result is nonzero.
If four points are chosen at random within a unit cube, they will almost surely 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, in 3D space a very small perturbation of two parallel or intersecting lines will almost certainly turn them into skew lines. In this sense, skew lines are the "usual" case, and parallel or intersecting lines are special cases.
[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 there exist multiple non-isotopic configurations of three or more lines in three dimensions (Viro & Viro 1990). The number of nonisotopic configurations of n lines in R3, starting at n = 1, is
[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. For instance, the three hyperboloids visible in the illustration can be formed in this way by rotating a line L around the central white vertical line L'. The copies of L within this surface make it a ruled surface; it also contains a second family of lines that are also skew to L' at the same distance as L 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 & Cohn-Vossen 1952).
[edit] Distance between two skew lines
To calculate the distance between two skew lines the lines are expressed using vectors,
.
.The cross product of b and d is perpendicular to the lines, as is the unit vector
(if |b × d| is zero the lines are parallel and this method cannot be used). The distance between the lines is then[1]
.
.[edit] Skew flats in higher dimensions
In d-dimensional space, an i-flat and a j-flat may be skew if i + j < d. As in the plane, skew flats are those that are neither parallel nor intersect.
In affine d-space, two flats of any dimension may be parallel. However, in projective space, parallelism does not exist; two flats must either intersect or be skew. Let I be the set of points on an i-flat, and let J be the set of points on a j-flat. In projective d-space, if i + j ≥ d then the intersection of I and J must contain a (i+j−d)-flat.
In either geometry, if I and J intersect at a k-flat, for k ≥ 0, then the points of I ∪ J determine a (i+j−k)-flat.
[edit] Notes
- ^ Weisstein, Eric W., "Line-Line Distance" from MathWorld.
[edit] References
- Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), Chelsea, pp. 13–17, ISBN 0-8284-1087-9.
- Viro, Julia Drobotukhina; Viro, Oleg (1990), "Configurations of skew lines" (in Russian), Leningrad Math. J. 1 (4): 1027–1050, http://www.math.uu.se/~julia/SplSkrPr.pdf. Revised version in English: arXiv:math.GT/0611374.
