- Skew lines
In
geometry , skew lines are two lines that do not intersect but are notparallel . Equivalently, they are lines that are not both in the sameplane . A simple example of a pair of skew lines is the pair of lines through opposite edges of aregular tetrahedron (or other non-degenerate tetrahedron). Lines that arecoplanar either intersect or are parallel, so skew lines exist only in three or moredimension s.Explanation
If each line is defined by two
point s, then these four points must not be coplanar, so they must be the vertices of atetrahedron of nonzerovolume ; 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" = (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 willalmost 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.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, ... Revised version in English: arxiv | archive = math.GT | id = 0611374.
External links
* [http://mathworld.wolfram.com/SkewLines.html Mathworld: Skew Lines]
* [http://www.netcomuk.co.uk/~jenolive/skew.html Finding the shortest distance between two skew lines]
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