Skew lines
If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines.
However, the plane through the first three points forms a subset of measure zero of the cube, and the probability that the fourth point lies on this plane is zero.
Therefore, any four points in general position always form skew lines.
Expressing the two lines as vectors: The cross product of
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.
[2] The number of nonisotopic configurations of n lines in R3, starting at n = 1, is 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.
[3] If three skew lines all meet three other skew lines, any transversal of the first set of three meets any transversal of the second set.
[4][5] In higher-dimensional space, a flat of dimension k is referred to as a k-flat.
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.
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.
