Intersecting Lines are Skew
Intersecting Lines are Skew
The lines which do not intersect, they are not parallel as well and which are not
coplanar are known as skew lines. Now the question arises that the Intersecting Lines
are skew or not?
By definition of skew lines we can say that skew lines never intersect each other and
also not parallel.
For example: Regular tetrahedron is best example of skew lines, it is composed of
four triangular faces, out of all faces three faces of regular tetrahedron meet at same
Point. The lines which are coplanar either intersect or are parallel, so we can say that
the skew lines exist in three or more dimensions.
Now we will see how to find the distance between two skew lines:
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;
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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.
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
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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).


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