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In three-dimensional geometry, skew lines are two
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
s that do not intersect and are not
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
. A simple example of a pair of skew lines is the pair of lines through opposite edges of a
regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
s. Two lines are skew if and only if they are not coplanar.


General position

If four points are chosen at random uniformly within a unit
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
, they will
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
define a pair of skew lines. 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. 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. If it does not, the lines defined by the points will be skew. Similarly, in three-dimensional space a very small perturbation of any two parallel or intersecting lines will almost certainly turn them into skew lines. Therefore, any four points in
general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
always form skew lines. In this sense, skew lines are the "usual" case, and parallel or intersecting lines are special cases.


Formulas


Testing for skewness

If each line in a pair of skew lines is defined by two points that it passes through, then these four points must not be coplanar, so they must be the vertices of a
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
of nonzero
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
. 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 define skew lines is to apply the formula for the volume of a tetrahedron in terms of its four vertices. Denoting one point as the 1×3 vector whose three elements are the point's three coordinate values, and likewise denoting , , and for the other points, we can check if the line through and is skew to the line through and by seeing if the tetrahedron volume formula gives a non-zero result: :V=\frac\left, \det\left begin\mathbf-\mathbf \\ \mathbf-\mathbf \\ \mathbf-\mathbf \end\right.


Nearest points

Expressing the two lines as vectors: :\text \; \mathbf=\mathbf+t_1\mathbf :\text \; \mathbf=\mathbf+t_2\mathbf The
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of \mathbf and \mathbf is perpendicular to the lines. : \mathbf= \mathbf \times \mathbf The plane formed by the translations of Line 2 along \mathbf contains the point \mathbf and is perpendicular to \mathbf= \mathbf \times \mathbf. Therefore, the intersecting point of Line 1 with the above-mentioned plane, which is also the point on Line 1 that is nearest to Line 2 is given by : \mathbf=\mathbf+ \frac \mathbf Similarly, the point on Line 2 nearest to Line 1 is given by (where \mathbf= \mathbf \times \mathbf ) : \mathbf=\mathbf+ \frac \mathbf


Distance

The nearest points \mathbf and \mathbf form the shortest line segment joining Line 1 and Line 2: : d = \Vert \mathbf - \mathbf \Vert. The distance between nearest points in two skew lines may also be expressed using other vectors: : \mathbf = \mathbf + \lambda \mathbf; : \mathbf = \mathbf + \mu \mathbf. Here the 1×3 vector represents an arbitrary point on the line through particular point with representing the direction of the line and with the value of the real number \lambda determining where the point is on the line, and similarly for arbitrary point on the line through particular point in direction . The
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of b and d is perpendicular to the lines, as is the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
: \mathbf = \frac The
perpendicular distance In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both. The distance from a point to a line is the distance to the nearest point on that line. Th ...
between the lines is then : d = , \mathbf \cdot (\mathbf - \mathbf), . (if , b × d, is zero the lines are parallel and this method cannot be used).


More than two lines


Configurations

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. The number of nonisotopic configurations of ''n'' lines in R3, starting at ''n'' = 1, is :1, 1, 2, 3, 7, 19, 74, ... .


Ruled surfaces

If one rotates a line ''L'' around another line ''M'' skew but not perpendicular to it, the
surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
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 ''M''. The copies of ''L'' within this surface form a
regulus Regulus is the brightest object in the constellation Leo and one of the brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinized to Alpha Leonis, and abbreviated Alpha Leo or α Leo. Re ...
; the hyperboloid also contains a second family of lines that are also skew to ''M'' at the same distance as ''L'' from it but with the opposite angle that form the opposite regulus. The two reguli display the hyperboloid as a ruled surface. An
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
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 In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plan ...
. 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.


Gallucci's theorem

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.


Skew flats in higher dimensions

In higher-dimensional space, a flat of dimension ''k'' is referred to as a ''k''-flat. Thus, a line may also be called a 1-flat. Generalizing the concept of ''skew lines'' to ''d''-dimensional space, an ''i''-flat and a ''j''-flat may be skew if . As with lines in 3-space, 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 In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
, parallelism does not exist; two flats must either intersect or be skew. Let be the set of points on an ''i''-flat, and let be the set of points on a ''j''-flat. In projective ''d''-space, if then the intersection of and must contain a (''i''+''j''−''d'')-flat. (A ''0''-flat is a point.) In either geometry, if and intersect at a ''k''-flat, for , then the points of determine a (''i''+''j''−''k'')-flat.


See also

* Distance between two parallel lines * Petersen–Morley theorem


References


External links

*{{mathworld, urlname=SkewLines, title=Skew Lines, mode=cs2 Elementary geometry Euclidean solid geometry Multilinear algebra Orientation (geometry)