Skew lines
Encyclopedia
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 dimension
s.
s, 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 and define skew lines is to apply the formula for the volume of a tetrahedron, , 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.
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 .
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
and a j-flat may be skew if
. 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 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.
In either geometry, if and intersect at a k-flat, for , then the points of determine a (i+j−k)-flat.
Solid geometry
In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry...
, skew lines are two lines that do not intersect and are not parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
. 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 dimension
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
s.
Explanation
If each line in a pair of skew lines is defined by two pointPoint (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
s, then these four points must not be coplanar, so they must be the vertices of a tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
of nonzero volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
; 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 and define skew lines is to apply the formula for the volume of a tetrahedron, , and testing whether the result is nonzero.
If four points are chosen at random within a unit cube
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. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, they will almost surely
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
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 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, ... .
Skew lines and ruled surfaces
If one rotates a line L around another line L' skew but not perpendicular to it, the surface of revolutionSurface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane ....
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
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
; 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
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
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 .
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
-
.
Skew flats in higher dimensions
In d-dimensional space, an i-flatFlat (geometry)
In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension. The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes....
and a j-flat may be skew if
. As in the plane, skew flats are those that are neither parallel nor intersect.
In affine d-space
Affine geometry
In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
, two flats of any dimension may be parallel.
However, in projective space
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, 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.
In either geometry, if and intersect at a k-flat, for , then the points of determine a (i+j−k)-flat.