In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. 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 dimensions. Two lines are skew if and only if they are not coplanar.
If four points are chosen at random uniformly within a unit cube, they will almost surely 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 always form skew lines.
In this sense, skew lines are the "usual" case, and parallel or intersecting lines are special cases.
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 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 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 a whose three elements are the point's three coordinate values, and likewise denoting b, c, and d for the other points, we can check if the line through a and b is skew to the line through c and d by seeing if the tetrahedron volume formula gives a non-zero result:
Expressing the two lines as vectors:
The cross product of and is perpendicular to the lines.
The plane formed by the translations of Line 2 along contains the point and is perpendicular to .
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
Similarly, the point on Line 2 nearest to Line 1 is given by (where )
The nearest points and form the shortest line segment joining Line 1 and Line 2:
The distance between nearest points in two skew lines may also be expressed using other vectors:
Here the 1×3 vector x represents an arbitrary point on the line through particular point a with b representing the direction of the line and with the value of the real number determining where the point is on the line, and similarly for arbitrary point y on the line through particular point c in direction d.
The cross product of b and d is perpendicular to the lines, as is the unit vector
The perpendicular distance between the lines is then [1]
(if |b × d| is zero the lines are parallel and this method cannot be used).
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. [2] The number of nonisotopic configurations of n lines in R3, starting at n = 1, is
If one rotates a line L around another line M 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 M. The copies of L within this surface form a regulus; 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 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. 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 i + j<d. 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, parallelism does not exist; two flats must either intersect or be skew. Let I be the set of points on an i-flat, and let J be the set of points on a j-flat. In projective d-space, if i + j ≥ d then the intersection of I and J must contain a (i+j−d)-flat. (A 0-flat is a point.)
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.
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—parallelepiped and cube in three dimensions, parallelogram and square in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only parallelograms and parallelepipeds exist. Three equivalent definitions of parallelepiped are
In geometry, a tetrahedron, 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 the ordinary convex polyhedra.
In Euclidean geometry, an affine transformation or affinity is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space . A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. While a pair of real numbers suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space .
In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
A 3D projection is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object for viewing capability on a simpler plane.
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space.
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimension spaces. The word line may also refer to a line segment in everyday life, which has two points to denote its ends (endpoints). Lines can be referred by two points that lay on it or by a single letter.
Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms.
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane.
In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the Euclidean space of dimension three that models physical space. More general three-dimensional spaces are called 3-manifolds.
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection.
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point, which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set of pairs of real numbers augmented by appropriate structure often serves as the canonical example.
In Euclidean geometry, Apollonius' problem is to construct all the circles that are tangent to three given circles. Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to:
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.
In geometry, a quadrisecant or quadrisecant line of a space curve is a line that passes through four points of the curve. This is the largest possible number of intersections that a generic space curve can have with a line, and for such curves the quadrisecants form a discrete set of lines. Quadrisecants have been studied for curves of several types:
In differential geometry Dupin's theorem, named after the French mathematician Charles Dupin, is the statement:
