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skew lines wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 general position 2 formulas toggle formulas subsection 2 1 testing for skewness 2 2 nearest points 2 2 1 distance 3 more than two lines toggle more than two lines subsection 3 1 configurations 3 2 ruled surfaces 3 3 gallucci s theorem 4 skew flats in higher dimensions 5 see also 6 references 7 external links toggle the table of contents skew lines 25 languages العربية български català čeština чӑвашла deutsch ελληνικά español eesti euskara فارسی magyar 日本語 한국어 norsk nynorsk norsk bokmål polski português română русский simple english svenska தமிழ் українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia lines not in the same plane rectangular parallelepiped the line through segment ad and the line through segment b 1 b are skew lines because they are not in the same plane 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 general position edit 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 formulas edit pq the shortest distance between two skew lines ab and cd is perpendicular to both ab and cd further information line line intersection formulas testing for skewness edit 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 v 1 6 det a b b c c d displaystyle v frac 1 6 left det left begin matrix mathbf a mathbf b mathbf b mathbf c mathbf c mathbf d end matrix right right nearest points edit see also line line intersection nearest points to skew lines see also triangulation computer vision mid point method expressing the two lines as vectors line 1 v 1 p 1 t 1 d 1 displaystyle text line 1 mathbf v_ 1 mathbf p_ 1 t_ 1 mathbf d_ 1 line 2 v 2 p 2 t 2 d 2 displaystyle text line 2 mathbf v_ 2 mathbf p_ 2 t_ 2 mathbf d_ 2 the cross product of d 1 displaystyle mathbf d_ 1 and d 2 displaystyle mathbf d_ 2 is perpendicular to the lines n d 1 d 2 displaystyle mathbf n mathbf d_ 1 times mathbf d_ 2 the plane formed by the translations of line 2 along n displaystyle mathbf n contains the point p 2 displaystyle mathbf p_ 2 and is perpendicular to n 2 d 2 n displaystyle mathbf n_ 2 mathbf d_ 2 times mathbf n 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 c 1 p 1 p 2 p 1 n 2 d 1 n 2 d 1 displaystyle mathbf c_ 1 mathbf p_ 1 frac mathbf p_ 2 mathbf p_ 1 cdot mathbf n_ 2 mathbf d_ 1 cdot mathbf n_ 2 mathbf d_ 1 similarly the point on line 2 nearest to line 1 is given by where n 1 d 1 n displaystyle mathbf n_ 1 mathbf d_ 1 times mathbf n c 2 p 2 p 1 p 2 n 1 d 2 n 1 d 2 displaystyle mathbf c_ 2 mathbf p_ 2 frac mathbf p_ 1 mathbf p_ 2 cdot mathbf n_ 1 mathbf d_ 2 cdot mathbf n_ 1 mathbf d_ 2 distance edit the nearest points c 1 displaystyle mathbf c_ 1 and c 2 displaystyle mathbf c_ 2 form the shortest line segment joining line 1 and line 2 d c 1 c 2 displaystyle d vert mathbf c_ 1 mathbf c_ 2 vert the distance between nearest points in two skew lines may also be expressed using other vectors x a λ b displaystyle mathbf x mathbf a lambda mathbf b y c μ d displaystyle mathbf y mathbf c mu mathbf d 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 λ displaystyle lambda 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 n b d b d displaystyle mathbf n frac mathbf b times mathbf d mathbf b times mathbf d the perpendicular distance between the lines is then 1 d n c a displaystyle d mathbf n cdot mathbf c mathbf a if b d is zero the lines are parallel and this method cannot be used more than two lines edit see also line line intersection more than two lines configurations edit 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 r 3 starting at n 1 is 1 1 2 3 7 19 74 sequence a110887 in the oeis ruled surfaces edit a fibration of projective space by skew lines on nested hyperboloids 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 r 3 lie on exactly one ruled surface of one of these types 3 gallucci s theorem edit 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 skew flats in higher dimensions edit 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 6 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 see also edit distance between two parallel lines petersen morley theorem references edit weisstein eric w line line distance mathworld viro julia drobotukhina viro oleg 1990 configurations of skew lines pdf leningrad math j in russian 1 4 1027 1050 archived from the original pdf on 2021 11 09 retrieved 2006 10 24 revised version in english arxiv math gt 0611374 hilbert david cohn vossen stephan 1952 geometry and the imagination 2nd ed chelsea pp 13 17 isbn 0 8284 1087 9 citation isbn date incompatibility help coxeter h s m 1969 introduction to geometry 2nd ed john wiley sons p 257 g gallucci 1906 studio della figura delle otto rette e sue applicazioni alla geometria del tetraedro ed alla teoria della configurazioni rendiconto dell accademia della scienza fisiche e matematiche 3rd series 12 49 79 dupré arthur m kass seymour 1992 07 01 distance and parallelism between flats in r n pdf linear algebra and its applications 171 9 doi 10 1016 0024 3795 92 90252 6 retrieved 2025 01 30 external links edit weisstein eric w skew lines mathworld retrieved from https en wikipedia org w index php title skew_lines oldid 1357123873 categories elementary geometry euclidean solid geometry multilinear algebra orientation geometry line geometry hidden categories articles with short description short description is different from wikidata cs1 errors isbn date cs1 russian language sources ru this page was last edited on 31 may 2026 at 21 52 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy 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