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y parallel lines are coplanar straight lines that do not intersect at any point parallel planes are planes in the same three dimensional space that never meet parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance in three dimensional euclidean space a line and a plane that do not share a point are also said to be parallel however two noncoplanar lines are called skew lines parallel lines are the subject of euclid s parallel postulate 1 parallelism is primarily a property of affine geometries and euclidean geometry is a special instance of this type of geometry in some other geometries such as hyperbolic geometry lines can have analogous properties that are referred to as parallelism contents 1 symbol 2 euclidean parallelism 2 1 two lines in a plane 2 1 1 conditions for parallelism 2 1 2 history 2 1 3 construction 2 1 4 distance between two parallel lines 2 2 two lines in three dimensional space 2 3 a line and a plane 2 4 two planes 3 extension to non euclidean geometry 3 1 hyperbolic geometry 3 2 spherical or elliptic geometry 4 reflexive variant 5 see also 6 notes 7 references 8 further reading 9 external links symbol edit the parallel symbol is displaystyle parallel 2 3 for example a b c d displaystyle ab parallel cd indicates that line ab is parallel to line cd in the unicode character set the parallel and not parallel signs have codepoints u 2225 and u 2226 respectively in addition u 22d5 represents the relation equal and parallel to 4 the same symbol is used for a binary function in electrical engineering the parallel operator it is distinct from the double vertical line brackets that indicate a norm as well as from the logical or operator in several programming languages euclidean parallelism edit two lines in a plane edit conditions for parallelism edit as shown by the tick marks lines a and b are parallel this can be proved because the transversal t produces congruent corresponding angles θ displaystyle theta shown here both to the right of the transversal one above and adjacent to line a and the other above and adjacent to line b given parallel straight lines l and m in euclidean space the following properties are equivalent every point on line m is located at exactly the same minimum distance from line l equidistant lines line m is in the same plane as line l but does not intersect l recall that lines extend to infinity in either direction when lines m and l are both intersected by a third straight line a transversal in the same plane the corresponding angles of intersection with the transversal are congruent since these are equivalent properties any one of them could be taken as the definition of parallel lines in euclidean space but the first and third properties involve measurement and so are more complicated than the second thus the second property is the one usually chosen as the defining property of parallel lines in euclidean geometry 5 the other properties are then consequences of euclid s parallel postulate another property that also involves measurement is that lines parallel to each other have the same gradient slope history edit the definition of parallel lines as a pair of straight lines in a plane which do not meet appears as definition 23 in book i of euclid s elements 6 alternative definitions were discussed by other greeks often as part of an attempt to prove the parallel postulate proclus attributes a definition of parallel lines as equidistant lines to posidonius and quotes geminus in a similar vein simplicius also mentions posidonius definition as well as its modification by the philosopher aganis 6 at the end of the nineteenth century in england euclid s elements was still the standard textbook in secondary schools the traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non euclidean geometry so several new textbooks for the teaching of geometry were written at this time a major difference between these reform texts both between themselves and between them and euclid is the treatment of parallel lines 7 these reform texts were not without their critics and one of them charles dodgson a k a lewis carroll wrote a play euclid and his modern rivals in which these texts are lambasted 8 one of the early reform textbooks was james maurice wilson s elementary geometry of 1868 9 wilson based his definition of parallel lines on the primitive notion of direction according to wilhelm killing 10 the idea may be traced back to leibniz 11 wilson without defining direction since it is a primitive uses the term in other definitions such as his sixth definition two straight lines that meet one another have different directions and the difference of their directions is the angle between them wilson 1868 p 2 in definition 15 he introduces parallel lines in this way straight lines which have the same direction but are not parts of the same straight line are called parallel lines wilson 1868 p 12 augustus de morgan reviewed this text and declared it a failure primarily on the basis of this definition and the way wilson used it to prove things about parallel lines dodgson also devotes a large section of his play act ii scene vi 1 to denouncing wilson s treatment of parallels wilson edited this concept out of the third and higher editions of his text 12 other properties proposed by other reformers used as replacements for the definition of parallel lines did not fare much better the main difficulty as pointed out by dodgson was that to use them in this way required additional axioms to be added to the system the equidistant line definition of posidonius expounded by francis cuthbertson in his 1874 text euclidean geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line this can not be proved and must be assumed to be true 13 the corresponding angles formed by a transversal property used by w d cooley in his 1860 text the elements of geometry simplified and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles then all transversals must do so again a new axiom is needed to justify this statement construction edit the three properties above lead to three different methods of construction 14 of parallel lines the problem draw a line through a parallel to l property 1 line m has everywhere the same distance to line l property 2 take a random line through a that intersects l in x move point x to infinity property 3 both l and m share a transversal line through a that intersect them at 90 distance between two parallel lines edit main article distance between two lines because parallel lines in a euclidean plane are equidistant there is a unique distance between the two parallel lines given the equations of two non vertical non horizontal parallel lines y m x b 1 displaystyle y mx b_ 1 y m x b 2 displaystyle y mx b_ 2 the distance between the two lines can be found by locating two points one on each line that lie on a common perpendicular to the parallel lines and calculating the distance between them since the lines have slope m a common perpendicular would have slope 1 m and we can take the line with equation y x m as a common perpendicular solve the linear systems y m x b 1 y x m displaystyle begin cases y mx b_ 1 y x m end cases and y m x b 2 y x m displaystyle begin cases y mx b_ 2 y x m end cases to get the coordinates of the points the solutions to the linear systems are the points x 1 y 1 b 1 m m 2 1 b 1 m 2 1 displaystyle left x_ 1 y_ 1 right left frac b_ 1 m m 2 1 frac b_ 1 m 2 1 right and x 2 y 2 b 2 m m 2 1 b 2 m 2 1 displaystyle left x_ 2 y_ 2 right left frac b_ 2 m m 2 1 frac b_ 2 m 2 1 right these formulas still give the correct point coordinates even if the parallel lines are horizontal i e m 0 the distance between the points is d x 2 x 1 2 y 2 y 1 2 b 1 m b 2 m m 2 1 2 b 2 b 1 m 2 1 2 displaystyle d sqrt left x_ 2 x_ 1 right 2 left y_ 2 y_ 1 right 2 sqrt left frac b_ 1 m b_ 2 m m 2 1 right 2 left frac b_ 2 b_ 1 m 2 1 right 2 which reduces to d b 2 b 1 m 2 1 displaystyle d frac b_ 2 b_ 1 sqrt m 2 1 when the lines are given by the general form of the equation of a line horizontal and vertical lines are included a x b y c 1 0 displaystyle ax by c_ 1 0 a x b y c 2 0 displaystyle ax by c_ 2 0 their distance can be expressed as d c 2 c 1 a 2 b 2 displaystyle d frac c_ 2 c_ 1 sqrt a 2 b 2 two lines in three dimensional space edit two lines in the same three dimensional space that do not intersect need not be parallel only if they are in a common plane are they called parallel otherwise they are called skew lines two distinct lines l and m in three dimensional space are parallel if and only if the distance from a point p on line m to the nearest point on line l is independent of the location of p on line m this never holds for skew lines a line and a plane edit a line m and a plane q in three dimensional space the line not lying in that plane are parallel if and only if they do not intersect equivalently they are parallel if and only if the distance from a point p on line m to the nearest point in plane q is independent of the location of p on line m two planes edit similar to the fact that parallel lines must be located in the same plane parallel planes must be situated in the same three dimensional space and contain no point in common two distinct planes q and r are parallel if and only if the distance from a point p in plane q to the nearest point in plane r is independent of the location of p in plane q this will never hold if the two planes are not in the same three dimensional space extension to non euclidean geometry edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed find sources parallel geometry news newspapers books scholar jstor may 2017 learn how and when to remove this template message in non euclidean geometry it is more common to talk about geodesics than straight lines a geodesic is the shortest path between two points in a given geometry in physics this may be interpreted as the path that a particle follows if no force is applied to it in non euclidean geometry elliptic or hyperbolic geometry the three euclidean properties mentioned above are not equivalent and only the second one line m is in the same plane as line l but does not intersect l is useful in non euclidean geometries since it involves no measurements in general geometry the three properties above give three different types of curves equidistant curves parallel geodesics and geodesics sharing a common perpendicular respectively hyperbolic geometry edit see also hyperbolic geometry intersecting parallel and ultra parallel lines through a with respect to l in the hyperbolic plane the parallel lines appear to intersect l just off the image this is just an artifact of the visualisation on a real hyperbolic plane the lines will get closer to each other and meet in infinity while in euclidean geometry two geodesics can either intersect or be parallel in hyperbolic geometry there are three possibilities two geodesics belonging to the same plane can either be intersecting if they intersect in a common point in the plane parallel if they do not intersect in the plane but converge to a common limit point at infinity ideal point or ultra parallel if they do not have a common limit point at infinity in the literature ultra parallel geodesics are often called non intersecting geodesics intersecting at infinity are called limiting parallel as in the illustration through a point a not on line l there are two limiting parallel lines one for each direction ideal point of line l they separate the lines intersecting line l and those that are ultra parallel to line l ultra parallel lines have single common perpendicular ultraparallel theorem and diverge on both sides of this common perpendicular spherical or elliptic geometry edit see also spherical geometry and elliptic geometry on the sphere there is no such thing as a parallel line line a is a great circle the equivalent of a straight line in spherical geometry line c is equidistant to line a but is not a great circle it is a parallel of latitude line b is another geodesic which intersects a in two antipodal points they share two common perpendiculars one shown in blue in spherical geometry all geodesics are great circles great circles divide the sphere in two equal hemispheres and all great circles intersect each other thus there are no parallel geodesics to a given geodesic as all geodesics intersect equidistant curves on the sphere are called parallels of latitude analogous to the latitude lines on a globe parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center of the sphere reflexive variant edit if l m n are three distinct lines then l m m n l n displaystyle l parallel m land m parallel n implies l parallel n in this case parallelism is a transitive relation however in case l n the superimposed lines are not considered parallel in euclidean geometry the binary relation between parallel lines is evidently a symmetric relation according to euclid s tenets parallelism is not a reflexive relation and thus fails to be an equivalence relation nevertheless in affine geometry a pencil of parallel lines is taken as an equivalence class in the set of lines where parallelism is an equivalence relation 15 16 17 to this end emil artin 1957 adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common 18 then a line is parallel to itself so that the reflexive and transitive properties belong to this type of parallelism creating an equivalence relation on the set of lines in the study of incidence geometry this variant of parallelism is used in the affine plane see also edit clifford parallel collinearity concurrent lines limiting parallel parallel curve ultraparallel theorem notes edit although this postulate only refers to when lines meet it is needed to prove the uniqueness of parallel lines in the sense of playfair s axiom kersey the elder john 1673 algebra vol book iv london p 177 cajori florian 1993 september 1928 184 359 368 a history of mathematical notations notations in elementary mathematics vol 1 two volumes in one unaltered reprint ed chicago us open court publishing company pp 193 402 403 411 412 isbn 0 486 67766 4 lccn 93 29211 retrieved 2019 07 22 359 for parallel occurs in oughtred s opuscula mathematica hactenus inedita 1677 p 197 a posthumous work 184 368 signs for parallel lines when recorde s sign of equality won its way upon the continent vertical lines came to be used for parallelism we find for parallel in kersey 14 caswell jones 15 wilson 16 emerson 17 kambly 18 and th...