Editing Parallel (geometry)

{{Short description|Relation used in geometry}}
{{About|the geometry concept||Parallel (disambiguation)}}
{{Redirect2|Parallel lines|Parallel line}}
{{Use dmy dates|date=July 2019|cs1-dates=y}}
[[File:Parallel (PSF).png|thumb|Line art drawing of parallel lines and curves.|alt==]]

In [[geometry]], '''parallel lines''' are [[coplanar]] infinite straight [[line (geometry)|lines]] that do not [[intersecting lines|intersect]] at any point. '''Parallel planes''' are [[plane (geometry)|planes]] in the same [[three-dimensional space]] that never meet. ''[[Parallel curve]]s'' are [[curve]]s that do not [[tangent|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]]''. [[Line segment]]s and [[Euclidean vectors]] are parallel if they have the same [[direction (geometry)|direction]] or [[opposite direction (geometry)|opposite direction]] (not necessarily the same length).<ref name=HMCS>{{cite book |last1=Harris |first1=John W. |last2=Stöcker |first2=Horst |year=1998 |title=Handbook of mathematics and computational science |publisher=Birkhäuser |isbn=0-387-94746-9 |at=Chapter 6, p. 332 |url=https://books.google.com/books?id=DnKLkOb_YfIC&pg=PA332 }}</ref>

Parallel lines are the subject of [[Euclid]]'s [[parallel postulate]].<ref>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]].</ref> Parallelism is primarily a property of [[affine geometry|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.

== Symbol ==
{{redirect|∥||Vertical bar (disambiguation)}}

The parallel symbol is <math>\parallel</math>.<ref name="Kersey_1673"/><ref name="Cajori_1928"/> For example, <math>AB \parallel CD</math> indicates that line ''AB'' is parallel to line&nbsp;''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".<ref>{{cite web| title = Mathematical Operators – Unicode Consortium| url = https://www.unicode.org/charts/PDF/U2200.pdf| access-date = 2013-04-21}}</ref>

== Euclidean parallelism==

===Two lines in a plane===

====Conditions for parallelism====

[[Image:Parallel transversal.svg|thumb|right|300px|As shown by the tick marks, lines ''a'' and ''b'' are parallel. This can be proved because the transversal ''t'' produces congruent corresponding angles <math>\theta</math>, 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 (geometry)|transversal]]) in the same plane,  the [[corresponding angles]] of intersection with the transversal are [[Congruence (geometry)|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.<ref>{{harvnb|Wylie|1964|loc=pp. 92—94}}</ref> The other properties are then consequences of [[Euclid's Fifth Axiom|Euclid's Parallel Postulate]].

====History====
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]].<ref name=Euclid>{{harvnb|Heath|1956|loc=pp. 190–194}}</ref> 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 of Cilicia|Simplicius]] also mentions Posidonius' definition as well as its modification by the philosopher Aganis.<ref name=Euclid />

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.<ref>{{harvnb|Richards|1988|loc=Chap. 4: Euclid and the English Schoolchild. pp. 161–200}}</ref> 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.<ref>{{citation|first=Lewis|last=Carroll|title=Euclid and His Modern Rivals|date=2009|orig-year=1879|publisher=Barnes & Noble|isbn=978-1-4351-2348-9}}</ref>

One of the early reform textbooks was James Maurice Wilson's ''Elementary Geometry'' of 1868.<ref>{{harvnb|Wilson|1868}}</ref> Wilson based his definition of parallel lines on the [[primitive notion]] of ''direction''. According to [[Wilhelm Killing]]<ref>''Einführung in die Grundlagen der Geometrie, I'', p. 5</ref> the idea may be traced back to [[Gottfried Wilhelm Leibniz|Leibniz]].<ref>{{harvnb|Heath|1956|loc= p. 194}}</ref> 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." {{harvtxt|Wilson|1868|loc=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''." {{harvtxt|Wilson|1868|loc=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.<ref>{{harvnb|Richards|1988|loc=pp. 180–184}}</ref>

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.<ref>{{harvnb|Heath|1956|loc=p. 194}}</ref> 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 ====
The three properties above lead to three different methods of construction<ref>Only the third is a straightedge and compass construction, the first two are infinitary processes (they require an "infinite number of steps".)</ref> of parallel lines.
{{Clear|left}}
[[Image:Par-prob.png|thumb|left|250px|The problem: Draw a line through ''a'' parallel to ''l''.]]
{{Clear|left}}
<gallery widths="200px">
image:Par-equi.png|Property 1: Line ''m'' has everywhere the same distance to line ''l''.
image:Par-para.png|Property 2: Take a random line through ''a'' that intersects ''l'' in ''x''. Move point ''x'' to infinity.
image:Par-perp.png|Property 3: Both ''l'' and ''m'' share a transversal line through ''a'' that intersect them at 90°.
</gallery>

==== Distance between two parallel lines ====
{{Main|Distance between two parallel 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,
:<math>y = mx+b_1\,</math>
:<math>y = mx+b_2\,,</math>
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
:<math>\begin{cases}
y = mx+b_1 \\
y = -x/m
\end{cases}</math>
and
:<math>\begin{cases}
y = mx+b_2 \\
y = -x/m
\end{cases}</math>
to get the coordinates of the points. The solutions to the linear systems are the points
:<math>\left( x_1,y_1 \right)\ = \left( \frac{-b_1m}{m^2+1},\frac{b_1}{m^2+1} \right)\,</math>
and
:<math>\left( x_2,y_2 \right)\ = \left( \frac{-b_2m}{m^2+1},\frac{b_2}{m^2+1} \right).</math>
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
:<math>d = \sqrt{\left(x_2-x_1\right)^2 + \left(y_2-y_1\right)^2} = \sqrt{\left(\frac{b_1m-b_2m}{m^2+1}\right)^2 + \left(\frac{b_2-b_1}{m^2+1}\right)^2}\,,</math>
which reduces to
:<math>d = \frac{|b_2-b_1|}{\sqrt{m^2+1}}\,.</math>

When the lines are given by the general form of the equation of a line (horizontal and vertical lines are included):
:<math>ax+by+c_1=0\,</math>
:<math>ax+by+c_2=0,\,</math>
their distance can be expressed as
:<math>d = \frac{|c_2-c_1|}{\sqrt {a^2+b^2}}.</math>

===Two lines in three-dimensional space===
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===
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===
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.

== In non-Euclidean geometry ==

In [[non-Euclidean geometry]], the concept of a straight line is replaced by the more general concept of a [[geodesic]], a curve which is [[local property|locally]] straight with respect to the [[metric tensor|metric]] (definition of distance) on a [[Riemannian manifold]], a surface (or higher-dimensional space) which may itself be curved. In [[general relativity]], particles not under the influence of external forces follow geodesics in [[spacetime]], a four-dimensional manifold with 3 spatial dimensions and 1 time dimension.<ref>{{Cite web |last=Church |first=Benjamin |date=December 3, 2022 |title=A Not So Gentle Introduction to General Relativity |url=https://web.stanford.edu/~bvchurch/assets/files/talks/GR.pdf}}</ref>

In non-Euclidean geometry ([[Elliptic 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 ===
{{See also|Hyperbolic geometry}}
[[Image:HyperParallel.png|thumb|300px|right|'''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.<ref>{{Cite web |date=2021-10-30 |title=5.3: Theorems of Hyperbolic Geometry |url=https://math.libretexts.org/Bookshelves/Geometry/An_IBL_Introduction_to_Geometries_(Mark_Fitch)/05:_Hyperbolic_Geometry/5.03:_New_Page |access-date=2024-08-22 |website=Mathematics LibreTexts |language=en}}</ref>

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.

<!-- This section needs major copyediting before being put back in the article
=== Hyperbolic ===
In the [[hyperbolic geometry|hyperbolic plane]], there are two lines through a given point that intersect a given line in the limit to infinity. While in Euclidean geometry a geodesic intersects its parallels in both directions in the limit to infinity, in hyperbolic geometry both directions have their own line of parallelism. When visualized on a plane a geodesic is said to have a '''left-handed parallel''' and a '''right-handed parallel''' through a given point. The angle the parallel lines make with the perpendicular from that point to the given line is called the [[angle of parallelism]]. The angle of parallelism depends on the distance of the point to the line with respect to the [[curvature]] of the space. The angle is also present in the Euclidean case, there it is always 90° so the left and right-handed parallels [[Coincident|coincide]]. The parallel lines divide the set of geodesics through the point in two sets: '''intersecting geodesics''' that intersect the given line in the hyperbolic plane, and '''ultra parallel geodesics''' that do not intersect even in the limit to infinity (in either direction). In the Euclidean limit the latter set is empty.
[[Image:HyperParallel.png|thumb|300px|left|'''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 an artifact of the visualisation. It is not possible to isometrically embed the hyperbolic plane in three dimensions. In a real hyperbolic space the lines will get closer to each other and 'touch' in infinity.]]
{{Clear}}
-->

=== Spherical or elliptic geometry ===
{{See also| Spherical geometry|Elliptic geometry}}

[[File:SphereParallel.png|thumb|300px|right|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 [[Sphere|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.
{{Clear}}

==Reflexive variant==
If ''l, m, n'' are three distinct lines, then <math>l \parallel m \ \land \  m \parallel n \ \implies \ l \parallel n .</math>

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.<ref>[[H. S. M. Coxeter]] (1961) ''Introduction to Geometry'', p 192, [[John Wiley & Sons]]</ref><ref>[[Wanda Szmielew]] (1983) ''From Affine to Euclidean Geometry'', p 17, [[D. Reidel]] {{isbn|90-277-1243-3}}</ref><ref>Andy Liu (2011) "Is parallelism an equivalence relation?", [[The College Mathematics Journal]] 42(5):372</ref>

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.<ref>[[Emil Artin]] (1957) [https://archive.org/details/geometricalgebra033556mbp/page/n63/mode/2up?view=theater ''Geometric Algebra'', page 52] via [[Internet Archive]]</ref>
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 (incidence geometry)|affine plane]].

== See also ==
*[[Clifford parallel]]
*[[Collinearity]]
*[[Concurrent lines]]
*[[Limiting parallel]]
*[[Parallel curve]]
*[[Ultraparallel theorem]]

==Notes==
{{Reflist|refs=
<ref name="Kersey_1673">{{cite book |author-first=John |author-last=Kersey (the elder) |author-link=John Kersey the elder |title=Algebra |location=London |date=1673 |volume=Book IV |page=177}}</ref>
<ref name="Cajori_1928">{{cite book |author-first=Florian |author-last=Cajori |author-link=Florian Cajori |title=A History of Mathematical Notations - Notations in Elementary Mathematics |chapter=§ 184, § 359, § 368 |volume=1 |orig-year=September 1928 |publisher=[[Open court publishing company]] |location=Chicago, US |date=1993 |edition=two volumes in one unaltered reprint |pages=[https://archive.org/details/historyofmathema00cajo_0/page/193 193, 402–403, 411–412] |isbn=0-486-67766-4 |lccn=93-29211 |chapter-url=https://archive.org/details/historyofmathema00cajo_0/page/193 |access-date=2019-07-22 |quote=§359. […] ∥ for parallel occurs in [[William Oughtred|Oughtred]]'s ''Opuscula mathematica hactenus inedita'' (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when [[Robert Recorde|Recorde]]'s sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in [[John Kersey the elder|Kersey]],[14] [[John Caswell|Caswell]], [[William Jones (mathematician)|Jones]],[15] Wilson,[16] [[William Emerson (mathematician)|Emerson]],[17] Kambly,[18] and the writers of the last fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often […] Hall and Stevens[1] use "par[1] or ∥" for parallel […] [14] [[John Kersey the elder|John Kersey]], ''Algebra'' (London, 1673), Book IV, p. 177. [15] [[William Jones (mathematician)|W. Jones]], ''Synopsis palmarioum matheseos'' (London, 1706). [16] John Wilson, ''Trigonometry'' (Edinburgh, 1714), characters explained. [17] [[William Emerson (mathematician)|W. Emerson]], ''Elements of Geometry'' (London, 1763), p. 4. [18] {{ill|Ludwig Kambly{{!}}L.<!-- Ludwig --> Kambly|de|Ludwig Kambly}}, ''Die Elementar-Mathematik'', Part 2: ''Planimetrie'', 43. edition (Breslau, 1876), p. 8. […] [1] H. S.<!-- Henry Sinclair --> Hall and F. H.<!-- Frederick Haller --> Stevens, ''Euclid's Elements'', Parts I and II (London, 1889), p. 10. […] }} [https://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf]</ref>
}}

==References==
* {{citation
 | last = Heath
 | first = Thomas L.
 | author-link = T. L. Heath
 | title = The Thirteen Books of Euclid's Elements
 | edition = 2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925]
 | date = 1956
 | publisher = Dover Publications
 | location = New York
 }}
: (3 vols.): {{isbn|0-486-60088-2}} (vol. 1), {{isbn|0-486-60089-0}} (vol. 2), {{isbn|0-486-60090-4}} (vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.
* {{citation|last=Richards|first=Joan L.|author-link= Joan L. Richards |title=Mathematical Visions: The Pursuit of Geometry in Victorian England|date=1988|publisher=Academic Press|location=Boston|isbn=0-12-587445-6}}
* {{citation|first=James Maurice|last=Wilson|title=Elementary Geometry|edition=1st|date=1868|place=London|publisher=Macmillan and Co.}}
* {{citation|first=C. R. Jr.|last=Wylie|title=Foundations of Geometry|date=1964|publisher=McGraw–Hill}}

==Further reading==
* {{citation
 | last1 = Papadopoulos
 | first1 = Athanase
 | last2=Théret
 |first2= Guillaume
 | title = La théorie des parallèles de Johann Heinrich Lambert : Présentation, traduction et commentaires
 | date = 2014
 | publisher = Collection Sciences dans l'histoire, Librairie Albert Blanchard
 | place=Paris
 |isbn=978-2-85367-266-5}}

[[Category:Elementary geometry]]
[[Category:Affine geometry]]
[[Category:Orientation (geometry)]]