Editing Parallel (geometry) (section)

{{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.