Editing Parallel (geometry) (section)

== 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.

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=== 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.]]
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=== 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.
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