Parallel (geometry)
In geometry, parallel lines are coplanar infinite 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. Line segments and Euclidean vectors are parallel if they have the same direction (not necessarily the same length).
This article is about the geometry concept. For other uses, see Parallel (disambiguation).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.
Symbol[edit]
The parallel symbol is .[2][3] For example, 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, U+2016 (‖), that indicate a norm (e.g. ), as well as from the logical or operator (||) in several programming languages.
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°.
Reflexive variant[edit]
If l, m, n are three distinct lines, then
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.[16][17][18]
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.[19]
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.
