Parallel (geometry)

This article is about the geometry concept. For other uses, see Parallel (disambiguation).
"Parallel lines" and "Parallel line" redirect here. For other uses, see Parallel lines (disambiguation).

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Line art drawing of parallel lines and curves.

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 or opposite direction (not necessarily the same length).[1]

Parallel lines are the subject of Euclid's parallel postulate.[2] 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.

  1. ^ Harris, John W.; Stöcker, Horst (1998). Handbook of mathematics and computational science. Birkhäuser. Chapter 6, p. 332. ISBN 0-387-94746-9.
  2. ^ 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.