Definition:Parallel (Geometry)/Lines

Definition


The contemporary interpretation of the concept of parallelism declares that a straight line is parallel to itself.

Comment
Different geometries allow different conditions for the existence of parallel lines.


 * Euclidean geometry allows that exactly one line through a given point can be constructed parallel to a given line;
 * Hyperbolic geometry allows for an infinite number of such;
 * Elliptical geometry does not allow construction of such lines.

Reflexivity
An attempt can be made to define parallelism by suggesting that the perpendiculars dropped from one (line or plane) to another (line or plane) are the same length everywhere along the line or plane, but this interpretation does not work in the context of non-Euclidean geometries, and is in fact no more than a derivable consequence of the definition of parallel as given here.