In the words of Euclid:
- Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.
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.
The contemporary interpretation of the concept of parallelism declares that a line (or a plane) is parallel to itself. This coincides with the mathematical definition of parallelism from the viewpoint of analytic geometry and allows the relation is parallel to to be an equivalence relation.
However, it is clear that a line intersects itself everywhere and therefore does not adhere to the standard definition of parallel. Therefore the above "special case" needs to be made.
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.