Editing Non-Euclidean geometry (section)

==Principles==
The essential difference between the metric geometries is the nature of [[Parallel (geometry)|parallel]] lines. [[Euclid]]'s fifth postulate, the [[parallel postulate]], is equivalent to [[Playfair's Postulate|Playfair's postulate]], which states that, within a two-dimensional plane, for any given line {{mvar|l}} and a point ''A'', which is not on {{mvar|l}}, there is exactly one line through ''A'' that does not intersect {{mvar|l}}. In hyperbolic geometry, by contrast, there are [[Infinite set|infinitely]] many lines through ''A'' not intersecting {{mvar|l}}, while in elliptic geometry, any line through ''A'' intersects {{mvar|l}}.

Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both [[perpendicular]] to a third line (in the same plane):
* In Euclidean geometry, the lines remain at a constant [[distance]] from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.
* In hyperbolic geometry, they diverge from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called [[Hyperbolic geometry#Non-intersecting / parallel lines|ultraparallels]].
* In elliptic geometry, the lines converge toward each other and intersect.