Editing Hyperbolic geometry (section)

====Non-intersecting / parallel lines====
[[File:Hyperbolic.svg|frame|right|Lines through a given point ''P'' and asymptotic to line ''R'']]

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in [[Euclidean geometry]]:

:For any line ''R'' and any point ''P'' which does not lie on ''R'', in the plane containing line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''.

This implies that there are through ''P'' an infinite number of coplanar lines that do not intersect ''R''.

These non-intersecting lines are divided into two classes:
* Two of the lines (''x'' and ''y'' in the diagram) are [[limiting parallel]]s (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the [[ideal point]]s at the "ends" of ''R'', asymptotically approaching ''R'', always getting closer to ''R'', but never meeting it.
* All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, and are called ''ultraparallel'', ''diverging parallel'' or sometimes ''non-intersecting.''
Some geometers simply use the phrase "''parallel'' lines" to mean "''limiting parallel'' lines", with ''ultraparallel'' lines meaning just ''non-intersecting''.

These [[limiting parallel]]s make an angle ''θ'' with ''PB''; this angle depends only on the [[Gaussian curvature]] of the plane and the distance ''PB'' and is called the [[angle of parallelism]].

For ultraparallel lines, the [[ultraparallel theorem]] states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines.