# parallel lines in hyperbolic geometry

In hyperbolic geometry, there are two kinds of parallel lines. If two lines do not intersect within a model of hyperbolic geometry but they do intersect on its boundary, then the lines are called asymptotically parallel or hyperparallel. (Note that, in the upper half plane model, any two vertical rays are asymptotically parallel. Thus, for consistency, $\infty$ is considered to be part of the boundary.) Any other set of parallel lines is called disjointly parallel or ultraparallel.

Below is an example of asymptotically parallel lines in the Beltrami-Klein model:

Below are some examples of asymptotically parallel lines in the Poincaré disc model:

Below are some examples of asymptotically parallel lines in the upper half plane model:

Below is an example of disjointly parallel lines in the Beltrami-Klein model:

Below is an example of disjointly parallel lines in the Poincaré disc model:

Below are some examples of disjointly parallel lines in the upper half plane model:

 Title parallel lines in hyperbolic geometry Canonical name ParallelLinesInHyperbolicGeometry Date of creation 2013-03-22 17:06:43 Last modified on 2013-03-22 17:06:43 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 11 Author Wkbj79 (1863) Entry type Topic Classification msc 51-00 Classification msc 51M10 Defines asymptotically parallel Defines asymptotically parallel lines Defines hyperparallel Defines hyperparallel lines Defines disjointly parallel Defines disjointly parallel lines Defines ultraparallel Defines ultraparallel lines