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, $\mathrm{\infty }$ 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