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# 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:

## Mathematics Subject Classification

51-00*no label found*51M10

*no label found*

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## Comments

## world editing

For the time being, I am leaving this open for world editing, as I am unsure about the clarity of the presentation of the two kinds of parallel lines (which has the added complication that each kind has at least two names). Also, if someone wants to add pstricks pictures about how these relate to other models of hyperbolic geometry, feel free to do so! Otherwise, I would greatly appreciate it if you would not edit the pictures without obtaining my consent first.

In any case, I plan on adding entries on the Poincar\'{e} disc model and the Poincar\'{e} upper half plane model later on today and, if it has not been done already, I will add what these parallel lines can look like in these models.

Warren

## no world editing

I have decided to remove the option of world editing from this entry. If anyone adds alternative models of hyperbolic geometry and can incorporate graphics, I will allow him/her to edit this entry appropriately.