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parallel lines in hyperbolic geometry
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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,  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:
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"parallel lines in hyperbolic geometry" is owned by Wkbj79.
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asymptotically parallel, asymptotically parallel lines, hyperparallel, hyperparallel lines, disjointly parallel, disjointly parallel lines, ultraparallel, ultraparallel lines |
This object's parent.
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Cross-references: Poincaré disc model, Beltrami-Klein model, rays, upper half plane model, boundary, intersect, lines, parallel lines, hyperbolic geometry
There are 5 references to this entry.
This is version 8 of parallel lines in hyperbolic geometry, born on 2007-05-20, modified 2007-06-06.
Object id is 9411, canonical name is ParallelLinesInHyperbolicGeometry.
Accessed 4029 times total.
Classification:
| AMS MSC: | 51-00 (Geometry :: General reference works ) | | | 51M10 (Geometry :: Real and complex geometry :: Hyperbolic and elliptic geometries and generalizations) |
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Pending Errata and Addenda
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