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:

单词	ParallelLinesInHyperbolicGeometry
释义	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:
随便看	MotionInCentralforceField motion in central-force field MotionOfContinuum motion of continuum MotivationForEuclideanDomains motivation for Euclidean domains MotivationForVonNeumannOrdinals motivation for von Neumann ordinals MotivationOfDefinitionOfConstructibleNumbers motivation of definition of constructible numbers MotivicCohomology motivic cohomology motivic cohomology MotivicCohomology1 MotzkinNumber Motzkin number MoufangLoop Moufang loop MoufangPlane Moufang plane MountainPassTheorem mountain pass theorem MovingAverage moving average MovingFrame