Ultraparallel theorem: Difference between revisions

Revision as of 23:45, 9 August 2019

Revision as of 23:45, 10 August 2019

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[[File:Ultraparallel.png|thumb|200px|right|Poincaré disc model: The pink line is ultraparallel to the blue line and the green lines are limiting parallel to the blue line.]]

In [[hyperbolic geometry]], two non-intersecting lines that are ''not'' [[limiting parallel]] are called '''ultraparallel lines'''. The '''ultraparallel theorem''' states that every pair of ultraparallel lines has a unique common [[perpendicular]] hyperbolic line.

In [[hyperbolic geometry]], two lines may intersect, be '''ultraparallel''', or be [[limiting parallel]].

In conformal models of the [[hyperbolic geometryplane]], twosuch non-intersectingas linesthe thatPoincaré are ''not''models, [[limitingright parallelangle]]s aremay calledbe '''ultraparallelrecognized between intersecting lines'''. TheIn such models, the '''ultraparallel theorem''' states that every pair of ultraparallel lines has a unique common [[perpendicular]] hyperbolic line.

==Hilbert's construction==