Editing Non-Euclidean geometry (section)

===Hyperbolic geometry===
{{main|Hyperbolic geometry}}
Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for [[hyperbolic geometry]]?". The model for [[hyperbolic geometry]] was answered by [[Eugenio Beltrami]], in 1868, who first showed that a surface called the [[pseudosphere]] has the appropriate [[curvature]] to model a portion of [[hyperbolic space]] and in a second paper in the same year, defined the [[Klein model]], which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were [[equiconsistency|equiconsistent]] so that hyperbolic geometry was [[logically consistent]] if and only if Euclidean geometry was. (The reverse implication follows from the [[horosphere]] model of Euclidean geometry.)

In the hyperbolic model, within a two-dimensional plane, for any given line {{mvar|l}} and a point ''A'', which is not on {{mvar|l}}, there are [[Infinite set|infinitely]] many lines through ''A'' that do not intersect {{mvar|l}}.

In these models, the concepts of non-Euclidean geometries are represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are represented by Euclidean curves that visually bend. This "bending" is not a property of the non-Euclidean lines, only an artifice of the way they are represented.