Non-Euclidean geometry: Difference between revisions

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The simplest model for elliptic geometry is a sphere, where lines are "[[great circle]]s" (such as the [[equator]] or the [[meridian (geography)|meridian]]s on a [[globe]]), and points opposite each other are identified (considered to be the same).
Even after the work of Lobachevski, Gauss, and Bolyai, the question remained: does such a model exist for hyperbolic geometry?
This question 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]], the [[Poincaré disk model]], and the [[Poincaré half-plane model]] which model 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 Euclidean geometry was. (The reverse implication follows from the [[horosphere]] model of Euclidean geometry.)
 
The development of non-Euclidean geometries proved very important to physics in the [[20th century]]. Given the limitation of the [[speed of light]], velocity additions necessitate the use of [[hyperbolic geometry]].