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Hyperbolic geometry

In [mathematics] , hyperbolic geometry (also called [Lobachevskian] geometry or [Bolyai] -Lobachevskian geometry ) is a [non-Euclidean geometry] , meaning that the [parallel postulate] of [Euclidean geometry] is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l , there is exactly one line through P that does not intersect l ; i.e., that is parallel to l . In hyperbolic geometry there are at least two distinct lines through P which do not intersect l , so the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid.

Because there is no precise hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and related terms varies among writers. In this article, the two limiting lines are called asymptotic and lines sharing a common perpendicular are called ultraparallel ; the simple word parallel may apply to both.

A characteristic property of hyperbolic geometry is that the angles of a [triangle] add to less than a [straight angle] (half circle). In the limit as the vertices go to infinity, there are even [ideal hyperbolic triangles] in which all three angles are 0°.

Non-intersecting lines

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