![]() Non-intersecting / parallel lines Lines through a given point P and asymptotic to line R. These properties are all independent of the model used, even if the lines may look radically different. For example, given two intersecting lines there are infinitely many lines that do not intersect either of the given lines. When a third line is introduced, then there can be properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting lines are supplementary. Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two points uniquely define a line, and line segments can be infinitely extended. Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry new concepts need to be introduced.įurther, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines. There are two kinds of absolute geometry, Euclidean and hyperbolic.Īll theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. Properties Relation to Euclidean geometry Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions See hyperbolic space for more information on hyperbolic geometry extended to three and more dimensions. ![]() This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. In the former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky. ![]() When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names Felix Klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry ( spherical geometry), parabolic geometry ( Euclidean geometry), and hyperbolic geometry. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.Ī modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. The hyperbolic plane is a plane where every point is a saddle point. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The parallel postulate of Euclidean geometry is replaced with:įor any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), along with two diverging ultra-parallel lines
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