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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