Nikolai lobachevsky hyperbolic geometry

Type of non-Euclidean geometry

For other uses, see Hyperbolic (disambiguation).

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry.

Nikolai lobachevsky hyperbolic geometry

The parallel postulate of Euclidean geometry is replaced with:

For 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.

(Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.)

The hyperbolic plane is a plane where every point is a saddle point.

Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.

The hyperboloid model of hyperbolic geometry provides a representation of eve