Hyperbolic Geometry

A non-Euclidean geometry, also called Lobachevsky-Bolyai-Gauss geometry, having constant sectional curvature . This geometry satisfies all of Euclid's postulates except the parallel postulate, which is modified to read: For any infinite straight line and any point not on it, there are many other infinitely extending straight lines that pass through and which do not intersect .

In hyperbolic geometry, the sum of angles of a triangle is less than , and triangles with the same angles have the same areas. Furthermore, not all triangles have the same angle sum (cf. the AAA theorem for triangles in Euclidean two-space). There are no similar triangles in hyperbolic geometry. The best-known example of a hyperbolic space are spheres in Lorentzian four-space. The Poincaré hyperbolic disk is a hyperbolic two-space. Hyperbolic geometry is well understood in two dimensions, but not in three dimensions.

Geometric models of hyperbolic geometry include the Klein-Beltrami model, which consists of an open disk in the Euclidean plane whose open chords correspond to hyperbolic lines. A two-dimensional model is the Poincaré hyperbolic disk. Felix Klein constructed an analytic hyperbolic geometry in 1870 in which a point is represented by a pair of real numbers with

(1)

(i.e., points of an open disk in the complex plane) and the distance between two points is given by

(2)

The geometry generated by this formula satisfies all of Euclid's postulates except the fifth. The metric of this geometry is given by the Cayley-Klein-Hilbert metric,

			(3)
			(4)
			(5)

Hilbert extended the definition to general bounded sets in a Euclidean space.

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