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"Given a line L and any point A not on L, at least two distinct lines exist which pass through A and are parallel to L." In this case parallel means that the lines do not intersect L, even when extended, rather than that they are a constant distance from L.

Hyperbolic geometry was initially explored by Saccheri in the 1700s, who nevertheless believed that it was inconsistent, and later by Bolyai, Gauss, and Lobachevsky, after whom it is sometimes named. (See article on non-Euclidean geometry for more history.)

There are three models commonly used for hyperbolic geometry. The Klein model uses the interior of a circle for the hyperbolic plane, and chords of the circle as lines. This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted. The PoincaréJules Henri Poincar ( April 29, 1854 July 17, 1912) was one of France's greatest mathematicians, theoretical scientists and a philosopher of science. Poincare is often described as the last "universalist" capable of understanding and contributing in virtu disc model also employs the interior of a circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle. The Poincaré half-plane modelIn mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. It is the domain of many functions of interest in complex analysis, especially modular forms. It is also a model of the hyperbolic plane. There is a group action takes one-half of the Euclidean plane, as determined by a Euclidean line B, to be the hyperbolic plane (B itself is not included). Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B.

Both Poincaré models preserve hyperbolic angles, and are thereby conformalIn mathematics, a mapping w f ''z is angle-preserving or (more usually) conformal at a point z if it preserves oriented angles between curves through z as well as their orientation, i. Conformal maps preserve both angles and the shapes of infinitesimally. All isometries within these models are therefore Möbius transformationMobius transformations should not be confused with the Mobius transform. Geometry In mathematics, a Mobius transformation named in honor of August Ferdinand Mobius, is a conformal mapping that is a bijection on the extended complex plane (that is, the coms.

A fourth model is the Alexander MacFarlaneAlexander MacFarlane ( 1851 1913) was a Scottish- Canadian mathematician. He was born at Blairgowrie, Scotland. During his life, he played a prominent role in his field. He was, at various times in his life, physics professor at the University of Texas, e model, which employs an 3-dimensional hyperboloidIn mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation: : (hyperboloid of one sheet), or : (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution . A hyperboloid of one sheet of revolution (of two sheets, but using one) embedded in 4-dimensional euclidean space. This model is sometimes ascribed to Karl Weierstrass. Macfarlane used hyperbolic quaternions to describe it in 1900.

Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate.