Hyperbolic geometry
Hyperbolic geometry, also called saddle geometry or Lobachevskian geometry, is the non-Euclidean geometry obtained by replacing the parallel postulate with the hyperbolic postulate, which states: "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 Poincaré 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 model 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 conformal. All isometries within these models are therefore Möbius transformations.
A fourth model is the Minkowski model, which employs an N-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in N+1-dimensional euclidean space. This model employs a metric whereby the distance between any two points on the hyperboloid is d2 = x12 + x22 + ... + xN2 - xN+12. This is the same metric as that used in special relativity for space-time.
Examples of the three models to come.
Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate.