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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.
In hyperbolic geometry, the term parallel only applies to lines that don't intersect in the hyperbolic plane but intersect at the circle at infinity. Lines that neither intersect in the hyperbolic plane nor the circle at infinity are called ultraparallel. One remarkable property of the hyperbolic plane is that there is a unique common perpendicular for each pair of ultraparallel lines (see Ultraparallel theorem).
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.)
The Klein model, also known as the projective disc model and Beltrami-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 known as the conformal 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.
A fourth model is the Lorentz model or hyperboloid model, which employs an 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré.
Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate.
Visualizing hyperbolic geometry
The famous circle limit III  and IV  drawings of M. C. Escher illustrate the unit disc version of the model quite well. In both one can clearly see the geodesics (in III they appear explicitly). It is also possible to see quite plainly the negative curvature of the hyperbolic plane, via its effect on the sum of angles in triangles and squares.
For example, in III every vertex is the intersection of three triangles and three squares. In normal Euclidean plane, this would sum up to 450°, leading to a contradiction. Hence we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is the fact that the hyperbolic plane has exponential growth. In IV, for example, one can see that the number of angels with a distance of n from the center rises exponentially. The angels have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.
Relationship to Riemann surfaces
Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Hyperbolic surfaces have a non-trivial fundamental group π1 = Γ, known as the Fuchsian group. The quotient space H/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface.
The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.
- Angle of parallelism
- Fuchsian group
- Fuchsian model
- Kleinian group
- Kleinian model
- Poincaré metric
- Riemann surface
- Stillwell, John (1996) Sources in Hyperbolic Geometry, volume 10 in AMS/LMS series History of Mathematics.
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