Draft in progress. Writing hyperbolic plane.
The hyperbolic plane is a two-dimensional surface with constant negative curvature equal to −1. According to the uniformization theorem in Riemannian geometry there are exactly three distinct simply connected surfaces with constant Gaussian curvature. These include
- the hyperbolic plane with curvature −1
- the Euclidean plane with curvature 0, and
- the unit sphere with curvature +1.
Topologically, the hyperbolic plane in equivalent to the regular Euclidean plane, R2, however the negative curvature makes its geometry quite different. In particular, the hyperbolic plane satisfies the axioms of hyperbolic geometry.
The hyperbolic plane is somewhat hard to visualize, as it cannot be isometrically embedded into 3 dimensional Euclidean space (R3). It can, however, be embedded into 2+1 dimensional Minkowski space, (R2,1). This model of the hyperbolic plane, sometimes called the Minkowski model is usually taken as the standard definition.
The hyperbolic plane, usually denote H2, is given as a 2-dimensional hyperboloid of revolution in R2,1:
Here r runs from 0 to and is periodic with period 2;. These coordinates cover the entire hyperbolic plane.
The hyperbolic metric
The metric on H2 is induced from the metric on R2,1 (this is what it means to be a isometric embedding). Explicitly, the tangent space to a point x H2 can be identified with the orthogonal complement of x in R2,1. The metric on the tangent space is obtained by simply restricting the metric on R2,1. In polar coordinates, the hyperbolic metric can be written
Geodesics in on the hyperbolic plane are given by the intersection of H2 with two-dimensional subspaces of R2,1. These curves are parametrized by arc length. The geodesic distance between two points (x0, x1, x2) and (y0, y1, y2) on H2 is given by
Note, in particular, that in polar coordinates r measures the distance of any point from the "origin" (1,0,0).
The Poincaré models
The Poincaré models (named for Henri Poincaré) of the hyperbolic plane make the complex structure of the plane explicit. Specifically, take any simply connected, open subset of the complex plane (which is not the entire plane) together with its induced complex structure. By the Riemann mapping theorem, all such subsets are conformally equivalent.
Each such subset of the complex plane comes equipped with a natural Riemannian metric compatible with the complex structure which is unique up to conformal equivalence. The metric can be fixed completely by demanding that it have constant curvature −1.
The upper half plane model
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
The unit disc model
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 Klein model
The Klein model (named for Felix Klein) uses the interior of the unit disc for the hyperbolic plane, and chords of the circle as geodesics. This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted.
The Klein model can be obtained from the Minkowski model in the following manner. Draw straight lines from the origin of R2,1 to points on H2. These lines will intersect the plane x0 = 1 in the interior of the unit disk on that plane. Take this disk as the Klein model. Explicity, the map from the Minkowski model to the Klein model is given by
The metric in the Klein model is given by pullback via the above map of the metric in the Minkowski model.
Points of Klein model are given in geodesic polar coordinates by
Note that the edge of the disc is infinitely far from the center in this metric.
Escher and the hyperbolic plane
The famous circle limit III  and IV  drawings of M. C. Escher illustrate the Poincaré disc version of the model quite well. In both one can clearly see the geodesics orthogonal to the disc (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.