Editing Hyperbolic geometry (section)

===Connection between the models===
[[File:Relation5models.png|thumb|upright=1.2|Poincaré disk, hemispherical and hyperboloid models are related by [[stereographic projection]] from −1. [[Beltrami–Klein model]] is [[orthographic projection]] from hemispherical model. [[Poincaré half-plane model]] here projected from the hemispherical model by rays from left end of Poincaré disk model.]]

All models essentially describe the same structure. The difference between them is that they represent different [[Atlas (topology)|coordinate charts]] laid down on the same [[metric space]], namely the hyperbolic plane. The characteristic feature of the hyperbolic plane itself is that it has a constant negative [[Gaussian curvature]], which is indifferent to the coordinate chart used. The [[geodesic]]s are similarly invariant: that is, geodesics map to geodesics under coordinate transformation. Hyperbolic geometry is generally introduced in terms of the geodesics and their intersections on the hyperbolic plane.<ref>Arlan Ramsay, Robert D. Richtmyer, ''Introduction to Hyperbolic Geometry'', Springer; 1 edition (December 16, 1995)</ref>

Once we choose a coordinate chart (one of the "models"), we can always [[Immersion (mathematics)|embed]] it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the curvature of Euclidean space is 0). The hyperbolic space can be represented by infinitely many different charts; but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics.

Since the four models describe the same metric space, each can be transformed into the other.

See, for example:
* [[Beltrami–Klein model#Relation to the hyperboloid model|the Beltrami–Klein model's relation to the hyperboloid model]],
* [[Beltrami–Klein model#Relation to the Poincaré disk model|the Beltrami–Klein model's relation to the Poincaré disk model]],
* and [[Poincaré disk model#Relation to the hyperboloid model|the Poincaré disk model's relation to the hyperboloid model]].