Editing Hyperbolic geometry (section)

== Models of the hyperbolic plane ==
Various [[pseudosphere]]s – surfaces with constant negative Gaussian curvature – can be embedded in 3-D space under the standard Euclidean metric, and so can be made into tangible models. Of these, the [[Pseudosphere#Tractroid|tractoid]] (or pseudosphere) is the best known; using the tractoid as a model of the hyperbolic plane is analogous to using a [[cone]] or [[cylinder]] as a model of the Euclidean plane. However, the entire hyperbolic plane cannot be embedded into Euclidean space in this way, and various other models are more convenient for abstractly exploring hyperbolic geometry.

There are four [[Mathematical model|model]]s commonly used for hyperbolic geometry: the [[Klein model]], the [[Poincaré disk model]], the [[Poincaré half-plane model]], and the Lorentz or [[hyperboloid model]]. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. Despite their names, the first three mentioned above were introduced as models of hyperbolic space by [[Eugenio Beltrami|Beltrami]], not by [[Henri Poincaré|Poincaré]] or [[Felix Klein|Klein]]. All these models are extendable to more dimensions.

===The Beltrami–Klein model===
{{main article|Beltrami–Klein model}}

The [[Beltrami–Klein model]], also known as the projective disk model, Klein disk model and [[Klein model]], is named after [[Eugenio Beltrami]] and [[Felix Klein]].

For the two dimensions this model uses the interior of the [[unit circle]] for the complete hyperbolic [[plane (mathematics)|plane]], and the [[chord (geometry)|chords]] of this circle are the hyperbolic lines.

For higher dimensions this model uses the interior of the [[unit ball]], and the [[chord (geometry)|chords]] of this ''n''-ball are the hyperbolic lines.
* This model has the advantage that lines are straight, but the disadvantage that [[angle]]s are distorted (the mapping is not [[Conformal map|conformal]]), and also circles are not represented as circles.
* The distance in this model is half the logarithm of the [[cross-ratio]], which was introduced by [[Arthur Cayley]] in [[projective geometry]].

=== The Poincaré disk model ===
[[File:Hyperbolic tiling omnitruncated 3-7.png|thumb|Poincaré disk model with [[truncated triheptagonal tiling]]]]
{{main article|Poincaré disk model}}

The [[Poincaré disk model]], also known as the conformal disk model, also employs the interior of the [[unit circle]], but lines are represented by arcs of circles that are [[orthogonal]] to the boundary circle, plus diameters of the boundary circle.
* This model preserves angles, and is thereby [[conformal map|conformal]]. All isometries within this model are therefore [[Möbius transformation]]s.
* Circles entirely within the disk remain circles although the Euclidean center of the circle is closer to the center of the disk than is the hyperbolic center of the circle.
* [[Horocycle]]s are circles within the disk which are [[tangent]] to the boundary circle, minus the point of contact.
* [[Hypercycle (hyperbolic geometry)|Hypercycle]]s are open-ended chords and circular arcs within the disc that terminate on the boundary circle at non-orthogonal angles.

=== The Poincaré half-plane model ===

{{main article|Poincaré half-plane model}}

The [[Poincaré half-plane model]] takes one-half of the Euclidean plane, bounded by a line ''B'' of the plane, to be a model of the hyperbolic plane.  The line ''B'' is not included in the model.

The Euclidean plane may be taken to be a plane with the [[Cartesian coordinate system]] and the [[x-axis]] is taken as line ''B'' and the half plane is the upper half (''y'' > 0 )  of this plane.
* Hyperbolic lines are then either half-circles orthogonal to ''B'' or rays perpendicular to ''B''.
* The length of an interval on a ray is given by [[logarithmic measure]] so it is invariant under a [[homothetic transformation]] <math>(x, y) \mapsto (\lambda x, \lambda y),\quad \lambda > 0 .</math>
* Like the Poincaré disk model, this model preserves angles, and is thus [[conformal map|conformal]]. All isometries within this model are therefore [[Möbius transformation]]s of the plane.
* The half-plane model is the limit of the Poincaré disk model whose boundary is tangent to ''B'' at the same point while the radius of the disk model goes to infinity.

=== The hyperboloid model ===
{{main article|hyperboloid model}}

The [[hyperboloid model]] or Lorentz model employs a 2-dimensional [[hyperboloid]] of revolution (of two sheets, but using one) embedded in 3-dimensional [[Minkowski space]]. This model is generally credited to Poincaré, but Reynolds<ref>{{aut|Reynolds, William F.}}, (1993) ''Hyperbolic Geometry on a Hyperboloid'', [[American Mathematical Monthly]] 100:442–455.</ref> says that [[Wilhelm Killing]] used this model in 1885
* This model has direct application to [[special relativity]], as Minkowski 3-space is a model for [[spacetime]], suppressing one spatial dimension. One can take the hyperboloid to represent the events (positions in spacetime) that various [[Inertial frame of reference|inertially]] moving observers, starting from a common event, will reach in a fixed [[proper time]].
* The hyperbolic distance between two points on the hyperboloid can then be identified with the relative [[rapidity]] between the two corresponding observers.
* The model generalizes directly to an additional dimension: a hyperbolic 3-space  three-dimensional hyperbolic geometry relates to Minkowski 4-space.

=== The hemisphere model ===

The [[Sphere#Hemisphere|hemisphere]] model is not often used as model by itself, but it functions as a useful tool for visualizing transformations between the other models.

The hemisphere model uses the upper half of the [[unit sphere]]:
<math> x^2 + y^2 +z^2 = 1 , z > 0. </math>

The hyperbolic lines are half-circles orthogonal to the boundary of the hemisphere.

The hemisphere model is part of a [[Riemann sphere]], and different projections give different models of the hyperbolic plane:
* [[Stereographic projection]] from  <math> (0,0, -1) </math> onto the plane <math> z = 0 </math>  projects corresponding points on the [[Poincaré disk model]]
* [[Stereographic projection]] from  <math> (0,0, -1) </math> onto the surface <math> x^2 + y^2 - z^2  = -1 , z > 0  </math>  projects corresponding points on the [[hyperboloid model]]
* [[Stereographic projection]] from <math> (-1,0,0) </math> onto the plane <math> x=1 </math> projects corresponding points on the [[Poincaré half-plane model]]
* [[Orthographic projection]] onto a plane <math> z = C </math> projects corresponding points on the [[Beltrami–Klein model]].
* [[Central projection]] from the centre of the sphere onto the plane <math> z = 1 </math> projects corresponding points on the [[Gans Model]]

===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]].



=== Other models of hyperbolic geometry ===

==== The Gans model ====
In 1966 David Gans proposed a [[flattened hyperboloid model]] in the journal ''[[American Mathematical Monthly]]''.<ref>{{cite journal|author=Gans David |title=A New Model of the Hyperbolic Plane |journal=American Mathematical Monthly |volume= 73 |issue= 3 |date=March 1966 |doi=10.2307/2315350 |pages=291–295|jstor=2315350 }}</ref> It is an [[orthographic projection]] of the hyperboloid model onto the xy-plane.
This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry.
* Unlike the Klein or the Poincaré models, this model utilizes the entire [[Euclidean plane]].
* The lines in this model are represented as branches of a [[hyperbola]].<ref>{{cite web|url=http://www.d.umn.edu/cs/thesis/kedar_bhumkar_ms.pdf|title=Computer Science Department|last=vcoit|date=8 May 2015}}</ref>

==== The conformal square model ====
[[File:Omnitruncated tiling on conformal square.png|thumb|Conformal square model with [[truncated triheptagonal tiling]]]]

The conformal square model of the hyperbolic plane arises from using [[Schwarz–Christoffel mapping]] to convert the [[Poincaré disk model|Poincaré disk]] into a square.<ref>{{cite conference|url=http://archive.bridgesmathart.org/2016/bridges2016-179.pdf|title=The Conformal Hyperbolic Square and Its Ilk |last=Fong| first = C.|year=2016 |conference=Bridges Finland Conference Proceedings}}</ref> This model has finite extent, like the Poincaré disk. However, all of the points are inside a square. This model is conformal, which makes it suitable for artistic applications.

==== The band model ====
{{main article|Band model}}
The band model employs a portion of the Euclidean plane between two parallel lines.<ref>{{cite book|url=http://matrixeditions.com/TeichmullerVol1.html|title=Teichmüller theory and applications to geometry, topology, and dynamics|year=2006–2016 |publisher=Matrix Editions|others=Hubbard, John Hamal |isbn=9780971576629|location=Ithaca, NY|oclc=57965863|chapter=2|chapter-url=http://matrixeditions.com/TVol1.Chap2.pdf|page=25}}</ref> Distance is preserved along one line through the middle of the band. Assuming the band is given by <math>\{z \in \mathbb C:|\operatorname {Im} z| < \pi / 2\}</math>, the metric is given by <math>|dz| \sec (\operatorname{Im} z)</math>.