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| [[Image:Klein model.png|thumb|240px|right|Lines in the projective model of the [[hyperbolic geometry|hyperbolic plane]].]]
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| [[File:Uniform tiling 73-t1 klein.png|thumb|240px|A hyperbolic [[triheptagonal tiling]] in a Beltrami–Klein model projection]]
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| [[File:Hyperbolic orthogonal dodecahedral honeycomb.png|thumb|240px|The regular hyperbolic [[order-4 dodecahedral honeycomb|dodecahedral honeycomb]], {5,3,4}]]
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| In geometry, the '''Beltrami–Klein model''', also called the '''projective model''', '''Klein disk model''', and the '''Cayley–Klein model''', is a model of ''n''-dimensional [[hyperbolic geometry]] in which points are represented by the points in the interior of the n-dimensional [[unit ball]] (or [[unit disk]], in two dimensions) and lines are represented by the [[chord (geometry)|chord]]s, straight line segments with endpoints on the boundary sphere. It made its first appearance in two memoirs of [[Eugenio Beltrami]] published in 1868, first for ''n'' = 2 and then for general ''n'', devoted to showing [[equiconsistency]] of hyperbolic geometry with ordinary [[Euclidean geometry]].<ref>{{cite journal|first=Eugenio|last=Beltrami|title=Saggio di interpretazione della geometria non-euclidea|journal=Giornale di Mathematiche|volume=VI|year=1868|pages=285–315}}</ref><ref>{{cite journal|first=Eugenio|last=Beltrami|title=Teoria fondamentale degli spazii di curvatura costante| journal=Annali. di Mat., ser II|volume=2|year=1868|pages=232–255|doi=10.1007/BF02419615}}</ref>
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| The Beltrami–Klein model is strongly analogous to the [[gnomonic projection]] of [[spherical geometry]], which maps [[great circle]]s to straight lines; the formulae relating these two to the [[hyperboloid model]] and the sphere, respectively, are very similar.
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| The [[metric (mathematics)|distance]] is given by the [[Cayley–Klein metric]] and was first written down by [[Arthur Cayley]] in the context of [[projective geometry|projective]] and [[spherical geometry|spherical]] geometry. [[Felix Klein]] recognized its importance for [[non-Euclidean geometry]] and popularized the subject. | |
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| == Distance formula ==
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| [[Arthur Cayley]] applied the [[cross-ratio]] from [[projective geometry]] to measurement of distances and angles in [[spherical geometry]].<ref>{{cite journal|first=Arthur|last=Cayley|title=A Sixth Memoire upon Quantics|journal=Philosophical Transactions of the Royal Society|year=1859|volume=159|pages=61–91|doi=10.1098/rstl.1859.0004}}</ref> Later, [[Felix Klein]] realized that Cayley's ideas give rise to a projective model of the non-Euclidean plane.<ref>{{cite journal|first=Felix|last=Klein|title=Ueber die sogenannte [[Non-Euclidean geometry|Nicht-Euklidische Geometrie]]|journal=Mathematische Annalen|year=1871|volume=4|pages=573–625|doi=10.1007/BF02100583}}</ref>
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| Given two distinct points ''p'' and ''q'' in the open unit ball, the unique straight line connecting them intersects the unit sphere in two points, ''a'' and ''b'', labeled so that the points are, in order, ''a'', ''p'', ''q'', ''b''. Then the hyperbolic distance between ''p'' and ''q'' is expressed as
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| :<math>d(p,q)=\frac{1}{2} \log \frac{|qa||bp|}{|pa||bq|}</math>
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| where the vertical bars indicate Euclidean distances. The factor of one half is needed to make the [[Gaussian curvature|curvature]] −1.
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| == Relation to the hyperboloid model ==
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| The [[hyperboloid model]] is a model of hyperbolic geometry within (''n'' + 1)-dimensional [[Minkowski space]]. The Minkowski inner product is given by
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| :<math>\mathbf{x} \cdot \mathbf{y} = x_0 y_0 - x_1 y_1 - \cdots - x_n y_n \, </math>
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| and the norm by <math>\|\mathbf{x}\| = \sqrt{\mathbf{x}\cdot\mathbf{x}}</math>. The hyperbolic plane is embedded in this space as the vectors '''x''' with ||'''x'''|| = 1 and ''x''<sub>0</sub> (the "timelike component") positive. The intrinsic distance (in the embedding) between points '''u''' and '''v''' is then given by
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| :<math>d(\mathbf{u},\mathbf{v}) = \cosh^{-1}(\mathbf{u} \cdot \mathbf{v}).\,</math>
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| This may also be written in the homogeneous form
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| :<math>d(\mathbf{u},\mathbf{v}) = \cosh^{-1}\left(\frac{\mathbf{u}}{\|\mathbf{u}\|} \cdot \frac{\mathbf{v}}{\|\mathbf{v}\|}\right)</math>
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| which allows the vectors to be rescaled for convenience.
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| The Beltrami–Klein model is obtained from the hyperboloid model by rescaling all vectors so that the timelike component is 1, that is, by projecting the hyperboloid embedding through the origin onto the plane ''x''<sub>0</sub> = 1. The distance function, in its homogeneous form, is unchanged. Since the intrinsic lines (geodesics) of the hyperboloid model are the intersection of the embedding with planes through the Minkowski origin, the intrinsic lines of the Beltrami–Klein model are the chords of the sphere.
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| In the [[gyrovector space]] approach to hyperbolic geometry, vector algebra in the Beltrami–Klein model can be developed using [[Velocity-addition formula|relativistic 3-velocities]] as the vectors, analogously to the use of ordinary vectors in [[Euclidean geometry]].
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| == Relation to the Poincaré disk model ==
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| Both the [[Poincaré disk model]] and the Beltrami–Klein model are models of the ''n''-dimensional hyperbolic space in the ''n''-dimensional unit ball in '''R'''<sup>''n''</sup>. If <math>u</math> is a vector of norm less than one representing a point of the Poincaré disk model, then the corresponding point of the Beltrami–Klein model is given by
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| :<math>s = \frac{2u}{1+u \cdot u}.</math> | |
| Conversely, from a vector <math>s</math> of norm less than one representing a point of the Beltrami–Klein model, the corresponding point of the Poincaré disk model is given by
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| :<math>u = \frac{s}{1+\sqrt{1-s \cdot s}} = | |
| \frac{\left(1-\sqrt{1-s \cdot s}\right)s}{s \cdot s}.</math>
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| Given two points on the boundary of the unit disk, which are traditionally called ''ideal points'', the straight line connecting them in the Beltrami–Klein model is the chord between them, while in the corresponding Poincaré model the line is a [[circular arc]] on the two-dimensional subspace generated by the two boundary point vectors, orthogonal to the boundary of the disk. The two models are related through a projection from the center of the disk; a ray from the center passing through a point of one model line passes through the corresponding point of the line in the other model.
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| ==See also==
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| *[[Poincaré half-plane model]]
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| *[[Poincaré disk model]]
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| *[[Poincaré metric]]
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| *[[Inversive geometry]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Luis Santaló (1961), ''[[Luis Santalo#Geometrias no Euclidianas (1961)|Geometrias no Euclidianas]], EUDEBA.
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| * {{cite book|first=Saul|last=Stahl|title=The Poincaré Half-Plane|publisher=Jones and Bartlett|year=1993}}
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| * {{cite article|first1=Frank|last1=Nielsen|first2=Richard|last2=Nock|title= Hyperbolic Voronoi diagrams made easy|publisher=arXiv:0903.3287 |year=2009|url=http://www.computer.org/portal/web/csdl/doi/10.1109/ICCSA.2010.37}}
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| {{DEFAULTSORT:Beltrami-Klein model}}
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| [[Category:Hyperbolic geometry]]
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Hello friend. Allow me introduce myself. I am Luther Aubrey. Managing individuals is what I do in my day occupation. The favorite hobby for my kids and me is taking part in crochet and now I'm trying to earn money with it. Some time ago he selected to live in Kansas.
My web site :: extended auto warranty (dig this)