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| [[File:HyperboloidProjection.png|thumb|Red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid.]]
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| In [[geometry]], the '''hyperboloid model''', also known as the '''Minkowski model''' or the '''Lorentz model''' (after [[Hermann Minkowski]] and [[Hendrik Lorentz]]), is a model of ''n''-dimensional [[hyperbolic geometry]] in which points are represented by the points on the forward sheet ''S''<sup>+</sup> of a two-sheeted [[hyperboloid]] in (''n''+1)-dimensional [[Minkowski space]] and ''m''-planes are represented by the intersections of the (''m''+1)-planes in Minkowski space with ''S''<sup>+</sup>. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the ''n''-dimensional hyperbolic space is closely related to the [[Beltrami–Klein model]] and to the [[Poincaré disk model]] as they are projective models in the sense that the [[isometry group]] is a subgroup of the [[projective group]].
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| == Minkowski quadratic form ==
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| {{main|Minkowski space}}
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| If (''x''<sub>0</sub>, ''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>) is a vector in the (''n''+1)-dimensional coordinate space '''R'''<sup>''n''+1</sup>, the '''Minkowski [[quadratic form]]''' is defined to be
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| :<math> Q(x_0, x_1, \ldots, x_n) = x_0^2 - x_1^2 - \ldots - x_n^2.</math>
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| The vectors ''v''∈ '''R'''<sup>''n''+1</sup> such that ''Q''(''v'') = 1 form an ''n''-dimensional [[hyperboloid]] ''S'' consisting of two [[connected space|connected component]]s, or ''sheets'': the forward, or future, sheet ''S''<sup>+</sup>, where ''x''<sub>0</sub>>0 and the backward, or past, sheet ''S''<sup>−</sup>, where ''x''<sub>0</sub><0. The points of the ''n''-dimensional hyperboloid model are the points on the forward sheet ''S''<sup>+</sup>.
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| The '''Minkowski [[bilinear form]]''' ''B'' is the [[polarization identity|polarization]] of the Minkowski quadratic form ''Q'',
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| :<math>B(u, v) = (Q(u+v)-Q(u)-Q(v))/2.</math>
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| Explicitly,
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| :<math>B((x_0, x_1, \ldots, x_n), (y_0, y_1, \ldots, y_n)) = x_0y_0 - x_1 y_1 - \ldots - x_n y_n</math>.
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| The '''hyperbolic distance''' between two points ''u'' and ''v'' of ''S''<sup>+</sup> is given by the formula
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| :<math>d(u, v) = \cosh^{-1}(B(u, v)).</math>
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| == Isometries ==
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| The [[indefinite orthogonal group]] O(1,''n''), also called the
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| (''n''+1)-dimensional [[Lorentz group]], is the [[Lie group]] of [[real number|real]] (''n''+1)×(''n''+1) [[Matrix (mathematics)|matrices]] which preserve the Minkowski bilinear form. In a different language, it is
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| the group of linear [[isometry|isometries]] of the [[Minkowski space]]. In particular, this group preserves the hyperboloid ''S''. Recall that indefinite orthogonal groups have four connected components, corresponding to reversing or preserving the orientation on each subspace (here 1-dimensional and ''n''-dimensional), and form a [[Klein four-group]]. The subgroup of O(1,''n'') that preserves the sign of the first coordinate is the '''[[orthochronous Lorentz group]]''', denoted O<sup>+</sup>(1,''n''), and has two components, corresponding to preserving or reversing the spacial dimension. Its subgroup SO<sup>+</sup>(1,''n'') consisting of matrices with [[determinant]] one is a connected Lie group of dimension ''n''(''n''+1)/2 which acts on ''S''<sup>+</sup> by linear automorphisms and preserves the hyperbolic distance. This action is transitive and the stabilizer of the vector (1,0,…,0) consists of the matrices of the form
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| :<math>\begin{pmatrix}
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| 1 & 0 & \ldots & 0 \\
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| 0 & & & \\
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| \vdots & & A & \\
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| 0 & & & \\
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| \end{pmatrix}</math>
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| Where <math> A </math> belongs to the compact [[special orthogonal group]] SO(''n'') (generalizing the [[rotation group SO(3)]] for {{nowrap|1=''n'' = 3}}). It follows that the ''n''-dimensional [[hyperbolic space]] can be exhibited as the [[homogeneous space]] and a [[Riemannian symmetric space]] of rank 1,
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| : <math> \mathbb{H}^n=\mathrm{SO}^{+}(1,n)/\mathrm{SO}(n).</math>
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| The group SO<sup>+</sup>(1,''n'') is the full group of orientation-preserving isometries of the ''n''-dimensional hyperbolic space.
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| ==History==
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| In 1880 [[Wilhelm Killing]] published "Die Rechnung in Nicht-Euclidischen Raumformen" in [[Crelle's Journal]] (89:265–87). This work discusses the hyperboloid model in a way that shows the analogy to the hemisphere model. Killing attributes the idea to [[Karl Weierstrass]] in a Berlin seminar some years before. Following on Killing’s attribution, the phrase '''Weierstrass coordinates''' has been associated with elements of the hyperboloid model as follows:
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| Given an [[inner product]] <math> \langle \cdot, \cdot \rangle </math> on R<sup>n</sup>,
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| the Weierstrass coordinates of ''x'' ∈ R<sup>n</sup> are:
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| :<math> (x , \sqrt {1 + \langle x,x \rangle}) \in R^{n+1}</math> compared to <math> (x, \sqrt {1 - \langle x,x \rangle})\in R^{n+1}</math>
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| for the hemispherical model. (See Elena Deza and [[Michel Deza]] (2006) ''Dictionary of Distances''.) | |
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| According to Jeremy Gray (1986),<ref>''Linear differential equations and group theory from Riemann to Poincaré'' (pages 271,2)</ref> [[Henri Poincaré|Poincaré]] used the hyperboloid model in his personal notes in 1880. Gray shows where the hyperboloid model is implicit in later writing by Poincaré.<ref>See also Poincaré: ''On the fundamental hypotheses of geometry'' 1887 Collected works vol.11, 71-91 and referred to in the book of B.A. Rosenfeld ''A History of Non-Euclidean Geometry'' p.266 in English version (Springer 1988).</ref>
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| For his part, W. Killing continued to publish on the hyperboloid model, particularly in 1885 in his ''Analytic treatment of non-Euclidean spaceforms''.<ref>''Die nicht-Euclidischen Raumformen in analytischer Behandlung'' J. reine angew. math.(Crelle) vol. 26</ref> Further exposure of the model was given by [[Alfred Clebsch]] and [[Ferdinand Lindemann]] in 1891 in ''Vorlesungen uber Geometrie'', page 524.
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| The hyperboloid was explored as a [[metric space]] by [[Alexander Macfarlane]] in his ''Papers in Space Analysis'' (1894). He noted that points on the hyperboloid could be written
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| :<math>\cosh A + \alpha \ \sinh A </math>
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| where α is a basis vector orthogonal to the hyperboloid axis. For example, he obtained the [[hyperbolic law of cosines]] through use of his
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| [[hyperbolic quaternion|Algebra of Physics]].<ref>A. Macfarlane (1894) ''[http://www.archive.org/details/principlesalgeb01macfgoog Papers on Space Analysis]'', B. Westerman, New York, weblink from [[archive.org]]</ref>
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| H. Jansen made the hyperboloid model the explicit focus of his 1909 paper "Representation of hyperbolic geometry on a two sheeted hyperboloid".<ref>''Abbildung hyperbolische Geometrie auf ein zweischaliges Hyperboloid'' Mitt. Math. Gesellsch Hamburg 4:409–440.</ref>
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| In 1993 W.F. Reynolds recounted some of the early history of the model in his article in the [[American Mathematical Monthly]].
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| Being a commonplace model by the twentieth century, it was identified with the ''Geschwindigkeitsvectoren'' (velocity vectors) by [[Hermann Minkowski]] in his [[Minkowski space]] of 1908. Scott Walter, in his 1999 paper "The Non-Euclidean Style of Special Relativity" recalls Minkowski’s awareness, but traces the lineage of the model to [[Hermann Helmholtz]] rather than Weierstrass and Killing. In the early years of relativity the hyperboloid model was used by [[Vladimir Varićak]] to explain the physics of velocity. In his speech to the German mathematical union in 1912 he referred to Weierstrass coordinates.
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| ==See also==
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| * [[Poincaré disk model]]
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| * [[Hyperbolic quaternion]]s
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| ==Notes and references==
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| <references/>
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| * {{Citation | last1=Alekseevskij | first1=D.V. | last2=Vinberg | first2=E.B. | authorlink2=Ernest Vinberg|last3=Solodovnikov | first3=A.S. | title=Geometry of Spaces of Constant Curvature | publisher=Springer-Verlag | location=Berlin, New York | series=Encyclopaedia of Mathematical Sciences | isbn=3-540-52000-7 | year=1993}}
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| * {{Citation | last1=Anderson | first1=James | title=Hyperbolic Geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Springer Undergraduate Mathematics Series | isbn=978-1-85233-934-0 | year=2005}}
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| * {{Citation | last1=Ratcliffe | first1=John G. | title=Foundations of hyperbolic manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94348-0 | year=1994}}, Chapter 3
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| * [[Miles Reid]] & Balázs Szendröi (2005) ''Geometry and Topology'', Figure 3.10, p 45, [[Cambridge University Press]], ISBN 0-521-61325-6, {{MathSciNet|id=2194744}}.
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| * Reynolds, William F. (1993) "Hyperbolic geometry on a hyperboloid", [[American Mathematical Monthly]] 100:442–55.
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| * {{Citation | last1=Ryan | first1=Patrick J. | title=Euclidean and non-Euclidean geometry: An analytical approach | publisher=[[Cambridge University Press]] | location=Cambridge, London, New York, New Rochelle, Melbourne, Sydney | isbn=0-521-25654-2 | year=1986 }}
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| *{{Citation | author=Varićak, V. | year=1912| title=[[s:On the Non-Euclidean Interpretation of the Theory of Relativity|On the Non-Euclidean Interpretation of the Theory of Relativity]]| journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume =21| pages =103–127}}
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| *{{Citation|author=Walter, Scott|year=1999|contribution=The non-Euclidean style of Minkowskian relativity|editor=J. Gray|title=The Symbolic Universe: Geometry and Physics|pages=91–127|publisher=Oxford University Press|contribution-url=http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf}}(see page 17 of e-link)
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| [[Category:Multi-dimensional geometry]]
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| [[Category:Hyperbolic geometry]]
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