132.64.42.23: /* 4/3 problem */

2014-08-26T21:30:19Z

4/3 problem

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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.]]~~		Greetings. The writer's name is Phebe and she feels comfy when people use the complete title. I utilized to be unemployed but now I am a librarian and the salary has been really satisfying. Minnesota is where he's been residing for years. Body building is one of the things I adore most.<br><br>my blog post at home std testing; [http://nxnn.info/user/G8498 simply click the up coming post],
	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]]~~.

	~~== Minkowski quadratic form ==~~
	~~{{main\|Minkowski space}}~~
	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

	~~:<math> Q(x_0, x_1, \ldots, x_n) = x_0^2 - x_1^2 - \ldots - x_n^2.</math>~~

	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>.

	~~The '''Minkowski [[bilinear form]]''' ''B'' is the [[polarization identity\|polarization]] of the Minkowski quadratic form ''Q'',~~

	~~:<math>B(u, v) = (Q(u+v)-Q(u)-Q(v))/2.</math>~~

	~~Explicitly,~~
	~~:<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>.~~

	~~The '''hyperbolic distance''' between two points ''u'' and ''v'' of ''S''<sup>+</sup> is given by the formula~~

	~~:<math>d(u, v) = \cosh^{-1}(B(u, v)).</math>~~

	~~== Isometries ==~~

	~~The [[indefinite orthogonal group]] O(1,''n''), also called the~~
	~~(''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~~
	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

	~~:<math>\begin{pmatrix}~~
	~~1 & 0 & \ldots & 0 \\~~
	~~0 & & & \\~~
	~~\vdots & & A & \\~~
	~~0 & & & \\~~
	~~\end{pmatrix}</math>~~

	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,

	~~: <math> \mathbb{H}^n=\mathrm{SO}^{+}(1,n)/\mathrm{SO}(n).</math>~~

	~~The group SO<sup>+</sup>(1,''n'') is the full group of orientation-preserving isometries of the ''n''-dimensional hyperbolic space.~~

	~~==History==~~
	~~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:~~
	~~Given an [[inner product]] <math> \langle \cdot, \cdot \rangle </math> on R<sup>n</sup>,~~
	~~the Weierstrass coordinates of ''x'' ∈ R<sup>n</sup> are:~~
	~~:<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>~~
	for ~~the hemispherical model. (See Elena Deza and [[Michel Deza]] (2006) ''Dictionary of Distances''~~.)

	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>

	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.~~

	~~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~~
	:<~~math~~>~~\cosh A + \alpha \ \sinh A~~ <~~/math~~>
	~~where &alpha~~; ~~is a basis vector orthogonal to the hyperboloid axis. For example, he obtained the [[hyperbolic law of cosines]] through use of his~~
	~~[[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>~~

	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>~~
	~~In 1993 W.F. Reynolds recounted some of the early history of the model in his article in the [[American Mathematical Monthly]].~~

	~~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.

	~~==See also==~~

	* [[Poincaré disk model]]
	* [[Hyperbolic quaternion]]s

	~~==Notes and references==~~
	~~<references/>~~
	* {{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}}
	* {{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}}
	* {{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
	* [[Miles Reid]] & Balázs Szendröi (2005) ''Geometry and Topology'', Figure 3.10, p 45, [[Cambridge University Press]], ISBN 0-521-61325-6, {{MathSciNet\|id=2194744}}.
	* Reynolds, William F. (1993) "Hyperbolic geometry on a hyperboloid", [[American Mathematical Monthly]] 100:442–55.
	* {{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 }}
	*{{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}}
	*{{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)

	~~[[Category:Multi-dimensional geometry]]~~
	~~[[Category:Hyperbolic geometry]]~~

en>Quondum: fixing a few hyphens/nen-dashes

2014-01-13T04:55:19Z

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132.64.42.23: /* 4/3 problem */

en>Quondum: fixing a few hyphens/nen-dashes

en>Quondum: /* Mass–energy equivalence */ Removing undue capitalization