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| In [[geometry]], a '''motion''' is an [[isometry]] of a [[metric space]]. For instance, a [[plane (geometry)|plane]] with [[Euclidean distance]] as [[metric (mathematics)|metric]] is a metric space in which a mapping associating [[congruence (geometry)|congruent]] figures is a motion.<ref>Gunter Ewald (1971) ''Geometry: An Introduction'', p. 179, Belmont: Wadsworth ISBN0-534-0034-7</ref> More generally, the term ''motion'' is a synonym for [[surjective]] isometry in metric geometry,<ref>M.A. Khamsi & W.A. Kirk (2001) ''An Introduction to Metric Spaces and Fixed Point Theorems'', p. 15, [[John Wiley & Sons]] ISBN 0-471-41825-0</ref> including [[elliptic geometry]] and [[hyperbolic geometry]]. In the latter case, [[hyperbolic motion]]s provide an approach to the subject for beginners.
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| In [[differential geometry]], a [[diffeomorphism]] is called a motion if it induces an isometry between the tangent space at a [[manifold]] point and the [[tangent space]] at the image of that point.<ref>A.Z. Petrov (1969) ''Einstein Spaces'', p. 60, [[Pergamon Press]]</ref><ref>B.A. Dubrovin, A.T. Fomenko, S.P Novikov (1992) ''Modern Geometry – Methods and Applications'', second edition, p 24, Springer, ISBN 0-387-97993-9</ref>
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| Given a geometry, the set of motions forms a [[group (mathematics)|group]] under composition of mappings. This '''group of motions''' is noted for its properties. When the underlying space is a [[Riemannian manifold]], the group of motions is a [[Lie group]]. Furthermore, the manifold has [[constant curvature]] if and only if, for every pair of points and every isometry, there is a motion taking one point to the other for which the motion induces the isometry.<ref>D.V. Alekseevskij, E.B. Vinberg, A.S. Solodonikov (1993) ''Geometry II'', p. 9, Springer, ISBN 0-387-52000-7</ref>
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| The idea of a group of motions for [[special relativity]] has been advanced as Lorentzian motions. For example, fundamental ideas have been laid out for a plane characterized by the [[quadratic form]] <math>x^2 - y^2 \ </math> in [[American Mathematical Monthly]].<ref>Graciela S. Birman & [[Katsumi Nomizu]] (1984) "Trigonometry in Lorentzian geometry", [[American Mathematical Monthly]] 91(9):543–9, group of motions: p 545</ref>
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| ==History==
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| An early appreciation of the role of motion in geometry was given by [[Alhazen]] (965 to 1039). His work "Space and its Nature"<ref>''Ibn Al_Haitham: Proceedings of the Celebrations of the 1000th Anniversary'', Hakim Mohammed Said editor, pages 224-7, Hamdard National Foundation, Karachi: The Times Press</ref> uses comparisons of the dimensions of a mobile body to quantify the vacuum of imaginary space.
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| In the 19th century [[Felix Klein]] became a proponent of [[group theory]] as a means to classify geometries according to their "groups of motions". He proposed using [[symmetry group]]s in his [[Erlangen program]], a suggestion that was widely adopted. He noted that every Euclidean congruence is an [[affine mapping]], and each of these is a [[projective transformation]]; therefore the group of projectivities contains the group of affine maps, which in turn contains the group of Euclidean congruencies. The term ''motion'', shorter than ''transformation'', puts more emphasis on the adjectives: projective, affine, Euclidean. The context was thus expanded, so much that "In [[topology]], the allowed movements are continuous invertible deformations that might be called elastic motions."<ref>Ari Ben-Menahem (2009) ''Historical Encyclopedia of the Natural and Mathematical Sciences'', v. I, p. 1789</ref>
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| The science of [[kinematics]] is dedicated to rendering [[motion (physics)|physical motion]] into expression as mathematical transformation. Frequently the transformation can be written using vector algebra and linear mapping. A simple example is a [[turn (geometry)#Kinematics of turns|turn]] written as a [[complex number]] multiplication: <math>z \mapsto \omega z \ </math> where <math>\ \omega = \cos \theta + i \sin \theta, \quad i^2 = -1</math>. Rotation in [[space]] is achieved by [[quaternions and spatial rotation|use of quaternions]], and [[Lorentz transformation]]s of [[spacetime]] by use of [[biquaternion]]s. Early in the 20th century, [[hypercomplex number]] systems were examined. Later their [[automorphism group]]s led to exceptional groups such as [[G2 (mathematics)|G2]].
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| In the 1890s logicians were reducing the [[primitive notion]]s of [[synthetic geometry]] to an absolute minimum. [[Giuseppe Peano]] and [[Mario Pieri]] used the expression ''motion'' for the congruence of point pairs. [[Alessandro Padoa]] celebrated the reduction of primitive notions to merely ''point'' and ''motion'' in his report to the 1900 [[International Congress of Philosophy]]. It was at this congress that [[Bertrand Russell]] was exposed to continental logic through Peano. In his book [[Principles of Mathematics]] (1903), Russell considered a motion to be a Euclidean isometry that preserves [[orientation (vector space)|orientation]].<ref>B. Russell (1903) [[Principles of Mathematics]] p 418. See also pp 406, 436</ref>
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| ==Notes and references==
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| {{reflist}}
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| * [[Tristan Needham]] (1997) ''Visual Complex Analysis'', Euclidean motion p 34, direct motion p 36, opposite motion p 36, spherical motion p 279, hyperbolic motion p 306, [[Clarendon Press]], ISBN 0-19-853447-7 .
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| * [[Miles Reid]] & Balázs Szendröi (2005) ''Geometry and Topology'', [[Cambridge University Press]], ISBN 0-521-61325-6, {{MathSciNet|id=2194744}}.
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| ==External links==
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| *[http://www.encyclopediaofmath.org/index.php?title=Motion&oldid=18181 Motion. I.P. Egorov (originator), ''Encyclopedia of Mathematics''.]
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| *[http://www.encyclopediaofmath.org/index.php?title=Group_of_motions&oldid=16960 Group of motions. I.P. Egorov (originator), ''Encyclopedia of Mathematics''.]
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| [[Category:Metric geometry]]
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| [[Category:Differential geometry]]
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| [[Category:Transformation (function)]]
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My name is Twila and I'm a 21 years old boy from France.
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