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| [[Image:Squeeze r=1.5.svg|thumb|right|''r'' = 3/2 squeeze mapping]]
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| In [[linear algebra]], a '''squeeze mapping''' is a type of [[linear map]] that preserves Euclidean [[area]] of regions in the [[Cartesian plane]], but is not a [[Euclidean motion]].
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| For a fixed positive real number ''r'', the mapping
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| :(''x'', ''y'') → (''rx'', ''y''/''r'' )
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| is the ''squeeze mapping'' with parameter ''r''. Since
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| :<math>\{ (u,v) \, : \, u v = \mathrm{constant}\}</math>
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| is a [[hyperbola]], if ''u'' = ''rx'' and ''v'' = ''y''/''r'', then ''uv'' = ''xy'' and the points of the image of the squeeze mapping are on the same hyperbola as (''x'',''y'') is. For this reason it is natural to think of the squeeze mapping as a '''hyperbolic rotation''', as did [[Émile Borel]] in 1913, by analogy with ''circular rotations'' which preserve circles.
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| ==Group theory==
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| If ''r'' and ''s'' are positive real numbers, the [[Function composition|composition]] of their squeeze mappings is the squeeze mapping of their product. Therefore the collection of squeeze mappings forms a [[one-parameter group]] isomorphic to the [[multiplicative group]] of positive real numbers. An additive view of this group arises from consideration of [[hyperbolic sector]]s and [[hyperbolic angle]]s. In fact, the [[invariant measure]] of this group is hyperbolic angle.
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| From the point of view of the [[classical group]]s, the group of squeeze mappings is SO<sup>+</sup>(1,1), the [[identity component]] of the [[indefinite orthogonal group]] of [[2 × 2 real matrices]] preserving the [[quadratic form]] ''u''<sup>2</sup> − ''v''<sup>2</sup>. This is equivalent to preserving the form ''xy'' via the [[change of basis]]
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| :<math>x=u+v,\quad y=u-v\,,</math>
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| and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group SO(2) (the connected component of the definite [[orthogonal group]]) preserving quadratic form ''x''<sup>2</sup> + ''y''<sup>2</sup>) as being ''circular rotations''.
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| Note that the "SO<sup>+</sup>" notation corresponds to the fact that the reflections
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| :<math>u \mapsto -u,\quad v \mapsto -v</math>
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| are not allowed, though they preserve the form (in terms of ''x'' and ''y'' these are ''x'' ↦ ''y'', ''y'' ↦ ''x'' and ''x'' ↦ −''x'', ''y'' ↦ −''y''); the additional "+" in the hyperbolic case (as compared with the circular case) is necessary to specify the identity component because the group O(1,1) has 4 [[connected component (topology)|connected component]]s, while the group O(2) has 2 components: SO(1,1) has 2 components, while SO(2) only has 1. The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroups SO ⊂ SL – in this case SO(1,1) ⊂ [[SL2(R)|SL(2)]] – of the subgroup of hyperbolic rotations in the [[special linear group]] of transforms preserving area and orientation (a [[volume form]]). In the language of [[Möbius transform]]s, the squeeze transformations are the [[SL2(R)#Hyperbolic elements|hyperbolic elements]] in the [[SL2(R)#Classification of elements|classification of elements]].
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| ==Literature==
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| :[[File:Hyperbolic sector squeeze mapping.svg|250px|right|thumb|A squeeze mapping moves one purple [[hyperbolic sector]] to another with the same area. <br>It also squeezes blue and green [[rectangle]]s.]]
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| An early recognition of squeeze symmetry was the 1647 discovery by [[Grégoire de Saint-Vincent]] that the area under a hyperbola (concretely, the curve given by ''xy'' = ''k'') is the same over [''a'', ''b''] as over [''c'', ''d''] when ''a''/''b'' = ''c''/''d'' . This preservation of areas under a hyperbola with hyperbolic rotation, was a key step in the development of the [[logarithm]]. Formalization of the squeeze group required the theory of groups, which was not developed until the 19th century.
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| [[William Kingdon Clifford]] was the author of ''Common Sense and the Exact Sciences'', published in 1885. In the third chapter on Quantity he discusses ''area'' in three sections. Clifford uses the term "stretch" for magnification and the term "squeeze" for contraction. Taking a given square area as fundamental, Clifford relates other areas by stretch and squeeze. He develops this calculus to the point of illustrating the [[addition#Rational numbers (fractions)|addition]] of [[fraction (mathematics)|fractions]] in these terms in the second section. The third section is concerned with [[shear mapping]] as area-preserving.
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| In 1965 [[Rafael Artzy]] listed the squeeze mapping as a generator of [[affine group#Planar affine group|planar affine mappings]] in his book ''Linear Geometry'' (p 94).
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| The myth of [[Procrustes]] is linked with this mapping in a 1967 educational ([[School Mathematics Study Group|SMSG]]) publication:
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| :Among the linear transformations, we have considered ''similarities'', which preserve ratios of distances, but have not touched upon the more bizarre varieties, such as the '''Procrustean stretch''' (which changes a circle into an ellipse of the same area).
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| ::Coxeter & Greitzer, pp. 100, 101.
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| Attention had been drawn to this plane mapping by Modenov and Parkhomenko in their Russian book of 1961 which was translated in 1967 by Michael B. P. Slater. It included a diagram showing the squeezing of a circle into an ellipse.
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| Werner Greub of the [[University of Toronto]] includes "pseudo-Euclidean rotation" in the chapter on symmetric bilinear functions of his text on linear algebra. This treatment in 1967 includes in short order both the diagonal form and the form with sinh and cosh.
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| The ''Mathematisch Centrum Amsterdam'' published E.R. Paërl's ''Representations of the Lorentz group and Projective Geometry'' in 1969. The squeeze mapping, written as a 2 × 2 diagonal matrix, Paërl calls a "hyperbolic screw".
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| In his 1999 monograph ''Classical Invariant Theory'', Peter Olver discusses GL(2,'''R''') and calls the group of squeeze mappings by the name [[isobaric process#Variable density viewpoint|the isobaric subgroup]]. However, in his 1986 book ''Applications of Lie Groups to Differential Equations'' (p. 127) he uses the term "hyperbolic rotation" for an equivalent mapping.
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| In 2004 the [[American Mathematical Society]] published ''Transformation Groups for Beginners'' by S.V. Duzhin and B.D. Chebotarevsky which mentions ''hyperbolic rotation'' on page 225. There the parameter ''r'' is given as ''e<sup>t</sup>'' and the transformation group of squeeze mappings is used to illustrate the invariance of a [[differential equation]] under the group operation.
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| ==Applications==
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| In studying linear algebra there are the purely abstract applications such as illustration of the [[singular-value decomposition]] or in the important role of the squeeze mapping in the structure of [[2 × 2 real matrices]]. These applications are somewhat bland compared to two [[physics|physical]] and a [[philosophy|philosophical]] application.
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| ===Corner flow===
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| In [[fluid dynamics]] one of the fundamental motions of an [[incompressible flow]] involves [[Bifurcation theory|bifurcation]] of a flow running up against an immovable wall.
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| Representing the wall by the axis ''y'' = 0 and taking the parameter ''r'' = exp(''t'') where ''t'' is time, then the squeeze mapping with parameter ''r'' applied to an initial fluid state produces a flow with bifurcation left and right of the axis ''x'' = 0. The same [[mathematical model|model]] gives '''fluid convergence''' when time is run backward. Indeed, the [[area]] of any [[hyperbolic sector]] is [[invariant (mathematics)|invariant]] under squeezing.
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| For another approach to a flow with hyperbolic [[streamlines, streaklines and pathlines|streamlines]], see the article [[potential flow#Power law with n = 2|potential flow]], section "Power law with ''n'' = 2".
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| In 1989 Ottino described the "linear isochoric two-dimensional flow" as
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| :<math>v_1 = G x_2 \quad v_2 = K G x_1</math>
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| where K lies in the interval [−1, 1]. The streamlines follow the curves
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| :<math>x_2^2 - K x_1^2 = \mathrm{constant}</math>
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| so negative ''K'' corresponds to an [[ellipse]] and positive ''K'' to a hyperbola, with the rectangular case of the squeeze mapping corresponding to ''K'' = 1.
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| Stocker and Hosoi (2004) announced their approach to corner flow as follows:
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| :we suggest an alternative formulation to account for the corner-like geometry, based on the use of hyperbolic coordinates, which allows substantial analytical progress towards determination of the flow in a Plateau border and attached liquid threads. We consider a region of flow forming an angle of ''π''/2 and delimited on the left and bottom by symmetry planes.
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| Stocker and Hosoi then recall H.K. Moffatt's 1964 paper "Viscous and resistive eddies near a sharp corner" ([[Journal of Fluid Mechanics]] 18:1–18). Moffatt considers "flow in a corner between rigid boundaries, induced by an arbitrary disturbance at a large distance." According to Stocker and Hosoi,
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| :For a free fluid in a square corner, Moffatt's (antisymmetric) stream function ... [indicates] that hyperbolic coordinates are indeed the natural choice to describe these flows.
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| ===Relativistic spacetime===
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| Select (0,0) for a "here and now" in a spacetime. Light radiant left and right through this central event tracks two lines in the spacetime, lines that can be used to give coordinates to events away from (0,0). Trajectories of lesser velocity track closer to the original timeline (0,''t''). Any such velocity can be viewed as a zero velocity under a squeeze mapping called a [[Lorentz boost]]. This insight follows from a study of [[split-complex number]] multiplications and the "diagonal basis" which corresponds to the pair of light lines.
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| Formally, a squeeze preserves the hyperbolic metric expressed in the form ''xy''; in a different coordinate system. This application in the [[theory of relativity]] was noted in 1912 by Wilson and Lewis (see footnote p. 401 of reference), by Werner Greub in the 1960s, and in 1985 by [[Louis Kauffman]]. Furthermore, [[Wolfgang Rindler]], in his popular textbook on relativity, used the squeeze mapping form of Lorentz transformations in his demonstration of their characteristic property (see equation 29.5 on page 45 of the 1969 edition, or equation 2.17 on page 37 of the 1977 edition, or equation 2.16 on page 52 of the 2001 edition).
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| ===Bridge to transcendentals===
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| The area-preserving property of squeeze mapping has an application in setting the foundation of the transcendental functions [[natural logarithm]] and its inverse the [[exponential function]]:
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| '''Definition:''' Sector(''a,b'') is the [[hyperbolic sector]] obtained with central rays to (''a'', 1/''a'') and (''b'', 1/''b'').
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| '''Lemma:''' If ''bc'' = ''ad'', then there is a squeeze mapping that moves the sector(''a,b'') to sector(''c,d'').
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| Proof: Take parameter ''r'' = ''c''/''a'' so that (''u,v'') = (''rx'', ''y''/''r'') takes (''a'', 1/''a'') to (''c'', 1/''c'') and (''b'', 1/''b'') to (''d'', 1/''d'').
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| '''Theorem''' ([[Gregoire de Saint-Vincent]] 1647) If ''bc'' = ''ad'', then the quadrature of the hyperbola ''xy'' = 1 against the asymptote has equal areas between ''a'' and ''b'' compared to between ''c'' and ''d''.
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| Proof: An argument adding and subtracting triangles of area ½, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area along the asymptote. The theorem then follows from the lemma.
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| '''Theorem''' ([[Alphonse Antonio de Sarasa]] 1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asymptote increase in geometric sequence. Thus the areas form ''logarithms'' of the asymptote index.
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| For instance, for a standard position angle which runs from (1, 1) to (''x'', 1/''x''), one may ask "When is the hyperbolic angle equal to one?" The answer is the [[transcendental number]] x = [[e (mathematical constant)|e]].
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| A squeeze with ''r'' = e moves the unit angle to one between (''e'', 1/''e'') and (''ee'', 1/''ee'') which subtends a sector also of area one. The [[geometric progression]]
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| : ''e'', ''e''<sup>2</sup>, ''e''<sup>3</sup>, ..., ''e''<sup>''n''</sup>, ...
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| corresponds to the asymptotic index achieved with each sum of areas
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| : 1,2,3, ..., ''n'',...
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| which is a proto-typical [[arithmetic progression]] ''A'' + ''nd'' where ''A'' = 0 and ''d'' = 1 .
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| ==See also==
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| *[[2 × 2 real matrices#Equi-areal mapping|Equi-areal mapping]]
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| *[[Indefinite orthogonal group]]
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| *[[Isochoric process]]
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| *[[Lorentz transformation]]
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| ==References==
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| * HSM Coxeter & SL Greitzer (1967) ''Geometry Revisited'', Chapter 4 Transformations, A genealogy of transformation.
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| * [[Edwin Bidwell Wilson]] & [[Gilbert N. Lewis]] (1912) "The space-time manifold of relativity. The non-Euclidean geometry of mechanics and electromagnetics", Proceedings of the [[American Academy of Arts and Sciences]] 48:387–507.
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| * W. H. Greub (1967) ''Linear Algebra'', Springer-Verlag. See pages 272 to 274.
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| * [[Louis Kauffman]] (1985) "Transformations in Special Relativity", [[International Journal of Theoretical Physics]] 24:223–36.
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| * P. S. Modenov and A. S. Parkhomenko (1965) ''Geometric Transformations'', volume one. See pages 104 to 106.
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| * J. M. Ottino (1989) ''The Kinematics of Mixing: stretching, chaos, transport'', page 29, [[Cambridge University Press]].
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| *{{Cite book|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 9 of e-link)
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| * Roman Stocker & A.E. Hosoi (2004) "Corner flow in free liquid films", ''Journal of Engineering Mathematics'' 50:267–88.
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| [[Category:Affine geometry]]
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| [[Category:Functions and mappings]]
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| [[Category:Linear algebra]]
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| [[Category:Articles containing proofs]]
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