Editing Squeeze mapping

{{Short description|Linear mapping permuting rectangles of the same area}}
[[Image:Squeeze r=1.5.svg|thumb|right|''a'' = 3/2 squeeze mapping]]
In [[linear algebra]], a '''squeeze mapping''', also called a '''squeeze transformation''', is a type of [[linear map]] that preserves Euclidean [[area]] of regions in the [[Cartesian plane]], but is ''not'' a [[rotation (mathematics)|rotation]] or [[shear mapping]].

For a fixed positive real number {{math|''a''}}, the mapping

:<math>(x, y) \mapsto (ax, y/a)</math>

is the ''squeeze mapping'' with parameter {{math|''a''}}. Since

:<math>\{ (u,v) \, : \, u v = \mathrm{constant}\}</math>

is a [[hyperbola]], if {{math|''u'' {{=}} ''ax''}} and {{math|''v'' {{=}} ''y''/''a''}}, then {{math|''uv'' {{=}} ''xy''}} and the points of the image of the squeeze mapping are on the same hyperbola as {{math|(''x'',''y'')}} is. For this reason it is natural to think of the squeeze mapping as a '''hyperbolic rotation''', as did [[Émile Borel]] in 1914,<ref>[[Émile Borel]] (1914) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=04710001 Introduction Geometrique à quelques Théories Physiques], page 29, Gauthier-Villars, link from [[Cornell University]] Historical Math Monographs</ref> by analogy with ''circular rotations'', which preserve circles.

==Logarithm and hyperbolic angle==
The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the [[area]] bounded by a hyperbola (such as {{math|''xy'' {{=}} 1)}} is one of [[quadrature (mathematics)|quadrature]]. The solution, found by [[Grégoire de Saint-Vincent]] and [[Alphonse Antonio de Sarasa]] in 1647, required the [[natural logarithm]] function, a new concept. Some insight into logarithms comes through [[hyperbolic sector]]s that are permuted by squeeze mappings while preserving their area. The area of a hyperbolic sector is taken as a measure of a [[hyperbolic angle]] associated with the sector. The hyperbolic angle concept is quite independent of the [[angle|ordinary circular angle]], but shares a property of invariance with it: whereas circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping. Both circular and hyperbolic angle generate [[invariant measure]]s but with respect to different transformation groups. The [[hyperbolic function]]s, which take hyperbolic angle as argument, perform the role that [[circular functions]] play with the circular angle argument.<ref>[[Mellen W. Haskell]] (1895) [http://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf On the introduction of the notion of hyperbolic functions] [[Bulletin of the American Mathematical Society]] 1(6):155–9,particularly equation 12, page 159</ref>

==Group theory==
[[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.]]
In 1688, long before abstract [[group theory]], the squeeze mapping was described by [[Euclid Speidell]] in the terms of the day: "From a Square and an infinite company of Oblongs on a Superficies, each Equal to that square, how a curve is begotten which shall have the same properties or  affections of any Hyperbola inscribed within a Right Angled Cone."<ref>Euclid Speidell (1688) [https://books.google.com/books?id=9l6zSrUQL0UC&q=logarithmotechnia Logarithmotechnia: the making of numbers called logarithms] from [[Google Books]]
</ref>

If {{math|''r''}} and {{math|''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 sectors and their hyperbolic angles.

From the point of view of the [[classical group]]s, the group of squeeze mappings is {{math|SO<sup>+</sup>(1,1)}}, the [[identity component]] of the [[indefinite orthogonal group]] of 2×2 real matrices preserving the [[quadratic form]] {{math|''u''<sup>2</sup> − ''v''<sup>2</sup>}}. This is equivalent to preserving the form {{math|''xy''}} via the [[change of basis]]

:<math>x=u+v,\quad y=u-v\,,</math>

and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group {{math|SO(2)}} (the connected component of the definite [[orthogonal group]]) preserving quadratic form {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} as being ''circular rotations''.

Note that the "{{math|SO<sup>+</sup>}}" notation corresponds to the fact that the reflections

:<math>u \mapsto -u,\quad v \mapsto -v</math>

are not allowed, though they preserve the form (in terms of {{math|''x''}} and {{math|''y''}} these are {{math|''x'' ↦ ''y'', ''y'' ↦ ''x''}} and {{math|''x'' ↦ −''x'', ''y'' ↦ −''y'')}}; the additional "{{math|+}}" in the hyperbolic case (as compared with the circular case) is necessary to specify the identity component because the group {{math|O(1,1)}} has {{math|4}} [[connected component (topology)|connected component]]s, while the group {{math|O(2)}} has {{math|2}} components: {{math|SO(1,1)}} has {{math|2}} components, while {{math|SO(2)}} only has 1. The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroups {{math|SO ⊂ SL}} &ndash; in this case {{math|SO(1,1)&nbsp;⊂&nbsp;[[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 transformation]]s, the squeeze transformations are the [[SL2(R)#Hyperbolic elements|hyperbolic elements]] in the [[SL2(R)#Classification of elements|classification of elements]].

A [[geometric transformation]] is called '''conformal''' when it preserves  angles. [[Hyperbolic angle]] is defined using area under ''y'' = 1/''x''. Since squeeze mappings preserve areas of transformed regions such as [[hyperbolic sector]]s, the angle measure of sectors is preserved.  Thus squeeze mappings are ''conformal'' in the sense of preserving hyperbolic angle.

==Applications==
Here some applications are summarized with historic references.

===Relativistic spacetime===
[[File:Orthogonality and rotation.svg|thumb|350px|Euclidean [[orthogonality]] is preserved by rotation in the left diagram; [[hyperbolic orthogonality]] with respect to hyperbola (B) is preserved by squeeze mapping in the right diagram]]
Spacetime geometry is conventionally developed as follows: 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 [[split-complex number#The diagonal basis|diagonal basis]] which corresponds to the pair of light lines.
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,<ref>[[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&ndash;507, footnote p.&nbsp;401</ref> by Werner Greub,<ref>W. H. Greub (1967) ''Linear Algebra'', Springer-Verlag. See pages 272 to 274</ref> and by [[Louis Kauffman]].<ref>[[Louis Kauffman]] (1985) "Transformations in Special Relativity", [[International Journal of Theoretical Physics]] 24:223&ndash;36</ref> Furthermore, the squeeze mapping form of Lorentz transformations was used by [[Gustav Herglotz]] (1909/10)<ref>{{Citation|author=Herglotz, Gustav|year=1910|orig-year=1909|title=Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper|trans-title=Wikisource translation: [[s:Translation:On bodies that are to be designated as "rigid"|On bodies that are to be designated as "rigid" from the standpoint of the relativity principle]]|journal=Annalen der Physik|volume=336|issue=2 |pages=408|doi=10.1002/andp.19103360208|bibcode = 1910AnP...336..393H |url=https://zenodo.org/record/1424161}}</ref> while discussing [[Born rigidity]], and was popularized by [[Wolfgang Rindler]] in his textbook on relativity, who used it in his demonstration of their characteristic property.<ref>[[Wolfgang Rindler]], ''Essential Relativity'', 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</ref>

The term ''squeeze transformation'' was used in this context in an article connecting the [[Lorentz group]] with [[Jones calculus]] in optics.<ref>Daesoo Han, Young Suh Kim & Marilyn E. Noz (1997) "Jones-matrix formalism as a representation of the Lorentz group", [[Journal of the Optical Society of America]] A14(9):2290–8</ref>

===Corner flow===
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.
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.

For another approach to a flow with hyperbolic [[streamlines, streaklines and pathlines|streamlines]], see {{section link|Potential flow|Power laws with n {{=}} 2}}.

In 1989 Ottino<ref>J. M. Ottino (1989) ''The Kinematics of Mixing: stretching, chaos, transport'', page 29, [[Cambridge University Press]]</ref> described the "linear isochoric two-dimensional flow" as
:<math>v_1 = G x_2 \quad v_2 = K G x_1</math>
where K lies in the interval [&minus;1, 1]. The streamlines follow the curves
:<math>x_2^2 - K x_1^2 = \mathrm{constant}</math>
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.

Stocker and Hosoi<ref>Roman Stocker & [[Anette Hosoi|A.E. Hosoi]] (2004) "Corner flow in free liquid films", ''Journal of Engineering Mathematics'' 50:267&ndash;88</ref> described their approach to corner flow as follows:
: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.
Stocker and Hosoi then recall Moffatt's<ref>H.K. Moffatt (1964) "Viscous and resistive eddies near a sharp corner", [[Journal of Fluid Mechanics]] 18:1&ndash;18</ref> consideration of "flow in a corner between rigid boundaries, induced by an arbitrary disturbance at a large distance." According to Stocker and Hosoi,
: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.

===Bridge to transcendentals===
The area-preserving property of squeeze mapping has an application in setting the foundation of the [[transcendental function]]s [[natural logarithm]] and its inverse the [[exponential function]]:

'''Definition:''' Sector(''a,b'') is the [[hyperbolic sector]] obtained with central rays to (''a'', 1/''a'') and (''b'', 1/''b'').

'''Lemma:''' If ''bc'' = ''ad'', then there is a squeeze mapping that moves the sector(''a,b'') to sector(''c,d'').

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'').

'''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''.

Proof: An argument adding and subtracting triangles of area {{frac|1|2}}, 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.

'''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.

For instance, for a standard position angle which runs from (1,&nbsp;1) to (''x'',&nbsp;1/''x''), one may ask "When is the hyperbolic angle equal to one?" The answer is the [[transcendental number]] x = [[e (mathematical constant)|e]].

A squeeze with ''r'' = e moves the unit angle to one between (''e'',&nbsp;1/''e'') and (''ee'',&nbsp;1/''ee'') which subtends a sector also of area one. The [[geometric progression]]
: ''e'', ''e''<sup>2</sup>, ''e''<sup>3</sup>, ..., ''e''<sup>''n''</sup>, ...
corresponds to the asymptotic index achieved with each sum of areas
: 1,2,3, ..., ''n'',...
which is a proto-typical [[arithmetic progression]] ''A'' + ''nd'' where ''A'' = 0 and ''d'' = 1 .

===Lie transform===
{{Further|History of Lorentz transformations#Lorentz transformation via squeeze mappings}}

Following [[Pierre Ossian Bonnet]]'s (1867) investigations on surfaces of constant curvatures, [[Sophus Lie]] (1879) found a way to derive new [[pseudospherical surface]]s from a known one. Such surfaces satisfy the [[Sine-Gordon equation]]:

:<math>\frac{d^{2}\Theta}{ds\ d\sigma}=K\sin\Theta ,</math>

where <math>(s,\sigma)</math> are asymptotic coordinates of two principal tangent curves and <math>\Theta</math> their respective angle. Lie showed that if <math>\Theta=f(s,\sigma)</math> is a solution to the Sine-Gordon equation, then the following squeeze mapping (now known as Lie transform<ref name=terng />) indicates other solutions of that equation:<ref>{{Cite journal|author=Lie, S.|year=1881|orig-year=1879|journal=Fortschritte der Mathematik|volume=11|title=Selbstanzeige: Über Flächen, deren Krümmungsradien durch eine Relation verknüpft sind|pages=529–531}} Reprinted in [https://archive.org/details/gesammabhand03lierich Lie's collected papers, Vol. 3, pp. 392–393].</ref>

:<math>\Theta=f\left(ms,\ \frac{\sigma}{m}\right) .</math>

Lie (1883) noticed its relation to two other transformations of pseudospherical surfaces:<ref>{{Cite journal|author=Lie, S.|year=1884|orig-year=1883|journal=Christ. Forh.|title=Untersuchungen über Differentialgleichungen IV}}. Reprinted in [https://archive.org/details/gesammabhand03lierich Lie's collected papers, Vol. 3, pp. 556–560].</ref> The [[Bäcklund transform]] (introduced by [[Albert Victor Bäcklund]] in 1883) can be seen as the combination of a Lie transform with a Bianchi transform (introduced by [[Luigi Bianchi]] in 1879.) Such transformations of pseudospherical surfaces were discussed in detail in the lectures on [[differential geometry]] by [[Gaston Darboux]] (1894),<ref>{{Cite book|author=Darboux, G.|year=1894|title=Leçons sur la théorie générale des surfaces. Troisième partie|publisher=Gauthier-Villars|location=Paris|url=https://archive.org/details/leonssurlathorie03darb|pages=[https://archive.org/details/leonssurlathorie03darb/page/381 381]–382}}</ref> [[Luigi Bianchi]] (1894),<ref>{{Cite book|author=Bianchi, L.|year=1894|title=Lezioni di geometria differenziale|publisher=Enrico Spoerri|location=Pisa|url=https://archive.org/details/lezionidigeomet00biangoog|pages=[https://archive.org/details/lezionidigeomet00biangoog/page/n443 433]–434}}</ref> or [[Luther Pfahler Eisenhart]] (1909).<ref>{{Cite book|author=Eisenhart, L. P.|year=1909|title=A treatise on the differential geometry of curves and surfaces|publisher=Ginn and Company|location=Boston|url=https://archive.org/details/treatonthediffer00eiserich|pages=[https://archive.org/details/treatonthediffer00eiserich/page/n306 289]–290}}</ref>

It is known that the Lie transforms (or squeeze mappings) correspond to Lorentz boosts in terms of [[light-cone coordinates]], as pointed out by Terng and Uhlenbeck (2000):<ref name=terng>{{Cite journal|author=Terng, C. L., & Uhlenbeck, K.|year=2000|journal=Notices of the AMS|volume=47|issue=1|title=Geometry of solitons|pages=17–25|url=https://www.ams.org/journals/notices/200001/fea-terng.pdf}}</ref>

:''Sophus Lie observed that the SGE [Sinus-Gordon equation] is invariant under Lorentz transformations. In asymptotic coordinates, which correspond to light cone coordinates, a Lorentz transformation is <math>(x,t)\mapsto\left(\tfrac{1}{\lambda}x,\lambda t\right)</math>.''

This can be represented as follows:

:<math>\begin{matrix}-c^{2}t^{2}+x^{2}=-c^{2}t^{\prime2}+x^{\prime2}\\
\hline \begin{align}ct' & =ct\gamma-x\beta\gamma &  & =ct\cosh\eta-x\sinh\eta\\
x' & =-ct\beta\gamma+x\gamma &  & =-ct\sinh\eta+x\cosh\eta
\end{align}
\\
\hline u=ct+x,\ v=ct-x,\ k=\sqrt{\tfrac{1+\beta}{1-\beta}}=e^{\eta}\\
u'=\frac{u}{k},\ v'=kv\\
\hline u'v'=uv
\end{matrix}</math>

where ''k'' corresponds to the Doppler factor in [[Bondi k-calculus|Bondi ''k''-calculus]], ''η'' is the [[rapidity]].

==See also==
{{commons category|Squeeze (geometry)}}
*[[Indefinite orthogonal group]]
*[[Isochoric process]]

==References==
{{Reflist}}
* [[HSM Coxeter]] & SL Greitzer (1967) ''Geometry Revisited'', Chapter 4 Transformations, A genealogy of transformation.
* P. S. Modenov and A. S. Parkhomenko (1965) ''Geometric Transformations'', volume one. See pages 104 to 106.
*{{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)
* {{Wikiversity inline|Reciprocal Eigenvalues}}

[[Category:Affine geometry]]
[[Category:Conformal mappings]]
[[Category:Linear algebra]]
[[Category:Articles containing proofs]]
[[Category:Minkowski spacetime]]