Squeeze mapping

Inlinear algebra,asqueeze mapping,also called asqueeze transformation,is a type oflinear mapthat preserves Euclideanareaof regions in theCartesian plane,but isnotarotationorshear mapping.

a= 3/2 squeeze mapping

For a fixed positive real numbera,the mapping

$(x,y)\mapsto (ax,y/a)$

is thesqueeze mappingwith parametera.Since

$\{(u,v)\,:\,uv=\mathrm {constant} \}$

is ahyperbola,ifu=axandv=y/a,thenuv=xyand 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 ahyperbolic rotation,as didÉmile Borelin 1914,by analogy withcircular 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 theareabounded by a hyperbola (such asxy= 1)is one ofquadrature.The solution, found byGrégoire de Saint-VincentandAlphonse Antonio de Sarasain 1647, required thenatural logarithmfunction, a new concept. Some insight into logarithms comes throughhyperbolic sectorsthat are permuted by squeeze mappings while preserving their area. The area of a hyperbolic sector is taken as a measure of ahyperbolic angleassociated with the sector. The hyperbolic angle concept is quite independent of theordinary 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 generateinvariant measuresbut with respect to different transformation groups. Thehyperbolic functions,which take hyperbolic angle as argument, perform the role thatcircular functionsplay with the circular angle argument.

Group theory

A squeeze mapping moves one purplehyperbolic sectorto another with the same area.
It also squeezes blue and greenrectangles.

In 1688, long before abstractgroup theory,the squeeze mapping was described byEuclid Speidellin 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."

Ifrandsare positive real numbers, thecompositionof their squeeze mappings is the squeeze mapping of their product. Therefore, the collection of squeeze mappings forms aone-parameter groupisomorphic to themultiplicative groupofpositive real numbers.An additive view of this group arises from consideration of hyperbolic sectors and their hyperbolic angles.

From the point of view of theclassical groups,the group of squeeze mappings isSO⁺(1,1),theidentity componentof theindefinite orthogonal groupof 2×2 real matrices preserving thequadratic formu²−v².This is equivalent to preserving the formxyvia thechange of basis

$x=u+v,\quad y=u-v\,,$

and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the groupSO(2)(the connected component of the definiteorthogonal group) preserving quadratic formx²+y²as beingcircular rotations.

Note that the "SO⁺"notation corresponds to the fact that the reflections

$u\mapsto -u,\quad v\mapsto -v$

are not allowed, though they preserve the form (in terms ofxandythese arex↦y,y↦xandx↦ −x,y↦ −y);the additional "+"in the hyperbolic case (as compared with the circular case) is necessary to specify the identity component because the groupO(1,1)has4connected components,while the groupO(2)has2components:SO(1,1)has2components, whileSO(2)only has 1. The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroupsSO ⊂ SL– in this caseSO(1,1) ⊂SL(2)– of the subgroup of hyperbolic rotations in thespecial linear groupof transforms preserving area and orientation (avolume form). In the language ofMöbius transformations,the squeeze transformations are thehyperbolic elementsin theclassification of elements.

Ageometric transformationis calledconformalwhen it preserves angles.Hyperbolic angleis defined using area undery= 1/x.Since squeeze mappings preserve areas of transformed regions such ashyperbolic sectors,the angle measure of sectors is preserved. Thus squeeze mappings areconformalin the sense of preserving hyperbolic angle.

Applications

Here some applications are summarized with historic references.

Relativistic spacetime

Euclideanorthogonalityis preserved by rotation in the left diagram;hyperbolic orthogonalitywith 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 aLorentz boost.This insight follows from a study ofsplit-complex numbermultiplications and thediagonal basiswhich corresponds to the pair of light lines. Formally, a squeeze preserves the hyperbolic metric expressed in the formxy;in a different coordinate system. This application in thetheory of relativitywas noted in 1912 by Wilson and Lewis,by Werner Greub,and byLouis Kauffman.Furthermore, the squeeze mapping form of Lorentz transformations was used byGustav Herglotz(1909/10)while discussingBorn rigidity,and was popularized byWolfgang Rindlerin his textbook on relativity, who used it in his demonstration of their characteristic property.

The termsqueeze transformationwas used in this context in an article connecting theLorentz groupwithJones calculusin optics.

Corner flow

Influid dynamicsone of the fundamental motions of anincompressible flowinvolvesbifurcationof a flow running up against an immovable wall. Representing the wall by the axisy= 0 and taking the parameterr= exp(t) wheretis time, then the squeeze mapping with parameterrapplied to an initial fluid state produces a flow with bifurcation left and right of the axisx= 0. The samemodelgivesfluid convergencewhen time is run backward. Indeed, theareaof anyhyperbolic sectorisinvariantunder squeezing.

For another approach to a flow with hyperbolicstreamlines,seePotential flow § Power laws with n = 2.

In 1989 Ottinodescribed the "linear isochoric two-dimensional flow" as

$v_{1}=Gx_{2}\quad v_{2}=KGx_{1}$

where K lies in the interval [−1, 1]. The streamlines follow the curves

$x_{2}^{2}-Kx_{1}^{2}=\mathrm {constant}$

so negativeKcorresponds to anellipseand positiveKto a hyperbola, with the rectangular case of the squeeze mapping corresponding toK= 1.

Stocker and Hosoidescribed 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 ofπ/2 and delimited on the left and bottom by symmetry planes.

Stocker and Hosoi then recall Moffatt'sconsideration 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 thetranscendental functionsnatural logarithmand its inverse theexponential function:

Definition:Sector(a,b) is thehyperbolic sectorobtained with central rays to (a,1/a) and (b,1/b).

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

Proof: Take parameterr=c/aso 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-Vincent1647) Ifbc=ad,then the quadrature of the hyperbolaxy= 1 against the asymptote has equal areas betweenaandbcompared to betweencandd.

Proof: An argument adding and subtracting triangles of area1⁄2,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 Sarasa1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asymptote increase in geometric sequence. Thus the areas formlogarithmsof the asymptote index.

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 thetranscendental numberx =e.

A squeeze withr= e moves the unit angle to one between (e,1/e) and (ee,1/ee) which subtends a sector also of area one. Thegeometric progression

e,e²,e³,...,eⁿ,...

corresponds to the asymptotic index achieved with each sum of areas

1,2,3,...,n,...

which is a proto-typicalarithmetic progressionA+ndwhereA= 0 andd= 1.

Lie transform

Further information:History of Lorentz transformations § Lorentz transformation via squeeze mappings

FollowingPierre Ossian Bonnet's (1867) investigations on surfaces of constant curvatures,Sophus Lie(1879) found a way to derive newpseudospherical surfacesfrom a known one. Such surfaces satisfy theSine-Gordon equation:

${\frac {d^{2}\Theta }{ds\ d\sigma }}=K\sin \Theta,$

where$(s,\sigma )$are asymptotic coordinates of two principal tangent curves and$\Theta$their respective angle. Lie showed that if$\Theta =f(s,\sigma )$is a solution to the Sine-Gordon equation, then the following squeeze mapping (now known as Lie transform) indicates other solutions of that equation:

$\Theta =f\left(ms,\ {\frac {\sigma }{m}}\right).$

Lie (1883) noticed its relation to two other transformations of pseudospherical surfaces:TheBäcklund transform(introduced byAlbert Victor Bäcklundin 1883) can be seen as the combination of a Lie transform with a Bianchi transform (introduced byLuigi Bianchiin 1879.) Such transformations of pseudospherical surfaces were discussed in detail in the lectures ondifferential geometrybyGaston Darboux(1894),Luigi Bianchi(1894),orLuther Pfahler Eisenhart(1909).

It is known that the Lie transforms (or squeeze mappings) correspond to Lorentz boosts in terms oflight-cone coordinates,as pointed out by Terng and Uhlenbeck (2000):

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$(x,t)\mapsto \left({\tfrac {1}{\lambda }}x,\lambda t\right)$.

This can be represented as follows:

${\begin{matrix}-c^{2}t^{2}+x^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}\\\hline {\begin{aligned}ct'&=ct\gamma -x\beta \gamma &&=ct\cosh \eta -x\sinh \eta \\x'&=-ct\beta \gamma +x\gamma &&=-ct\sinh \eta +x\cosh \eta \end{aligned}}\\\hline u=ct+x,\ v=ct-x,\ k={\sqrt {\tfrac {1+\beta }{1-\beta }}}=e^{\eta }\\u'={\frac {u}{k}},\ v'=kv\\\hline u'v'=uv\end{matrix}}$

wherekcorresponds to the Doppler factor inBondik-calculus,ηis therapidity.

See also

• Indefinite orthogonal group
• Isochoric process

References

• 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.
• Walter, Scott (1999)."The non-Euclidean style of Minkowskian relativity"(PDF).In J. Gray (ed.).The Symbolic Universe: Geometry and Physics.Oxford University Press. pp. 91–127.(see page 9 of e-link)
• Learning materials related toReciprocal Eigenvaluesat Wikiversity