Squeeze mapping

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.

For a fixed positive real number "r", the mapping

:("x","y") → ("r x", "y" / "r" )

is the "squeeze mapping" with parameter "r". Since

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

is a hyperbola, if "u" = "r x" 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.

Group theory

If "r" and "s" are positive real numbers, the 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 hyperbolic angles.

Literature

The myth of Procrustes is linked with this mapping in an educational (SMSG) publication::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).::Coxeter & Greitzer, pp. 100, 101.In his 1999 monograph "Classical Invariant Theory", Peter Olver discusses GL(2,R) and calls the group of squeeze mappings by the name the isobaric subgroup.

Applications

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 real matrices (2 x 2). These applications are somewhat bland compared to two physical and a philosophical application:

Fluid flow

In fluid dynamics one of the fundamental motions of an incompressible flow involves 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 model gives fluid convergence when time is run backward. Indeed, the area of any hyperbolic sector is invariant under squeezing.

For another approach to this flow with hyperbolic streamlines, see the article potential flow, section "Power law with n = 2".

Relativistic spacetime

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.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), and in 1985 by Louis Kauffman.

Bridge to transcendentals

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:

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 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.

Theorem (Alphonse Antonio de Sarasa 1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asyptote increase in geometric sequence. Thus the areas form "logarithms" of the asymptote index.

For instance, on may ask “When is the hyperbolic angle in standard position equal to one?” The standard position angle runs from (1,1) to ("x", 1/"x"). The answer is “When "x" = E (mathematical constant)" which is a transcendental number. 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 sequence: 1,e, e², e³, … eⁿ, …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 .

ee also

*Equi-areal mapping
*Isochoric process

References

* HSM Coxeter & SL Greitzer (1967) "Geometry Revisited", Chapter 4 Transformations, A genealogy of transformation.
* 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. See [http://ca.geocities.com/cocklebio/synsptm.html Synthetic Spacetime] for an excerpt.
* Louis Kauffman (1985) "Transformations in Special Relativity", International Journal of Theoretical Physics 24:223-36.