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) → (rx,y/r)

is the squeeze mapping with parameter r. Since

{(u,v):uv = constant}

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.

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.

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 physical applications:

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 evolution of the fluid shows 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 ; 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).

See 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.

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