Squeeze Mapping - Literature

Literature

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 x y = k) is the same over as over 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.

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 of fractions in these terms in the second section. The third section is concerned with shear mapping as area-preserving.

In 1965 Rafael Artzy listed the squeeze mapping as a generator of planar affine mappings in his book Linear Geometry (p 94).

The myth of Procrustes is linked with this mapping in a 1967 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.

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.

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.

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".

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. However, in his 1986 book Applications of Lie Groups to Differential Equations (p. 127) he uses the term "hyperbolic rotation" for an equivalent mapping.

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 et and the transformation group of squeeze mappings is used to illustrate the invariance of a differential equation under the group operation.