Squeeze mapping
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In mathematics, a squeeze mapping in linear algebra 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.
Formally, a squeeze preserves the hyperbolic metric expressed in the form xy; in a different coordinate system, it is a Lorentz boost.
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.
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.
[edit] See also
[edit] References
HSM Coxeter & SL Greitzer (1967) Geometry Revisited, Chapter 4 Transformations, A genealogy of transformation.
