Editing Squeeze mapping (section)

{{Short description|Linear mapping permuting rectangles of the same area}}
[[Image:Squeeze r=1.5.svg|thumb|right|''a'' = 3/2 squeeze mapping]]
In [[linear algebra]], a '''squeeze mapping''', also called a '''squeeze transformation''', is a type of [[linear map]] that preserves Euclidean [[area]] of regions in the [[Cartesian plane]], but is ''not'' a [[rotation (mathematics)|rotation]] or [[shear mapping]].

For a fixed positive real number {{math|''a''}}, the mapping

:<math>(x, y) \mapsto (ax, y/a)</math>

is the ''squeeze mapping'' with parameter {{math|''a''}}. Since

:<math>\{ (u,v) \, : \, u v = \mathrm{constant}\}</math>

is a [[hyperbola]], if {{math|''u'' {{=}} ''ax''}} and {{math|''v'' {{=}} ''y''/''a''}}, then {{math|''uv'' {{=}} ''xy''}} and the points of the image of the squeeze mapping are on the same hyperbola as {{math|(''x'',''y'')}} is. For this reason it is natural to think of the squeeze mapping as a '''hyperbolic rotation''', as did [[Émile Borel]] in 1914,<ref>[[Émile Borel]] (1914) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=04710001 Introduction Geometrique à quelques Théories Physiques], page 29, Gauthier-Villars, link from [[Cornell University]] Historical Math Monographs</ref> by analogy with ''circular rotations'', which preserve circles.