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 or shear mapping.

For a fixed positive real number a, the mapping

${\displaystyle (x,y)\mapsto (ax,y/a)}$

is the squeeze mapping with parameter a. Since

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

is a hyperbola, if u = ax and v = y/a, 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 1914, by analogy with circular rotations, which preserve circles.