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An affine transformation of the plane under which each point is shifted towards the x-axis, parallel to the y-axis, by a distance proportional to its ordinate. In a Cartesian coordinate system a contraction is given by the relations

$$x'=x,\quad y'=ky,\quad k>0.$$

A contraction of space towards the $xy$-plane, parallel to the $z$-axis, is given by the relations

$$x'=x,\quad y'=y,\quad z'=kz,\quad k>0.$$


More usually, a contraction is defined as a transformation of a metric space that reduces distances. The notion defined above has no established name in Western literature, but is sometimes called a compression or compression-expansion.

How to Cite This Entry:
Contraction(2). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.V. Reveryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article