Similarity in Euclidean Space
One of the meanings of the terms similarity and similarity transformation (also called dilation) of a Euclidean space is a function f from the space into itself that multiplies all distances by the same positive scalar r, so that for any two points x and y we have
where "d(x,y)" is the Euclidean distance from x to y. Two sets are called similar if one is the image of the other under such a similarity.
A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image. A similarity is a composition of a homothety and an isometry. Therefore, in general Euclidean spaces every similarity is an affine transformation, because the Euclidean group E(n) is a subgroup of the affine group.
Viewing the complex plane as a 2-dimensional space over the reals, the 2D similarity transformations expressed in terms of complex arithmetic are and where a and b are complex numbers, a ≠ 0.
Read more about this topic: Similarity (geometry)
Famous quotes containing the words similarity and/or space:
“Incompatibility. In matrimony a similarity of tastes, particularly the taste for domination.”
—Ambrose Bierce (1842–1914)
“In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
—Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)