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)
“The true gardener then brushes over the ground with slow and gentle hand, to liberate a space for breath round some favourite; but he is not thinking about destruction except incidentally. It is only the amateur like myself who becomes obsessed and rejoices with a sadistic pleasure in weeds that are big and bad enough to pull, and at last, almost forgetting the flowers altogether, turns into a Reformer.”
—Freya Stark (1893–1993)