Editing Similarity (geometry) (section)

==In Euclidean space==
A '''similarity''' (also called a '''similarity transformation''' or '''similitude''') of a [[Euclidean space]] is a [[bijection]] {{mvar|f}} from the space onto itself that multiplies all distances by the same positive [[real number]] {{mvar|r}}, so that for any two points {{mvar|x}} and {{mvar|y}} we have

:<math>d(f(x),f(y)) = r\, d(x,y), </math>

where {{math|''d''(''x'',''y'')}} is the [[Euclidean distance]] from {{mvar|x}} to {{mvar|y}}.{{sfn|Smart|1998|p=92}} The [[scalar (mathematics)|scalar]] {{mvar|r}} has many names in the literature including; the ''ratio of similarity'', the ''stretching factor'' and the ''similarity coefficient''. When {{math|1=''r'' = 1}} a similarity is called an [[Euclidean plane isometry|isometry]] ([[rigid transformation]]). Two sets are called '''similar''' if one is the image of the other under a similarity.

As a map {{tmath|f : \R^n \to \R^n,}} a similarity of ratio {{mvar|r}} takes the form

:<math>f(x) = rAx + t,</math>

where {{tmath|A \in O^n(\R)}} is an {{math|''n'' × ''n''}} [[orthogonal matrix]] and {{tmath|t \in \R^n}} is a translation vector.

Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments.{{sfn|Yale|1968|p=47 Theorem 2.1}} Similarities preserve angles but do not necessarily preserve orientation, ''direct similitudes'' preserve orientation and ''opposite similitudes'' change it.{{sfn|Pedoe|1988|pp=179–181}}

The similarities of Euclidean space form a [[group (mathematics)|group]] under the operation of composition called the ''similarities group'' {{mvar|S}}.{{sfn|Yale|1968|p=46}} The direct similitudes form a [[normal subgroup]] of {{mvar|S}} and the [[Euclidean group]] {{math|''E''(''n'')}} of isometries also forms a normal subgroup.{{sfn|Pedoe|1988|p=182}} The similarities group {{mvar|S}} is itself a subgroup of the [[affine group]], so every similarity is an [[affine transformation]].
<!-- FOR LATER INCLUSION
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 orthogonal transformation. -->

One can view the Euclidean plane as the [[complex plane]],{{efn|This traditional term, as explained in its article, is a misnomer. This is actually the 1-dimensional complex line.}} that is, as a 2-dimensional space over the [[real number|reals]]. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by 
*<math>f(z) = az + b</math> (direct similitudes), and 
*<math>f(z) = a\overline z + b</math> (opposite similitudes), 
where {{mvar|a}} and {{mvar|b}} are complex numbers, {{math|''a'' ≠ 0}}. When {{math|1={{abs|''a''}}= 1}}, these similarities are isometries.