Editing Similarity (geometry) (section)

==Similarity with a center==
{{multiple image
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 | image1 = 3 iterations of similarities applied to 3 regular polygons.svg
 | image2 = Academ Example of similarity with ratio square root of 2.svg
 | caption1 = Example where each similarity<br />[[function composition|composed]] with itself several times successively<br />has a '''center''' at the center of a [[regular polygon]] that it shrinks.
 | caption2 = {{anchor|caption2OfSection}}Example of direct similarity of center {{mvar|S}}<br />[[function composition|decomposed]] into a rotation of 135° angle<br />and a homothety that halves [[area]]s.
 | footer = <span style="font-size:120%">Examples of direct similarities that have each a '''center'''.</span>
 | footer_align = center
}}
{{Clear}}
If a similarity has exactly one [[invariant (mathematics)|invariant point]]: a point that the similarity keeps unchanged, then this only point is called "'''center'''" of the similarity.

On the first image below the title, on the left, one or another similarity shrinks a [[regular polygon]] into a [[concentric objects|concentric one]], the vertices of which are each on a side of the previous polygon. This rotational reduction [[iterated function|is repeated]], so the initial polygon is extended into an [[mise en abyme|abyss]] of regular polygons. The '''center''' of the similarity is the common center of the successive polygons. A red [[line segment|segment]] joins a vertex of the initial polygon to its [[Image (mathematics)#Image of an element|image]] under the similarity, followed by a red segment going to the following image of vertex, and so on to form a [[spiral]]. Actually we can see more than three direct similarities on this first image, because every regular polygon is invariant under certain direct similarities, more precisely certain rotations the center of which is the center of the polygon, and a composition of direct similarities is also a direct similarity. For example we see the image of the initial regular [[pentagon]] under a [[homothety]] of negative ratio {{mvar|−k}}, which is a similarity of ±180° angle and a positive ratio equal to {{mvar|k}}.

Below the title on the right, the second image shows a similarity [[function composition|decomposed]] into a [[rotation (mathematics)|rotation]] and a homothety. Similarity and rotation have the same angle of +135 degrees [[modular arithmetic|modulo 360 degrees]]. Similarity and homothety have the same ratio of {{tmath|\tfrac{\sqrt 2}{2},}} [[multiplicative inverse]] of the ratio {{tmath|\sqrt 2}} ([[square root of 2]]) of the [[inverse function|inverse]] similarity. Point {{mvar|S}} is the common '''center''' of the three transformations: rotation, homothety and similarity. For example point {{mvar|W}} is the image of {{mvar|F}} under the rotation, and point {{mvar|T}} is the image of {{mvar|W}} under the homothety, more briefly
<math display=block>T = H(W) = (R(F)) = (H \circ R)(F) = D(F),</math>
by naming {{mvar|R}}, {{mvar|H}} and {{mvar|D}} the previous rotation, homothety and similarity, with “{{mvar|D}}" like "Direct".

This direct similarity that transforms triangle {{math|△''EFA''}} into triangle {{math|△''ATB''}} can be decomposed into a rotation and a homothety of same center {{mvar|S}} in several manners. For example, {{math|1=''D'' = ''R'' ○ ''H'' = ''H'' ○ ''R''}}, the last decomposition being only represented on the image. To get {{mvar|D}} we can also compose in any order a rotation of −45° angle and a homothety of ratio {{tmath|\tfrac{- \sqrt 2}{2}.}}

With "{{mvar|M}}" like "Mirror" and "{{mvar|I}}" like "Indirect", if {{mvar|M}} is the [[reflection (mathematics)|reflection]] with respect to line {{mvar|CW}}, then {{math|1=''M'' ○ ''D'' = ''I''}} is the '''indirect''' similarity that transforms segment {{mvar|{{overline|BF}}}} like {{mvar|D}} into segment {{mvar|{{overline|CT}}}}, but transforms point  {{mvar|E}} into {{mvar|B}} and point  {{mvar|A}} into  {{mvar|A}} itself. Square {{mvar|ACBT}} is the image of {{mvar|ABEF}} under similarity {{mvar|I}} of ratio {{tmath|\tfrac{1}{\sqrt 2}.}} Point  {{mvar|A}} is the center of this similarity because any point {{mvar|K}} being invariant under it fulfills <math>AK = \tfrac{AK}{\sqrt 2},</math> only possible if {{math|1=''AK'' = 0}}, otherwise written {{math|1=''A'' = ''K''}}.

How to [[#caption2OfSection|construct the center {{mvar|S}}]] of direct similarity {{mvar|D}} from square {{mvar|ABEF}}, how to find point {{mvar|S}} center of a rotation of +135° angle that transforms ray {{tmath|\overset{}\overrightarrow{SE} }} into ray {{tmath|\overset{}\overrightarrow{SA} }}? This is an [[inscribed angle]] problem plus a question of [[orientation (vector space)|orientation]]. The set of points {{mvar|P}} such that <math>\overset{}{ \overrightarrow{PE}, \overrightarrow{PA} = +135^\circ }</math> is an arc of circle {{mvar|{{overarc|EA}}}} that joins  {{mvar|E}} and  {{mvar|A}}, of which the two radius leading to  {{mvar|E}} and  {{mvar|A}} form a [[central angle]] of {{math|1=2(180° − 135°) = 2 × 45° = 90°}}.  This set of points is the blue quarter of circle of center {{mvar|F}} inside square {{mvar|ABEF}}. In the same manner, point {{mvar|S}} [[set (mathematics)|is a member]] of the blue quarter of circle of center {{mvar|T}} inside square {{mvar|BCAT}}. So point {{mvar|S}} is the [[intersection (Euclidean geometry)|intersection]] point of these two quarters of circles.