Editing Homothety (section)

{{Short description|Generalized scaling operation in geometry}}
[[File:Zentr-streck-T-e.svg|thumb|upright=1|Homothety: Example with {{math|''k'' > 0}}. {{math|1=''k'' = 1}} corresponds to {{em|identity}} (no point is moved); {{math|''k'' > 1}} an {{em|enlargement}}; {{math|''k'' < 1}} a {{em|reduction}}]]
[[File:Zentr-streck-T-nk-e.svg|thumb|upright=1|Example with {{math|''k'' < 0}}. {{math|1= k = −1}} corresponds to a [[point reflection]] at point {{mvar|S}}]]
[[File:Zentr-streck-pyram-e.svg|thumb|upright=1.2|Homothety of a pyramid]]
In [[mathematics]], a '''homothety''' (or '''homothecy''', or '''homogeneous dilation''') is a [[Transformation (mathematics)|transformation]] of an [[affine space]] determined by a point {{mvar|S}} called its ''center'' and a nonzero number {{mvar|k}} called its ''ratio'', which sends point {{mvar|X}} to a point {{mvar|{{prime|X}}}} by the rule,{{sfnp|Hadamard|1906|p=[https://archive.org/details/leonsdegomtriel04hadagoog/page/n155/ 134]}}
: <math>\overrightarrow{SX'}=k\overrightarrow{SX}</math> for a fixed number <math>k\ne 0</math>.
Using position vectors:
:<math>\mathbf x'=\mathbf s + k(\mathbf x -\mathbf s)</math>.
In case of <math>S=O</math> (Origin):
:<math>\mathbf x'=k\mathbf x</math>,
which is a [[uniform scaling]] and shows the meaning of special choices  for <math>k</math>:
:for <math>k=1</math> one gets the ''identity'' mapping,
:for <math>k=-1</math> one gets the ''reflection'' at the center,
For <math>1/k</math> one gets the ''inverse'' mapping defined by <math>k</math>.

In [[Euclidean geometry]] homotheties are the [[Similarity (geometry)|similarities]] that fix a point and either preserve (if <math>k>0</math>) or reverse (if <math>k<0</math>) the direction of all vectors. Together with the [[Translation (geometry)|translations]], all homotheties of an affine (or Euclidean) space form a [[group (mathematics)|group]], the group of '''dilations''' or '''homothety-translations'''. These are precisely the [[affine transformation]]s with the property that the image of every line ''g'' is a line [[parallel (geometry)|parallel]] to ''g''.

In [[projective geometry]], a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise [[invariant (mathematics)|invariant]].{{sfnp|Tuller|1967|p=119}}

In Euclidean geometry, a homothety of ratio <math>k</math> multiplies ''distances'' between points by <math>|k|</math>, ''areas'' by <math>k^2</math> and volumes by <math>|k|^3</math>.  Here <math>k</math> is the ''ratio of magnification'' or ''dilation factor'' or ''scale factor'' or ''similitude ratio''. Such a transformation can be called an '''enlargement''' if the scale factor exceeds&nbsp;1. The above-mentioned fixed point ''S'' is called ''[[homothetic center]]'' or ''center of similarity'' or ''center of similitude''.

The term, coined by French mathematician [[Michel Chasles]], is derived from two [[Greek language|Greek]] elements: the prefix {{Transliteration|grc|homo-}} ({{lang|grc|όμο}} {{gloss|similar}}}; and {{Translation|grc|thesis}} ({{lang|grc|Θέσις}}) {{gloss|position}}). It describes the relationship between two figures of the same shape and orientation. For example, two [[Matryoshka doll|Russian dolls]] looking in the same direction can be considered homothetic.

Homotheties are used to scale the contents of computer screens; for example, smartphones, notebooks, and laptops.