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[[Image:Geom podobnost stejnolehlest.svg|thumb|right|334px|Two similar geometric figures related by a homothetic transformation with respect to a [[homothetic center]] '''S'''. The angles at corresponding points are the same and have the same sense; for example, the angles ABC and A'B'C' are both clockwise and equal in magnitude.]]
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In [[mathematics]], a '''homothety''' (or '''homothecy''', or '''homogeneous dilation''') is a [[Transformation (mathematics)|transformation]] of an [[affine space]] determined by a point ''S'' called its ''center'' and a nonzero number ''λ'' called its ''ratio'', which sends
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:<math> M \mapsto S + \lambda \overrightarrow{SM}, </math>
 
in other words it fixes ''S'', and sends any ''M'' to another point ''N'' such that the segment ''SN'' is on the same line as ''SM'', but scaled by a factor ''λ''.<ref>{{harvtxt|Hadamard|p=145}}</ref> In [[Euclidean geometry]] homotheties are the [[Similarity (geometry)|similarities]] that fix a point and either preserve (if {{nowrap|''λ'' &gt; 0}}) or reverse (if {{nowrap|''λ'' &lt; 0}}) 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 ''L'' is a line [[parallel (geometry)|parallel]] to ''L''.
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In [[projective geometry]], a homothetic transformation is a similarity transformation that leaves the line at infinity pointwise [[invariant (mathematics)|invariant]].<ref>{{harvtxt|Tuller|p=119}}</ref>
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In Euclidean geometry, a homothety of ratio ''λ'' multiplies distances between points by |''λ''| and all areas by ''λ''<sup>2</sup>. The first number is called 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''
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==Homothety and uniform scaling==
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If the [[homothetic center]] ''S'' happens to coincide with the [[Origin (mathematics)|origin]] ''O'' of the vector space (''S'' ≡ ''O''), then every homothety with scale factor ''λ'' is equivalent to a [[uniform scaling]] by the same factor, which sends
 
 
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As a consequence, in the specific case in which ''S'' ≡ ''O'', the homothety becomes a [[linear transformation]], which preserves not only the collinearity of points (straight lines are mapped to straight lines), but also vector addition and scalar multiplication.
 
 
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==See also==
 
*[[Scaling (geometry)]] a similar notion in vector spaces
</ul>
*[[Homothetic center]], the center of a homothetic transformation taking one of a pair of shapes into the other
*The [[Hadwiger conjecture (combinatorial geometry)|Hadwiger conjecture]] on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it
*[[Homothetic function (economics)]], a function of the form ''U''(''f''(''y'')) in which ''f'' is a [[homogeneous function]] and ''U'' is a [[monotonic function|monotonically increasing function]].
 
== Notes ==
<references/>
 
==References==
* {{ citation | first1 = J. | last1 = Hadamard | title = Lessons in Plane Geometry
}}.
* {{ citation | first1 = Bruce E. | last1 = Meserve | year =1955 | title = Fundamental Concepts of Geometry | chapter = Homothetic transformations | pages = 166-169 | publisher = [[Addison-Wesley]] }}.
* {{ citation | last1 = Tuller | first1 =  Annita | title = A Modern Introduction to Geometries }}.
 
==External links==
* [http://www.cut-the-knot.org/Curriculum/Geometry/Homothety.shtml Homothety], interactive applet from [[Cut-the-Knot]].
 
[[Category:Transformation (function)]]

Latest revision as of 02:58, 9 November 2014

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