View source for Homothety

{{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.

== Properties ==
The following properties hold in any dimension.

=== Mapping lines, line segments and angles ===
A homothety has the following properties:
* A ''line'' is mapped onto a parallel line. Hence: ''angles'' remain unchanged.
* The ''ratio of two line segments'' is preserved.
Both properties show: 
* A homothety is a ''[[Similarity (geometry)|similarity]]''.

''Derivation of the properties:''
In order to make calculations easy it is assumed that the center <math>S</math>  is the origin: <math>\mathbf x \to k\mathbf x</math>. A line  <math>g</math> with parametric representation <math>\mathbf x=\mathbf p +t\mathbf v</math> is mapped onto the point set <math>g'</math> with equation <math>\mathbf x=k(\mathbf p+t\mathbf v)= k\mathbf p+tk\mathbf v</math>, which is a line parallel to <math>g</math>.

The distance of two points <math>P:\mathbf p,\;Q:\mathbf q</math> is <math>|\mathbf p -\mathbf q|</math> and <math>|k\mathbf p -k\mathbf q|=|k||\mathbf p-\mathbf q|</math> the distance between their images. Hence, the ''ratio'' (quotient) of two line segments remains unchanged.

In case of <math>S\ne O</math> the calculation is analogous but a little extensive.

Consequences: A triangle is mapped on a [[Similarity (geometry)|similar]] one. The homothetic image of a [[circle]] is a circle. The image of an [[ellipse]] is a similar one. i.e. the ratio of the two axes is unchanged.

[[File:Zentr-streck-T-S-e.svg|thumb|upright=0.8|With [[intercept theorem]]]]

=== Graphical constructions ===

==== using the intercept theorem ====
If for a homothety with center <math>S</math> the image <math>Q_1</math> of a point <math>P_1</math> is given (see diagram) then the image <math>Q_2</math> of a second point <math>P_2</math>, which lies not on line <math>SP_1</math>  can be constructed graphically using the intercept theorem: <math>Q_2</math> is the common point th two lines <math>\overline{P_1P_2}</math> and <math>\overline{SP_2}</math>. The image of a point collinear with <math>P_1,Q_1</math> can be determined using <math>P_2,Q_2</math>.

[[File:Pantograph animation.gif|thumb|Pantograph]]
[[File:Pantograf-konstr-e.svg|thumb|upright=1.2|Geometrical background]]
[[File:Pantograph01.jpg|thumb|Pantograph 3d rendering]]

==== using a pantograph ====
Before computers became ubiquitous, scalings of drawings were done by using a [[pantograph]], a tool similar to a [[Compass (drawing tool)|compass]].

''Construction and geometrical background:''

#Take 4 rods and assemble a mobile ''parallelogram'' with vertices <math>P_0,Q_0,H,P</math> such that the two rods meeting at <math>Q_0</math> are prolonged at the other end as shown in the diagram. Choose the ''ratio'' <math>k</math>.
#On the prolonged rods mark the two points <math>S,Q</math> such that <math>|SQ_0|=k|SP_0|</math> and <math>|QQ_0|=k|HQ_0|</math>. This is the case if <math>|SQ_0|=\tfrac{k}{k-1}|P_0Q_0|.</math> (Instead of <math>k</math> the location of the center <math>S</math> can be prescribed. In this case the ratio is <math>k=|SQ_0|/|SP_0|</math>.) 
#Attach the mobile rods rotatable at point <math>S</math>.
#Vary the location of point <math>P</math> and mark at each time point <math>Q</math>.

Because of <math>|SQ_0|/|SP_0|=|Q_0Q|/|PP_0|</math> (see diagram) one gets from the ''intercept theorem'' that the points <math>S,P,Q</math> are collinear (lie on a line) and equation <math>|SQ|=k|SP|</math> holds. That shows: the mapping  <math>P\to Q</math> is a homothety with center <math>S</math> and ratio <math>k</math>.

=== Composition ===
[[File:Zentr-streck-TT-e.svg|thumb|upright=1|The composition of two homotheties with centers {{math|''S''{{sub|1}}, ''S''{{sub|2}}}} and ratios {{math|1=''k''{{sub|1}}, ''k''{{sub|2}} = 0.3}} mapping {{math|''P{{sub|i}}'' &rarrow;  ''Q{{sub|i}}'' &rarrow; ''R{{sub|i}}''}} is a homothety again with its center {{math|''S''{{sub|3}}}} on line {{math|{{overline|''S''{{sub|1}}{{thinsp}}''S''{{sub|2}}}}}} with ratio {{math|1=''{{thinsp|k|⋅|l}}'' = 0.6}}.]]
*The composition of two homotheties with the ''same center'' <math>S</math> is again a homothety with center <math>S</math>. The homotheties with center <math>S</math> form a [[Group (mathematics)|group]].
*The composition of two homotheties with ''different centers'' <math>S_1,S_2</math> and its ratios <math>k_1,k_2</math> is 
::in case of <math>k_1k_2\ne 1</math> a ''homothety'' with its center on line <math>\overline{S_1S_2}</math> and ratio <math>k_1k_2</math> or 
::in case of <math>k_1k_2= 1</math> a ''[[Translation (geometry)|translation]]'' in direction <math>\overrightarrow{S_1S_2}</math>. Especially, if <math>k_1=k_2=-1</math> ([[point reflection]]s).

''Derivation:''

For the composition <math>\sigma_2\sigma_1</math> of the two homotheties <math>\sigma_1,\sigma_2</math> with centers <math>S_1,S_2</math> with
:<math>\sigma_1: \mathbf x \to \mathbf s_1+k_1(\mathbf x -\mathbf s_1), </math>
:<math>\sigma_2: \mathbf x \to \mathbf s_2+k_2(\mathbf x -\mathbf s_2)\ </math> 
one gets by calculation for the image of point <math>X:\mathbf x</math>:
:<math>(\sigma_2\sigma_1)(\mathbf x)= \mathbf s_2+k_2\big(\mathbf s_1+k_1(\mathbf x-\mathbf s_1)-\mathbf s_2\big) </math>
:<math>\qquad \qquad \ =(1-k_1)k_2\mathbf s_1+(1-k_2)\mathbf s_2 + k_1k_2\mathbf x</math>.
Hence, the composition is 
:in case of <math>k_1k_2= 1</math> a translation in direction <math>\overrightarrow{S_1S_2}</math> by vector <math>\ (1-k_2)(\mathbf s_2-\mathbf s_1)</math>.
:in case of <math>k_1k_2\ne 1</math> point 
:<math>S_3: \mathbf s_3=\frac{(1-k_1)k_2\mathbf s_1+(1-k_2)\mathbf s_2}{1-k_1k_2}
=\mathbf s_1+\frac{1-k_2}{1-k_1k_2}(\mathbf s_2-\mathbf s_1) </math> 
is a ''fixpoint'' (is not moved) and  the composition
:<math>\sigma_2\sigma_1: \ \mathbf x \to \mathbf s_3 + k_1k_2(\mathbf x -\mathbf s_3)\quad </math>. 
is a ''homothety'' with center <math>S_3</math> and ratio <math>k_1k_2</math>. <math>S_3</math> lies on line <math>\overline{S_1S_2}</math>.
[[File:Zentr-streck-T-st-e.svg|thumb|upright=1|Composition with a translation]]
*The composition of a homothety and a translation is a homothety.

''Derivation:''

The composition of the homothety 
:<math>\sigma: \mathbf x \to \mathbf s +k(\mathbf x-\mathbf s),\; k\ne 1,\;</math> and the translation
:<math>\tau: \mathbf x \to \mathbf x +\mathbf v </math> is
:<math>\tau\sigma: \mathbf x \to \mathbf s +\mathbf v +k(\mathbf x-\mathbf s)</math>
:::<math>=\mathbf s +\frac{\mathbf v}{1-k}+k\left(\mathbf x-(\mathbf s+\frac{\mathbf v}{1-k})\right)</math>
which is a homothety with center <math>\mathbf s'=\mathbf s +\frac{\mathbf v}{1-k}</math> and ratio <math>k</math>.

=== In homogeneous coordinates ===
The homothety <math>\sigma: \mathbf x \to \mathbf s+k(\mathbf x -\mathbf s)</math> 
with center <math>S=(u,v)</math> can be written as the composition of a homothety with center <math>O</math> and a translation:
:<math>\mathbf x \to k\mathbf x + (1-k)\mathbf s</math>.
Hence <math>\sigma</math> can be represented  in [[homogeneous coordinates]] 
by the matrix:
:<math>\begin{pmatrix}
k & 0 & (1-k)u\\
0 & k & (1-k)v\\
0 & 0 & 1
\end{pmatrix} </math>

A pure homothety [[linear transformation]] is also [[conformal linear transformation|conformal]] because it is composed of translation and uniform scale.

==See also==
*[[Scaling (geometry)]] a similar notion in vector spaces
*[[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 {{math|''f''(''U''(''y''))}} in which {{mvar|U}} is a [[homogeneous function]] and {{mvar|f}} is a [[monotonic function|monotonically increasing function]].
*[[Menelaus%27s theorem#A proof using homotheties]]

== Notes ==
<references/>

==References==
* Coxeter, H. S. M. (1961), "Introduction to geometry", Wiley, p.&nbsp;94
* {{citation |last= Hadamard |first= J. |date= 1906 |language= fr |chapter= V: Homothétie et Similtude |trans-chapter= V: Homothety and Similarity |title= Leçons de Géométrie élémentaire. I: Géométrie plane |trans-title= Lessons in Elementary Geometry. I: Plane Geometry |edition= 2nd |publisher= Armand Colin |location= Paris |author-link= Jacques Hadamard |url= https://archive.org/details/leonsdegomtriel04hadagoog/ }}
* {{citation |last= Meserve |first= Bruce E. |date= 1955 |title= Fundamental Concepts of Geometry |chapter= Homothetic transformations |pages= 166–169 |publisher= [[Addison-Wesley]] }}
* {{citation |last= Tuller |first= Annita |date= 1967 |author-link= Annita Tuller |title= A Modern Introduction to Geometries |location= Princeton, New Jersey |publisher= D. Van Nostrand Co. |series= University Series in Undergraduate Mathematics }}

==External links==
* [http://www.cut-the-knot.org/Curriculum/Geometry/Homothety.shtml Homothety], interactive applet from [[Cut-the-Knot]].

[[Category:Transformation (function)]]