View source for Homothety

In [[Mathematics|mathematics]], a '''homothety''' (or '''homothecy''', or '''homogeneous dilation''') is a transformation of an [[Affine space|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 display="block">\overrightarrow{SX'}=k\overrightarrow{SX}</math>
for a fixed number {{tmath|k\ne 0}}. Using position vectors:
<math display="block">\mathbf x'=\mathbf s + k(\mathbf x -\mathbf s).</math>

In case of <math>S=O</math> (Origin):
<math display="block">\mathbf x'=k\mathbf x,</math>
which is a uniform scaling and shows the meaning of special choices for {{tmath|k}}:
* 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 {{tmath|k}}.

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

In [[Projective geometry|projective geometry]], a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the [[Line at infinity|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 {{tmath|\vert k \vert}}, ''areas'' by <math>k^2</math> and volumes by {{tmath|{\vert k \vert}^3 }}.  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 {{tmath|S}} is called ''[[Homothetic center|homothetic center]]'' or ''center of similarity'' or ''center of similitude''.

The term, coined by French mathematician [[Biography:Michel Chasles|Michel Chasles]], is derived from two [[Greek language|Greek]] elements: the prefix {{translit|grc|homo-}} ({{lang|grc|όμο}} {{gloss|similar}}); and {{translit|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 that 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: {{tmath|\mathbf x \to k\mathbf x}}. 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 display="block">\mathbf x=k(\mathbf p+t\mathbf v)= k\mathbf p+tk\mathbf v,</math>
which is a line parallel to {{tmath|g}}.

The distance of two points <math>P:\mathbf p,\;Q:\mathbf q</math> is <math>|\mathbf p -\mathbf q|</math> and {{tmath|\vert k\mathbf p -k\mathbf q\vert=\vert k\vert \vert\mathbf p-\mathbf q\vert}}, 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|circle]] is a circle. The image of an [[Ellipse|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|intercept theorem]]]]

=== Graphical construction (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 {{tmath|P_2}}, which lies not on line <math>SP_1</math>  can be constructed graphically using the [[Intercept theorem|intercept theorem]]: <math>Q_2</math> is the common point of two lines <math>\overline{P_1P_2}</math> and {{tmath|\overline{SP_2} }}. The image of a point collinear with <math>P_1,Q_1</math> can be determined using {{tmath|P_2,Q_2}}.

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

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

#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'' {{tmath|k}}.
#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 {{tmath|S}}.
#Vary the location of point <math>P</math> and mark at each time point {{tmath|Q}}.

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 {{tmath|k}}.

=== 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}} = 2, ''k''{{sub|2}} = 0.3}} mapping {{math|''P{{sub|i}}'' &rarr;  ''Q{{sub|i}}'' &rarr; ''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}}.]]

==== Composition of two homotheties with the same center ====
The composition of two homotheties with the ''same center'' <math>S</math> is again a homothety with center {{tmath|S}}. The homotheties with center <math>S</math> form a [[Group (mathematics)|group]].

==== Composition of two homotheties with different centers ====
The composition of two homotheties with ''different centers'' <math>S_1,S_2</math> and its ratios <math>k_1,k_2</math> is:
* for <math>k_1k_2\ne 1</math>, a ''homothety'' with its center on line <math>\overline{S_1S_2}</math> and ratio {{tmath|k_1k_2}}; or 
* for <math>k_1k_2= 1</math>, a ''[[Translation (geometry)|translation]]'' in direction {{tmath|\overrightarrow{S_1S_2} }}. Especially, if <math>k_1=k_2=-1</math> ([[Point reflection|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 display="block">\begin{align}
\sigma_1: \mathbf x &\to \mathbf s_1+k_1(\mathbf x -\mathbf s_1), \\
\sigma_2: \mathbf x &\to \mathbf s_2+k_2(\mathbf x -\mathbf s_2)
\end{align}</math> 
one gets by calculation for the image of point <math>X:\mathbf x</math>:
<math display="block">\begin{align}
(\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) \\
   &= (1-k_1)k_2\mathbf s_1+(1-k_2)\mathbf s_2 + k_1k_2\mathbf x.
\end{align}</math>
Hence, the composition is
<ul>
<li>in case of <math>k_1k_2= 1</math>, a translation in direction <math>\overrightarrow{S_1S_2}</math> by vector {{tmath|1= (1-k_2)(\mathbf s_2-\mathbf s_1) }};</li>
<li>in case of <math>k_1k_2\ne 1</math> point 
<math display="block">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 display="block">\sigma_2\sigma_1: \ \mathbf x \to \mathbf s_3 + k_1k_2(\mathbf x -\mathbf s_3)</math>
is a ''homothety'' with center <math>S_3</math> and ratio <math>k_1k_2</math>. <math>S_3</math> lies on line {{tmath|\overline{S_1S_2} }}.</li>
</ul>
[[File:Zentr-streck-T-st-e.svg|thumb|upright=1|Composition with a translation]]

==== Composition of a homothety and a translation ====
The composition of a homothety and a translation is a homothety.

===== Derivation =====
The composition of the homothety 
<math display="block">\sigma: \mathbf x \to \mathbf s +k(\mathbf x-\mathbf s),\; k\ne 1,</math>
and the translation
<math display="block">\tau: \mathbf x \to \mathbf x +\mathbf v </math>
is
<math display="block">\begin{align}
\tau\sigma: \mathbf x &\to \mathbf s +\mathbf v +k(\mathbf x-\mathbf s) \\
    &=\mathbf s +\frac{\mathbf v}{1-k}+k\left(\mathbf x-\left(\mathbf s+\frac{\mathbf v}{1-k}\right)\right)
\end{align}</math>
which is a homothety with center <math display="inline">\mathbf s'=\mathbf s +\frac{\mathbf v}{1-k}</math> and ratio {{tmath|k}}.

=== 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 display="block">\mathbf x \to k\mathbf x + (1-k)\mathbf s.</math>
Hence <math>\sigma</math> can be represented  in [[Homogeneous coordinates|homogeneous coordinates]] 
by the matrix:
<math display="block">\begin{pmatrix}
k & 0 & (1-k)u\\
0 & k & (1-k)v\\
0 & 0 & 1
\end{pmatrix} </math>

A pure homothety [[Linear transformation|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|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 |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 |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)]]

{{Sourceattribution|Homothety}}