Editing Homothety (section)

== 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.