Homothety

The following properties hold in any dimension.

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.

Derivation of the properties:

In order to make calculations easy it is assumed that the center $S$ is the origin: $\backslash mathbf\; x\; \backslash to\; k\backslash mathbf\; x$. A line $g$ with parametric representation $\backslash mathbf\; x=\backslash mathbf\; p\; +t\backslash mathbf\; v$ is mapped onto the point set $g\text{'}$ with equation $\backslash mathbf\; x=k(\backslash mathbf\; p+t\backslash mathbf\; v)=\; k\backslash mathbf\; p+tk\backslash mathbf\; v$, which is a line parallel to $g$.

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

In case of $S\backslash ne\; O$ the calculation is analogous but a little extensive.

Consequences: A triangle is mapped on a 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]]

If for a homothety with center $S$ the image $Q\_1$ of a point $P\_1$ is given (see diagram) then the image $Q\_2$ of a second point $P\_2$, which lies not on line $SP\_1$ can be constructed graphically using the intercept theorem: $Q\_2$ is the common point th two lines $\backslash overline\{P\_1P\_2\}$ and $\backslash overline\{SP\_2\}$. The image of a point collinear with $P\_1,Q\_1$ can be determined using $P\_2,Q\_2$.

File:Pantograph animation.gif

File:Pantograf-konstr-e.svg

File:Pantograph01.jpg

Before computers became ubiquitous, scalings of drawings were done by using a pantograph, a tool similar to a compass.

Construction and geometrical background:

1. Take 4 rods and assemble a mobile parallelogram with vertices $P\_0,Q\_0,H,P$ such that the two rods meeting at $Q\_0$ are prolonged at the other end as shown in the diagram. Choose the ratio $k$.
2. On the prolonged rods mark the two points $S,Q$ such that $|SQ\_0|=k|SP\_0|$ and $|QQ\_0|=k|HQ\_0|$. This is the case if $|SQ\_0|=\backslash tfrac\{k\}\{k-1\}|P\_0Q\_0|.$ (Instead of $k$ the location of the center $S$ can be prescribed. In this case the ratio is $k=|SQ\_0|/|SP\_0|$.)
3. Attach the mobile rods rotatable at point $S$.
4. Vary the location of point $P$ and mark at each time point $Q$.

Because of $|SQ\_0|/|SP\_0|=|Q\_0Q|/|PP\_0|$ (see diagram) one gets from the intercept theorem that the points $S,P,Q$ are collinear (lie on a line) and equation $|SQ|=k|SP|$ holds. That shows: the mapping $P\backslash to\; Q$ is a homothety with center $S$ and ratio $k$.

File:Zentr-streck-TT-e.svg

• The composition of two homotheties with the same center $S$ is again a homothety with center $S$. The homotheties with center $S$ form a group.
• The composition of two homotheties with different centers $S\_1,S\_2$ and its ratios $k\_1,k\_2$ is

::in case of $k\_1k\_2\backslash ne\; 1$ a homothety with its center on line $\backslash overline\{S\_1S\_2\}$ and ratio $k\_1k\_2$ or

::in case of $k\_1k\_2=\; 1$ a translation in direction $\backslash overrightarrow\{S\_1S\_2\}$. Especially, if $k\_1=k\_2=-1$ (point reflections).

Derivation:

For the composition $\backslash sigma\_2\backslash sigma\_1$ of the two homotheties $\backslash sigma\_1,\backslash sigma\_2$ with centers $S\_1,S\_2$ with

:$\backslash sigma\_1:\; \backslash mathbf\; x\; \backslash to\; \backslash mathbf\; s\_1+k\_1(\backslash mathbf\; x\; -\backslash mathbf\; s\_1),$

:$\backslash sigma\_2:\; \backslash mathbf\; x\; \backslash to\; \backslash mathbf\; s\_2+k\_2(\backslash mathbf\; x\; -\backslash mathbf\; s\_2)\backslash$

one gets by calculation for the image of point $X:\backslash mathbf\; x$:

:$(\backslash sigma\_2\backslash sigma\_1)(\backslash mathbf\; x)=\; \backslash mathbf\; s\_2+k\_2\backslash big(\backslash mathbf\; s\_1+k\_1(\backslash mathbf\; x-\backslash mathbf\; s\_1)-\backslash mathbf\; s\_2\backslash big)$

:$\backslash qquad\; \backslash qquad\; \backslash \; =(1-k\_1)k\_2\backslash mathbf\; s\_1+(1-k\_2)\backslash mathbf\; s\_2\; +\; k\_1k\_2\backslash mathbf\; x$.

Hence, the composition is

:in case of $k\_1k\_2=\; 1$ a translation in direction $\backslash overrightarrow\{S\_1S\_2\}$ by vector $\backslash \; (1-k\_2)(\backslash mathbf\; s\_2-\backslash mathbf\; s\_1)$.

:in case of $k\_1k\_2\backslash ne\; 1$ point

:$S\_3:\; \backslash mathbf\; s\_3=\backslash frac\{(1-k\_1)k\_2\backslash mathbf\; s\_1+(1-k\_2)\backslash mathbf\; s\_2\}\{1-k\_1k\_2\}$

=\mathbf s_1+\frac{1-k_2}{1-k_1k_2}(\mathbf s_2-\mathbf s_1)

is a fixpoint (is not moved) and the composition

:$\backslash sigma\_2\backslash sigma\_1:\; \backslash \; \backslash mathbf\; x\; \backslash to\; \backslash mathbf\; s\_3\; +\; k\_1k\_2(\backslash mathbf\; x\; -\backslash mathbf\; s\_3)\backslash quad$.

is a homothety with center $S\_3$ and ratio $k\_1k\_2$. $S\_3$ lies on line $\backslash overline\{S\_1S\_2\}$.

File:Zentr-streck-T-st-e.svg

• The composition of a homothety and a translation is a homothety.

Derivation:

The composition of the homothety

:$\backslash sigma:\; \backslash mathbf\; x\; \backslash to\; \backslash mathbf\; s\; +k(\backslash mathbf\; x-\backslash mathbf\; s),\backslash ;\; k\backslash ne\; 1,\backslash ;$ and the translation

:$\backslash tau:\; \backslash mathbf\; x\; \backslash to\; \backslash mathbf\; x\; +\backslash mathbf\; v$ is

:$\backslash tau\backslash sigma:\; \backslash mathbf\; x\; \backslash to\; \backslash mathbf\; s\; +\backslash mathbf\; v\; +k(\backslash mathbf\; x-\backslash mathbf\; s)$

:::$=\backslash mathbf\; s\; +\backslash frac\{\backslash mathbf\; v\}\{1-k\}+k\backslash left(\backslash mathbf\; x-(\backslash mathbf\; s+\backslash frac\{\backslash mathbf\; v\}\{1-k\})\backslash right)$

which is a homothety with center $\backslash mathbf\; s\text{'}=\backslash mathbf\; s\; +\backslash frac\{\backslash mathbf\; v\}\{1-k\}$ and ratio $k$.

The homothety $\backslash sigma:\; \backslash mathbf\; x\; \backslash to\; \backslash mathbf\; s+k(\backslash mathbf\; x\; -\backslash mathbf\; s)$

with center $S=(u,v)$ can be written as the composition of a homothety with center $O$ and a translation:

:$\backslash mathbf\; x\; \backslash to\; k\backslash mathbf\; x\; +\; (1-k)\backslash mathbf\; s$.

Hence $\backslash sigma$ can be represented in homogeneous coordinates

by the matrix:

:$\backslash begin\{pmatrix\}$

k & 0 & (1-k)u\\

0 & k & (1-k)v\\

0 & 0 & 1

\end{pmatrix}

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

• 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 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 monotonically increasing function.

• Coxeter, H. S. M. (1961), "Introduction to geometry", Wiley, p. 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 }}

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

Category:Transformation (function)