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A transformation of Euclidean space with respect to a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479001.png" /> that brings each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479002.png" /> in a one-to-one correspondence with a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479003.png" /> on the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479004.png" /> in accordance with the rule
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A transformation of Euclidean space with respect to a certain point $O$ that brings each point $M$ in a one-to-one correspondence with a point $M'$ on the straight line $OM$ in accordance with the rule
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479005.png" /></td> </tr></table>
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$$OM'=kOM,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479006.png" /> is a constant number, not equal to zero, which is known as the homothety ratio. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479007.png" /> is said to be the centre of the homothety. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479008.png" />, the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h0479009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790010.png" /> lie on the same ray; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790011.png" />, on different sides from the centre. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790012.png" /> corresponds to itself. A homothety is a special case of a [[Similarity|similarity]]. Two figures called homothetic (similar or similarly situated) if each one consists of points obtained from the other figure by a homothety with respect to some centre.
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where $k$ is a constant number, not equal to zero, which is known as the homothety ratio. The point $O$ is said to be the centre of the homothety. If $k>0$, the points $M$ and $M'$ lie on the same ray; if $k<0$, on different sides from the centre. The point $O$ corresponds to itself. A homothety is a special case of a [[Similarity|similarity]]. Two figures called homothetic (similar or similarly situated) if each one consists of points obtained from the other figure by a homothety with respect to some centre.
  
Simplest properties of a homothety. A homothety with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790013.png" /> is a one-to-one mapping of the Euclidean space onto itself, with one fixed point. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790014.png" />, the homothety is the identity transformation. A homothety maps a straight line (a plane) passing through its centre into itself; a straight line (a plane) not passing through its centre into a straight line (a plane) parallel to it; the angles between straight lines (planes) are preserved under this transformation. Under a homothety segments are mapped into parallel segments with a length which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790015.png" /> times the original length, i.e. a homothety is a contraction (expansion) of the Euclidean space at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790016.png" />. Under a homothety a sphere is mapped into another sphere, and the centre of the former is mapped to the centre of the latter.
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Simplest properties of a homothety. A homothety with $k\neq1$ is a one-to-one mapping of the Euclidean space onto itself, with one fixed point. If $k=1$, the homothety is the identity transformation. A homothety maps a straight line (a plane) passing through its centre into itself; a straight line (a plane) not passing through its centre into a straight line (a plane) parallel to it; the angles between straight lines (planes) are preserved under this transformation. Under a homothety segments are mapped into parallel segments with a length which is $|k|$ times the original length, i.e. a homothety is a contraction (expansion) of the Euclidean space at the point $O$. Under a homothety a sphere is mapped into another sphere, and the centre of the former is mapped to the centre of the latter.
  
 
A homothety is most often specified (geometrically) by the homothety centre and a pair of corresponding points or by two pairs of corresponding points. A homothety is an [[Affine transformation|affine transformation]] with one (and only one) fixed point.
 
A homothety is most often specified (geometrically) by the homothety centre and a pair of corresponding points or by two pairs of corresponding points. A homothety is an [[Affine transformation|affine transformation]] with one (and only one) fixed point.
  
In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790017.png" />-dimensional Euclidean space a homothety leaves the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790018.png" />-dimensional subspaces invariant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790019.png" />.
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In $n$-dimensional Euclidean space a homothety leaves the set of all $k$-dimensional subspaces invariant, $k<n$.
  
A homothety is defined in a similar manner in pseudo-Euclidean spaces. A homothety in Riemannian spaces and in pseudo-Riemannian spaces is defined as a transformation that transforms the metric of the space into itself, up to a constant factor. The set of homotheties forms a Lie group of transformations, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790020.png" />-parameter homothety group of a Riemannian space contains the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047900/h04790021.png" />-parameter normal subgroup of displacements.
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A homothety is defined in a similar manner in pseudo-Euclidean spaces. A homothety in Riemannian spaces and in pseudo-Riemannian spaces is defined as a transformation that transforms the metric of the space into itself, up to a constant factor. The set of homotheties forms a Lie group of transformations, and the $r$-parameter homothety group of a Riemannian space contains the $(r-1)$-parameter normal subgroup of displacements.
  
  
  
 
====Comments====
 
====Comments====
A homotopy is also called a central dilatation (cf. also [[Dilatation|Dilatation]]).
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A homothety is also called a central dilatation (cf. also [[Dilatation|Dilatation]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Artin,  "Geometric algebra" , Interscience  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Artin,  "Geometric algebra" , Interscience  (1957)</TD></TR></table>

Latest revision as of 19:50, 2 May 2019

A transformation of Euclidean space with respect to a certain point $O$ that brings each point $M$ in a one-to-one correspondence with a point $M'$ on the straight line $OM$ in accordance with the rule

$$OM'=kOM,$$

where $k$ is a constant number, not equal to zero, which is known as the homothety ratio. The point $O$ is said to be the centre of the homothety. If $k>0$, the points $M$ and $M'$ lie on the same ray; if $k<0$, on different sides from the centre. The point $O$ corresponds to itself. A homothety is a special case of a similarity. Two figures called homothetic (similar or similarly situated) if each one consists of points obtained from the other figure by a homothety with respect to some centre.

Simplest properties of a homothety. A homothety with $k\neq1$ is a one-to-one mapping of the Euclidean space onto itself, with one fixed point. If $k=1$, the homothety is the identity transformation. A homothety maps a straight line (a plane) passing through its centre into itself; a straight line (a plane) not passing through its centre into a straight line (a plane) parallel to it; the angles between straight lines (planes) are preserved under this transformation. Under a homothety segments are mapped into parallel segments with a length which is $|k|$ times the original length, i.e. a homothety is a contraction (expansion) of the Euclidean space at the point $O$. Under a homothety a sphere is mapped into another sphere, and the centre of the former is mapped to the centre of the latter.

A homothety is most often specified (geometrically) by the homothety centre and a pair of corresponding points or by two pairs of corresponding points. A homothety is an affine transformation with one (and only one) fixed point.

In $n$-dimensional Euclidean space a homothety leaves the set of all $k$-dimensional subspaces invariant, $k<n$.

A homothety is defined in a similar manner in pseudo-Euclidean spaces. A homothety in Riemannian spaces and in pseudo-Riemannian spaces is defined as a transformation that transforms the metric of the space into itself, up to a constant factor. The set of homotheties forms a Lie group of transformations, and the $r$-parameter homothety group of a Riemannian space contains the $(r-1)$-parameter normal subgroup of displacements.


Comments

A homothety is also called a central dilatation (cf. also Dilatation).

References

[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a3] E. Artin, "Geometric algebra" , Interscience (1957)
How to Cite This Entry:
Homothety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homothety&oldid=11584
This article was adapted from an original article by I.P. Egorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article