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A transformation of a Euclidean space in which for any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851602.png" /> and their transforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851604.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851605.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851606.png" /> is a positive number called the similarity coefficient.
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A transformation of a Euclidean space in which for any two points $A$ and $B$ and their transforms $A'$ and $B'$ one has $|A'B'|=k|AB|$, where $k$ is a positive number called the similarity coefficient.
  
Any [[Homothety|homothety]] is a similarity. An [[Isometric mapping|isometric mapping]] (including the identity) can also be regarded as a similarity with coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851607.png" /> equal to one. A figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851608.png" /> is said to be similar to a figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s0851609.png" /> if there exists a similarity under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516010.png" />. Similarity of figures is an equivalence relation, i.e. it has the properties of reflexivity, symmetry and transitivity. A similarity is a one-to-one mapping of a Euclidean space onto itself; a similarity also retains the order of points on a straight line, i.e. if a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516011.png" /> lies between points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516013.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516016.png" /> are the corresponding images under some similarity, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516017.png" /> lies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516019.png" />; points not lying on a straight line, under any similarity, become points not lying on one straight line. A similarity transforms a straight line into a straight line, a segment into a segment, a ray into a ray, an angle into an angle, and a circle into a circle. Under a similarity, an angle retains its value.
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Any [[Homothety|homothety]] is a similarity. An [[Isometric mapping|isometric mapping]] (including the identity) can also be regarded as a similarity with coefficient $k$ equal to one. A figure $F$ is said to be similar to a figure $F'$ if there exists a similarity under which $F\to F'$. Similarity of figures is an equivalence relation, i.e. it has the properties of reflexivity, symmetry and transitivity. A similarity is a one-to-one mapping of a Euclidean space onto itself; a similarity also retains the order of points on a straight line, i.e. if a point $B$ lies between points $A$ and $C$, while $B'$, $A'$ and $C'$ are the corresponding images under some similarity, then $B'$ lies between $A'$ and $C'$; points not lying on a straight line, under any similarity, become points not lying on one straight line. A similarity transforms a straight line into a straight line, a segment into a segment, a ray into a ray, an angle into an angle, and a circle into a circle. Under a similarity, an angle retains its value.
  
A similarity with coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516020.png" /> that transforms each straight line into a straight line parallel to it is a homothety with coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516022.png" />. Any similarity can be considered as a composition of a motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516023.png" /> and some homothety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516024.png" /> with a positive coefficient.
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A similarity with coefficient $k\neq1$ that transforms each straight line into a straight line parallel to it is a homothety with coefficient $k$ or $-k$. Any similarity can be considered as a composition of a motion $D$ and some homothety $\Gamma$ with a positive coefficient.
  
A similarity is called proper (improper) if the [[Motion|motion]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516025.png" /> is proper (improper). A proper similarity retains the orientation of a figure, while an improper similarity reverses the orientation.
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A similarity is called proper (improper) if the [[Motion|motion]] $D$ is proper (improper). A proper similarity retains the orientation of a figure, while an improper similarity reverses the orientation.
  
Analogously, one defines a similarity (with retention of the above properties) in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516026.png" />-dimensional Euclidean space, and also in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516027.png" />-dimensional Euclidean and pseudo-Euclidean spaces.
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Analogously, one defines a similarity (with retention of the above properties) in the $3$-dimensional Euclidean space, and also in the $n$-dimensional Euclidean and pseudo-Euclidean spaces.
  
A similarity is defined in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516028.png" />-dimensional Riemannian, pseudo-Riemannian or Finsler space as a transformation that converts the metric of the space into itself up to a constant factor.
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A similarity is defined in an $n$-dimensional Riemannian, pseudo-Riemannian or Finsler space as a transformation that converts the metric of the space into itself up to a constant factor.
  
The set of all similarities of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516029.png" />-dimensional Euclidean, pseudo-Euclidean, Riemannian, pseudo-Riemannian, or Finsler space constitutes an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516030.png" />-dimensional Lie transformation group, which is called the group of similarity (homothetic) transformations for the corresponding space. In each of the spaces of these types, the Lie group of similarity transformations with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516031.png" /> parameters contains a normal subgroup of isometries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085160/s08516032.png" /> dimensions.
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The set of all similarities of the $n$-dimensional Euclidean, pseudo-Euclidean, Riemannian, pseudo-Riemannian, or Finsler space constitutes an $r$-dimensional Lie transformation group, which is called the group of similarity (homothetic) transformations for the corresponding space. In each of the spaces of these types, the Lie group of similarity transformations with $r$ parameters contains a normal subgroup of isometries of $r-1$ dimensions.
  
 
Interest arises in metric spaces of vector densities with groups of similarities and isometries containing infinite-dimensional subgroups of isometries with common trajectories.
 
Interest arises in metric spaces of vector densities with groups of similarities and isometries containing infinite-dimensional subgroups of isometries with common trajectories.

Latest revision as of 14:19, 12 April 2014

A transformation of a Euclidean space in which for any two points $A$ and $B$ and their transforms $A'$ and $B'$ one has $|A'B'|=k|AB|$, where $k$ is a positive number called the similarity coefficient.

Any homothety is a similarity. An isometric mapping (including the identity) can also be regarded as a similarity with coefficient $k$ equal to one. A figure $F$ is said to be similar to a figure $F'$ if there exists a similarity under which $F\to F'$. Similarity of figures is an equivalence relation, i.e. it has the properties of reflexivity, symmetry and transitivity. A similarity is a one-to-one mapping of a Euclidean space onto itself; a similarity also retains the order of points on a straight line, i.e. if a point $B$ lies between points $A$ and $C$, while $B'$, $A'$ and $C'$ are the corresponding images under some similarity, then $B'$ lies between $A'$ and $C'$; points not lying on a straight line, under any similarity, become points not lying on one straight line. A similarity transforms a straight line into a straight line, a segment into a segment, a ray into a ray, an angle into an angle, and a circle into a circle. Under a similarity, an angle retains its value.

A similarity with coefficient $k\neq1$ that transforms each straight line into a straight line parallel to it is a homothety with coefficient $k$ or $-k$. Any similarity can be considered as a composition of a motion $D$ and some homothety $\Gamma$ with a positive coefficient.

A similarity is called proper (improper) if the motion $D$ is proper (improper). A proper similarity retains the orientation of a figure, while an improper similarity reverses the orientation.

Analogously, one defines a similarity (with retention of the above properties) in the $3$-dimensional Euclidean space, and also in the $n$-dimensional Euclidean and pseudo-Euclidean spaces.

A similarity is defined in an $n$-dimensional Riemannian, pseudo-Riemannian or Finsler space as a transformation that converts the metric of the space into itself up to a constant factor.

The set of all similarities of the $n$-dimensional Euclidean, pseudo-Euclidean, Riemannian, pseudo-Riemannian, or Finsler space constitutes an $r$-dimensional Lie transformation group, which is called the group of similarity (homothetic) transformations for the corresponding space. In each of the spaces of these types, the Lie group of similarity transformations with $r$ parameters contains a normal subgroup of isometries of $r-1$ dimensions.

Interest arises in metric spaces of vector densities with groups of similarities and isometries containing infinite-dimensional subgroups of isometries with common trajectories.


Comments

References

[a1] E. Artin, "Geometric algebra" , Interscience (1957) pp. Chapt. II
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 72–76
[a3] M. Berger, "Geometry" , I , Springer (1987)
[a4] A.L. Besse, "Einstein manifolds" , Springer (1987)
How to Cite This Entry:
Similarity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similarity&oldid=31636
This article was adapted from an original article by I.P. Egorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article