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A special kind of [[Motion|motion]], for which at least one point in space remains at rest. If the rotation is in a plane, the fixed point is called the centre of the rotation; if the rotation is in space, the fixed straight line is called the axis of rotation. A rotation in a Euclidean space is called proper (a rotation of the first kind) or improper (a rotation of the second kind) depending on whether or not the [[Orientation|orientation]] in space remains unchanged.
 
A special kind of [[Motion|motion]], for which at least one point in space remains at rest. If the rotation is in a plane, the fixed point is called the centre of the rotation; if the rotation is in space, the fixed straight line is called the axis of rotation. A rotation in a Euclidean space is called proper (a rotation of the first kind) or improper (a rotation of the second kind) depending on whether or not the [[Orientation|orientation]] in space remains unchanged.
  
A proper rotation in a plane can be analytically expressed in Cartesian orthogonal coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r0826201.png" /> by the formulas
+
A proper rotation in a plane can be analytically expressed in Cartesian orthogonal coordinates $  x, y $
 +
by the formulas
  
<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/r/r082/r082620/r0826202.png" /></td> </tr></table>
+
$$
 +
\widetilde{x}  = x  \cos  \phi - y  \sin  \phi ,\ \
 +
\widetilde{y}  = x  \sin  \phi + y  \cos  \phi ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r0826203.png" /> is the rotation angle and the centre of the rotation is the coordinate origin. A proper rotation through an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r0826204.png" /> may be represented as the product of two axial symmetries (reflections, cf. [[Reflection|Reflection]]) with axes forming an angle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r0826205.png" /> with each other. An improper rotation in a plane can be analytically expressed in Cartesian orthogonal coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r0826206.png" /> by the formulas
+
where $  \phi $
 +
is the rotation angle and the centre of the rotation is the coordinate origin. A proper rotation through an angle $  \phi $
 +
may be represented as the product of two axial symmetries (reflections, cf. [[Reflection|Reflection]]) with axes forming an angle of $  \phi / 2 $
 +
with each other. An improper rotation in a plane can be analytically expressed in Cartesian orthogonal coordinates $  x, y $
 +
by the formulas
  
<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/r/r082/r082620/r0826207.png" /></td> </tr></table>
+
$$
 +
\widetilde{x}  = x  \cos  \phi + y  \sin  \phi ,\ \
 +
\widetilde{y}  = x  \sin  \phi - y  \cos  \phi ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r0826208.png" /> is the rotation angle and the centre of the rotation is the coordinate origin. An improper rotation in a plane may be represented as a product of a proper rotation by an axial symmetry.
+
where $  \phi $
 +
is the rotation angle and the centre of the rotation is the coordinate origin. An improper rotation in a plane may be represented as a product of a proper rotation by an axial symmetry.
  
A rotation in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r0826209.png" />-dimensional Euclidean space can be analytically expressed by an [[Orthogonal matrix|orthogonal matrix]] in canonical form:
+
A rotation in an $  n $-
 +
dimensional Euclidean space can be analytically expressed by an [[Orthogonal matrix|orthogonal matrix]] in canonical form:
  
<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/r/r082/r082620/r08262010.png" /></td> </tr></table>
+
$$
 +
= \left \|
 +
 
 +
\begin{array}{lllllll}
 +
u _ {1}  &{}  &{}  &{}  &{}  &{}  & 0  \\
 +
{}  &\cdot  &{}  &{}  &{}  &{}  &{}  \\
 +
{}  &{}  &\cdot  &{}  &{}  &{}  &{}  \\
 +
{}  &{}  &{}  &\cdot  &{}  &{}  &{}  \\
 +
{}  &{}  &{}  &{}  &u _ {k}  &{}  &{}  \\
 +
{}  &{}  &{}  &{}  &{}  &\epsilon  ^ {p}  &{}  \\
 +
0  &{}  &{}  &{}  &{}  &{}  &- \epsilon  ^ {q}  \\
 +
\end{array}
 +
 
 +
\right \| ,
 +
$$
  
 
where
 
where
  
<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/r/r082/r082620/r08262011.png" /></td> </tr></table>
+
$$
 +
u _ {i}  = \
 +
\left \|
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262012.png" /> is the identity matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262014.png" />). The following cases are possible:
+
\begin{array}{rl}
 +
\cos  \phi _ {i}  &\sin  \phi _ {i}  \\
 +
- \sin  \phi _ {i}  &\cos  \phi _ {i}  \\
 +
\end{array}
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262015.png" /> — the identity transformation;
+
\right \| .
 +
$$
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262016.png" /> — the rotation is a central symmetry;
+
$  \epsilon  ^ {s} $
 +
is the identity matrix of order  $  s $(
 +
$  s= p, q $).
 +
The following cases are possible:
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262017.png" /> — the rotation is a symmetry with respect to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262018.png" />-plane (a reflection in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262019.png" />-plane);
+
1) $  p = n $—  
 +
the identity transformation;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262020.png" /> does not contain submatrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262022.png" /> — the rotation is called a rotation around a unique fixed point;
+
2) $  q = n $—  
 +
the rotation is a central symmetry;
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262023.png" /> contains the submatrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262025.png" /> but does not contain the submatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262026.png" /> — the rotation is a rotation around a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262028.png" />-plane;
+
3) $  p + q = n $—  
 +
the rotation is a symmetry with respect to a  $  p $-
 +
plane (a reflection in a $  p $-
 +
plane);
  
6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262029.png" /> contains the submatrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262031.png" /> but does not contain the submatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262032.png" /> — the rotation is called a rotational reflection in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262034.png" />-plane.
+
4) $  M $
 +
does not contain submatrices  $  \epsilon  ^ {p} $
 +
and  $  - \epsilon  ^ {q} $—  
 +
the rotation is called a rotation around a unique fixed point;
  
The rotations of a Euclidean space around a given point form a group with respect to multiplication of rotations. This group is isomorphic to the group of orthogonal transformations (cf. [[Orthogonal transformation|Orthogonal transformation]]) of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262035.png" /> or to the group of orthogonal matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262036.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262037.png" />. The rotation group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262038.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262039.png" />-dimensional Lie group with an intransitive action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262040.png" />.
+
5)  $  M $
 +
contains the submatrices  $  u _ {i} $
 +
and  $  \epsilon  ^ {p} $
 +
but does not contain the submatrix  $  - \epsilon  ^ {q} $—
 +
the rotation is a rotation around a  $  p $-
 +
plane;
 +
 
 +
6)  $  M $
 +
contains the submatrices  $  u _ {i} $
 +
and  $  - \epsilon  ^ {q} $
 +
but does not contain the submatrix  $  \epsilon  ^ {p} $—
 +
the rotation is called a rotational reflection in an  $  ( n - q) $-
 +
plane.
 +
 
 +
The rotations of a Euclidean space around a given point form a group with respect to multiplication of rotations. This group is isomorphic to the group of orthogonal transformations (cf. [[Orthogonal transformation|Orthogonal transformation]]) of the vector space $  \mathbf R  ^ {n} $
 +
or to the group of orthogonal matrices of order $  n $
 +
over the field $  \mathbf R $.  
 +
The rotation group of the space $  E _ {n} $
 +
is an $  n( n - 1)/2 $-
 +
dimensional Lie group with an intransitive action on $  E _ {n} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Multi-dimensional spaces" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.A. Shirokov,  "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Multi-dimensional spaces" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.A. Shirokov,  "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Greenberg,  "Euclidean and non-Euclidean geometry" , Freeman  (1980)  pp. 105</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Artmann,  "Lineare Algebra" , Birkhäuser  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.R. Halmos,  "Finite-dimensional vector spaces" , v. Nostrand  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Greenberg,  "Euclidean and non-Euclidean geometry" , Freeman  (1980)  pp. 105</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Artmann,  "Lineare Algebra" , Birkhäuser  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.R. Halmos,  "Finite-dimensional vector spaces" , v. Nostrand  (1958)</TD></TR></table>

Latest revision as of 14:55, 7 June 2020


A special kind of motion, for which at least one point in space remains at rest. If the rotation is in a plane, the fixed point is called the centre of the rotation; if the rotation is in space, the fixed straight line is called the axis of rotation. A rotation in a Euclidean space is called proper (a rotation of the first kind) or improper (a rotation of the second kind) depending on whether or not the orientation in space remains unchanged.

A proper rotation in a plane can be analytically expressed in Cartesian orthogonal coordinates $ x, y $ by the formulas

$$ \widetilde{x} = x \cos \phi - y \sin \phi ,\ \ \widetilde{y} = x \sin \phi + y \cos \phi , $$

where $ \phi $ is the rotation angle and the centre of the rotation is the coordinate origin. A proper rotation through an angle $ \phi $ may be represented as the product of two axial symmetries (reflections, cf. Reflection) with axes forming an angle of $ \phi / 2 $ with each other. An improper rotation in a plane can be analytically expressed in Cartesian orthogonal coordinates $ x, y $ by the formulas

$$ \widetilde{x} = x \cos \phi + y \sin \phi ,\ \ \widetilde{y} = x \sin \phi - y \cos \phi , $$

where $ \phi $ is the rotation angle and the centre of the rotation is the coordinate origin. An improper rotation in a plane may be represented as a product of a proper rotation by an axial symmetry.

A rotation in an $ n $- dimensional Euclidean space can be analytically expressed by an orthogonal matrix in canonical form:

$$ M = \left \| \begin{array}{lllllll} u _ {1} &{} &{} &{} &{} &{} & 0 \\ {} &\cdot &{} &{} &{} &{} &{} \\ {} &{} &\cdot &{} &{} &{} &{} \\ {} &{} &{} &\cdot &{} &{} &{} \\ {} &{} &{} &{} &u _ {k} &{} &{} \\ {} &{} &{} &{} &{} &\epsilon ^ {p} &{} \\ 0 &{} &{} &{} &{} &{} &- \epsilon ^ {q} \\ \end{array} \right \| , $$

where

$$ u _ {i} = \ \left \| \begin{array}{rl} \cos \phi _ {i} &\sin \phi _ {i} \\ - \sin \phi _ {i} &\cos \phi _ {i} \\ \end{array} \right \| . $$

$ \epsilon ^ {s} $ is the identity matrix of order $ s $( $ s= p, q $). The following cases are possible:

1) $ p = n $— the identity transformation;

2) $ q = n $— the rotation is a central symmetry;

3) $ p + q = n $— the rotation is a symmetry with respect to a $ p $- plane (a reflection in a $ p $- plane);

4) $ M $ does not contain submatrices $ \epsilon ^ {p} $ and $ - \epsilon ^ {q} $— the rotation is called a rotation around a unique fixed point;

5) $ M $ contains the submatrices $ u _ {i} $ and $ \epsilon ^ {p} $ but does not contain the submatrix $ - \epsilon ^ {q} $— the rotation is a rotation around a $ p $- plane;

6) $ M $ contains the submatrices $ u _ {i} $ and $ - \epsilon ^ {q} $ but does not contain the submatrix $ \epsilon ^ {p} $— the rotation is called a rotational reflection in an $ ( n - q) $- plane.

The rotations of a Euclidean space around a given point form a group with respect to multiplication of rotations. This group is isomorphic to the group of orthogonal transformations (cf. Orthogonal transformation) of the vector space $ \mathbf R ^ {n} $ or to the group of orthogonal matrices of order $ n $ over the field $ \mathbf R $. The rotation group of the space $ E _ {n} $ is an $ n( n - 1)/2 $- dimensional Lie group with an intransitive action on $ E _ {n} $.

References

[1] B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian)
[2] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)
[3] P.A. Shirokov, "Tensor calculus. Tensor algebra" , Kazan' (1961) (In Russian)

Comments

References

[a1] M. Berger, "Geometry" , I , Springer (1987)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a3] M. Greenberg, "Euclidean and non-Euclidean geometry" , Freeman (1980) pp. 105
[a4] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
[a5] B. Artmann, "Lineare Algebra" , Birkhäuser (1986)
[a6] P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958)
How to Cite This Entry:
Rotation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotation&oldid=49410
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article