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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 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|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|orthogonal matrix]] in canonical form:

$$ 
M  =  \left \|

where

$$ 
u _ {i}  = \ 
\left \|

$  \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|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====
<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====

====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>