# Rotation

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)