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} $.

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