View source for Rotation

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

<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>

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

<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>

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.

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:

<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>

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>

<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:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262015.png" /> — the identity transformation;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082620/r08262016.png" /> — the rotation is a central symmetry;

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);

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;

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;

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.

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" />.

====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>