Editing Rotation (mathematics) (section)

===In Euclidean geometry===<!-- caution: an internal #-link -->
{{further|Euclidean space #Rotations and reflections|Special orthogonal group}}
[[Image:Rotation4.svg|right|thumb|A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a [[Translation (geometry)|translation]].]]

A motion of a [[Euclidean space]] is the same as its [[isometry]]: it leaves [[Euclidean distance|the distance]] between any two points unchanged after the transformation. But a (proper) rotation also has to preserve the [[orientation (vector space)|orientation structure]]. The "[[improper rotation]]" term refers to isometries that reverse (flip) the orientation. In the language of [[group theory]] the distinction is expressed as ''direct'' vs ''indirect'' isometries in the [[Euclidean group]], where the former comprise the [[identity component]]. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation.

In [[one-dimensional space]], there are only [[triviality (mathematics)|trivial]] rotations. In [[Plane (mathematics)|two dimensions]], only a single [[angle]] is needed to specify a rotation about the [[origin (mathematics)|origin]] – the ''angle of rotation'' that specifies an element of the [[circle group]] (also known as {{math|U(1)}}). The rotation is acting to rotate an object [[counterclockwise]] through an angle {{mvar|θ}} about the [[origin (mathematics)|origin]]; see [[#Two dimensions|below]] for details. Composition of rotations [[summation|sums]] their angles [[modular arithmetic|modulo]] 1 [[turn (geometry)|turn]], which implies that all two-dimensional rotations about ''the same'' point [[Abelian group|commute]]. Rotations about ''different'' points, in general, do not commute. Any two-dimensional direct motion is either a translation or a rotation; see [[Euclidean plane isometry]] for details.

{{anchor|Euler angles}}[[Image:Praezession.svg|thumb|170px|left|Euler rotations of the Earth. Intrinsic (green), precession (blue) and nutation (red)]]
Rotations in [[three-dimensional space]] differ from those in two dimensions in a number of important ways. Rotations in three dimensions are generally not [[commutative]], so the order in which rotations are applied is important even about the same point. Also, unlike the two-dimensional case, a three-dimensional direct motion, in [[general position]], is not a rotation but a [[screw axis|screw operation]]. Rotations about the origin have three degrees of freedom (see [[rotation formalisms in three dimensions]] for details), the same as the number of dimensions.
A three-dimensional rotation can be specified in a number of ways. The most usual methods are:
* [[Euler angles]] (pictured at the left). Any rotation about the origin can be represented as the [[function composition|composition]] of three rotations defined as the motion obtained by changing one of the Euler angles while leaving the other two constant. They constitute a '''mixed axes of rotation''' system because angles are measured with respect to a mix of different [[reference frames]], rather than a single frame that is purely external or purely intrinsic. Specifically, the first angle moves the [[line of nodes]] around the external axis ''z'', the second rotates around the line of nodes and the third is an intrinsic rotation (a spin) around an axis fixed in the body that moves. Euler angles are typically denoted as [[Alpha|''&alpha;'']], [[Beta|''&beta;'']], [[Gamma|''&gamma;'']], or [[Phi|''&phi;'']], [[Theta|''&theta;'']], [[Psi (letter)|''&psi;'']]. This presentation is convenient only for rotations about a fixed point.
[[Image:Euler AxisAngle.png|thumb|right|104px]]
* [[Axis–angle representation]] (pictured at the right) specifies an angle with the axis about which the rotation takes place. It can be easily visualised. There are two variants to represent it:
** as a pair consisting of the angle and a [[unit vector]] for the axis, or
** as a [[Euclidean vector]] obtained by multiplying the angle with this unit vector, called the ''rotation vector'' (although, strictly speaking, it is a [[pseudovector]]).
* Matrices, versors (quaternions), and other [[algebra]]ic things: see the section [[#Linear and multilinear algebra formalism|''Linear and Multilinear Algebra Formalism'']] for details.

[[File:8-cell.gif|thumb|A perspective projection onto three-dimensions of a [[tesseract]] being rotated in four-dimensional Euclidean space.]]
A general rotation in [[four-dimensional space|four dimensions]] has only one fixed point, the centre of rotation, and no axis of rotation; see [[rotations in 4-dimensional Euclidean space]] for details. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for each [[plane of rotation]], through which points in the planes rotate. If these are {{math|''ω''<sub>1</sub>}} and {{math|''ω''<sub>2</sub>}} then all points not in the planes rotate through an angle between {{math|''ω''<sub>1</sub>}} and {{math|''ω''<sub>2</sub>}}. Rotations in four dimensions about a fixed point have six degrees of freedom. A four-dimensional direct motion in general position ''is'' a rotation about certain point (as in all [[even number|even]] Euclidean dimensions), but screw operations exist also.