Editing Euler angles (section)

==Properties==
{{See also|Charts on SO(3)|Quaternions and spatial rotation}}
The Euler angles form a [[chart (topology)|chart]] on all of [[SO(3)]], the [[special orthogonal group]] of rotations in 3D space. The chart is smooth except for a polar coordinate style singularity along {{Nowrap|1=''β'' = 0}}. See [[charts on SO(3)]] for a more complete treatment.

The space of rotations is called in general "The [[Quaternions and spatial rotation#Quaternion rotation operations|Hypersphere of rotations]]", though this is a misnomer: the group [[Spin group|Spin(3)]] is [[isometric embedding|isometric]] to the hypersphere ''S''<sup>3</sup>, but the rotation space SO(3) is instead isometric to the [[real projective space]] '''RP'''<sup>3</sup> which is a 2-fold [[Quotient space (topology)|quotient space]] of the hypersphere. This 2-to-1 ambiguity is the mathematical origin of [[Spin (physics)|spin in physics]].

A similar three angle decomposition applies to [[SU(2)]], the [[special unitary group]] of rotations in complex 2D space, with the difference that ''β'' ranges from 0 to 2{{pi}}. These are also called Euler angles.

The [[Haar measure]] for SO(3) in Euler angles is given by the Hopf angle parametrisation of SO(3), <math>\textrm{d}V \propto \sin \beta \cdot \textrm{d}\alpha \cdot \textrm{d}\beta \cdot \textrm{d}\gamma</math>,<ref>{{cite journal|at=Section 8 – Derivation of Hopf parametrisation |pmc=2896220 |year=2010 |last1=Yershova |first1=A. |last2=Jain |first2=S. |last3=Lavalle |first3=S. M. |last4=Mitchell |first4=J. C. |title=Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration |journal=The International Journal of Robotics Research |volume=29 |issue=7 |doi=10.1177/0278364909352700 |pmid=20607113 }}</ref> where <math>(\beta, \alpha)</math> parametrise <math>S^{2}</math>, the space of rotation axes.

For example, to generate uniformly randomized orientations, let ''α'' and ''γ'' be uniform from 0 to 2{{pi}}, let ''z'' be uniform from &minus;1 to 1, and let {{nowrap|1=''β'' = arccos(''z'')}}.

===Geometric algebra===
Other properties of Euler angles and rotations in general can be found from the [[geometric algebra]], a higher level abstraction, in which the quaternions are an even subalgebra. The principal tool in geometric algebra is the rotor <math> \mathbf{R} = [ \cos(\theta / 2) - I u \sin(\theta / 2) ] </math> where <math>\theta =</math>[[angle of rotation]], <math>\mathbf{u} </math> is the rotation axis (unitary vector) and <math>\mathbf{I}</math> is the pseudoscalar (trivector in <math>\mathbb{R}^3</math>)

===Higher dimensions===
It is possible to define parameters analogous to the Euler angles in dimensions higher than three.<ref>{{Citation
  |last = Hoffman
  |first = D. K.
  |title = Generalization of Euler Angles to N-Dimensional Orthogonal Matrices
  |journal = Journal of Mathematical Physics
 |publisher = [J. Math. Phys. 13, 528–533]
  |year = 1972
  |volume = 13
 |issue = 4
 |pages = 528–533
 |doi = 10.1063/1.1666011
  |bibcode = 1972JMP....13..528H
 |url = https://pubs.aip.org/aip/jmp/article/13/4/528/440286/Generalization-of-Euler-Angles-to-N-Dimensional
|url-access = subscription}}</ref>
<ref>{{in lang|it}} [http://ansi.altervista.org A generalization of Euler Angles to ''n''-dimensional real spaces]</ref>{{unreliable source?|date=December 2022}} In four dimensions and above, the concept of "rotation about an axis" loses meaning and instead becomes "rotation in a plane." The number of Euler angles needed to represent the group {{math|SO(''n'')}} is {{math|''n''(''n'' − 1)/2}}, equal to the number of planes containing two distinct coordinate axes in ''n''-dimensional Euclidean space.

In [[SO(4)]] a rotation matrix [[Rotation (mathematics)#In four dimensions|is defined by two unit quaternions]], and therefore has six degrees of freedom, three from each quaternion.