View source for Euler–Rodrigues formula

{{Short description|Math formula for 3D vector rotation}}
In [[mathematics]] and [[mechanics]], the '''Euler&ndash;Rodrigues formula''' describes the rotation of a vector in three dimensions. It is based on [[Rodrigues' rotation formula]], but uses a different parametrization.

The rotation is described by four '''Euler parameters''' due to [[Leonhard Euler]]. The [[Rodrigues' rotation formula]]  (named after [[Olinde Rodrigues]]), a method of calculating the position of a rotated point, is used in some software applications, such as [[flight simulator]]s and [[PC game|computer games]].

== Definition ==
A rotation about the origin is represented by four real numbers, {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} such that
:<math>a^2 + b^2 + c^2 + d^2 = 1.</math>
When the rotation is applied, a point at position <math>\vec x</math> rotates to its new position,<ref>e.g. [[Felix Klein]] (1897), ''The mathematical theory of the top'', New York: Scribner. p.[https://archive.org/details/mathematicaltheo00kleiuoft/page/4/mode/2up?view=theater 4]</ref>
:<math>\vec x' = \begin{pmatrix} a^2+b^2-c^2-d^2 & 2(bc-ad) & 2(bd + ac) \\
                                       2(bc+ad) & a^2+c^2-b^2-d^2 & 2(cd - ab) \\
                                       2(bd-ac) & 2(cd+ab) & a^2+d^2-b^2-c^2 \end{pmatrix}\vec x.</math>

=== Vector formulation ===
The parameter {{mvar|a}} may be called the ''scalar'' parameter and <math>\vec \omega = (b, c, d)</math> the ''vector'' parameter. In standard vector notation, the Rodrigues rotation formula takes the compact form{{Citation needed|date=August 2024}}
{{Equation box 1
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<math>\vec x ' = \vec x + 2a(\vec \omega\times\vec x) + 2\left (\vec \omega\times(\vec \omega\times\vec x)\right )</math>
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=== Symmetry ===
The parameters {{math|(''a'', ''b'', ''c'', ''d'')}} and {{math|(−''a'', −''b'', −''c'', −''d'')}} describe the same rotation. Apart from this symmetry, every set of four parameters describes a unique rotation in three-dimensional space.

=== Composition of rotations ===
The composition of two rotations is itself a rotation. Let {{math|(''a''<sub>1</sub>, ''b''<sub>1</sub>, ''c''<sub>1</sub>, ''d''<sub>1</sub>)}} and {{math|(''a''<sub>2</sub>, ''b''<sub>2</sub>, ''c''<sub>2</sub>, ''d''<sub>2</sub>)}} be the Euler parameters of two rotations. The parameters for the compound rotation (rotation 2 after rotation 1) are as follows:
:<math>\begin{align}
a & = a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2; \\
b & = a_1b_2 + b_1a_2 - c_1d_2 + d_1c_2; \\
c & = a_1c_2 + c_1a_2 - d_1b_2 + b_1d_2; \\
d & = a_1d_2 + d_1a_2 - b_1c_2 + c_1b_2.
\end{align}</math>

It is straightforward, though tedious, to check that {{math|''a''<sup>2</sup> + ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> {{=}} 1}}. (This is essentially [[Euler's four-square identity]].)

== Rotation angle and rotation axis ==

Any central rotation in three dimensions is uniquely determined by its axis of rotation (represented by a [[unit vector]] {{math|{{vec|''k''}} {{=}} (''k<sub>x</sub>'', ''k<sub>y</sub>'', ''k<sub>z</sub>'')}}) and the rotation angle {{math|''φ''}}. The Euler parameters for this rotation are calculated as follows:

: <math>\begin{align}

a & = \cos \frac{\varphi}{2}; \\
b & = k_x \sin \frac{\varphi}{2}; \\
c & = k_y \sin \frac{\varphi}{2}; \\
d & = k_z \sin \frac{\varphi}{2}.
\end{align}</math>
Note that if {{math|''φ''}} is increased by a full rotation of 360 degrees, the arguments of sine and cosine only increase by 180 degrees. The resulting parameters are the opposite of the original values, {{math|(−''a'', −''b'', −''c'', −''d'')}}; they represent the same rotation.

In particular, the identity transformation (null rotation, {{math|''φ'' {{=}} 0}}) corresponds to parameter values {{math|(''a'', ''b'', ''c'', ''d'') {{=}} (±1, 0, 0, 0)}}. Rotations of 180 degrees about any axis result in {{math|''a'' {{=}} 0}}.

== Connection with quaternions ==
The Euler parameters can be viewed as the coefficients of a [[quaternion]]; the scalar parameter {{mvar|a}} is the real part, the vector parameters {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are the imaginary parts.
Thus we have the quaternion
:<math>q = a + bi + cj + dk,</math>
which is a quaternion of unit length (or [[versor]]) since
:<math>\left\|q\right\|^2 = a^2 + b^2 + c^2 + d^2 = 1.</math>
Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions <math>q = q_2 \, q_1</math>. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with composition.

A rotation in 3D can thus be represented by a quaternion ''q'':
:<math>q = \cos \frac{\varphi}{2} + \vec{\omega} \sin \frac{\varphi}{2}, </math>
where:
* <math>\cos \tfrac{\varphi}{2} </math> is the scalar (real) part,
* <math>\vec{\omega} \sin \tfrac{\varphi}{2} </math> is the vector (imaginary) part,
* <math>\vec{\omega}</math> is a unit vector representing the axis of rotation.

For a pure quaternion <math> X = 0 + \vec{x}</math>, the rotated vector <math>\vec{x}'</math> is given by:
:<math> X' = qXq^{-1},</math>
where <math> q^{-1} = \cos \tfrac{\varphi}{2} - \vec{\omega} \sin \tfrac{\varphi}{2}.</math>

To derive the quaternionic equivalent of the Euler-Rodrigues equation, substitute <math>q = \cos \tfrac{\varphi}{2} + \vec{\omega} \sin \tfrac{\varphi}{2}</math> into <math> X' = qXq^{-1}</math>:
:<math> X' = \left( \cos \frac{\varphi}{2} + \vec{\omega} \sin \frac{\varphi}{2} \right) \left( 0 + \vec{x} \right) \left(\cos \frac{\varphi}{2} - \vec{\omega} \sin \frac{\varphi}{2} \right). </math>

Using quaternion multiplication and the fact that <math>\vec{\omega} \cdot (\vec{\omega} \times \vec{x}) = 0</math> (since <math> \vec{\omega}</math> is perpendicular to the cross product terms), the expanded form becomes:
:<math> X' = \vec{x} + 2 (\vec{\omega} \cdot \vec{x}) \vec{\omega} + 2(\vec{\omega} \times \vec{x}) \sin \frac{\varphi}{2} \cos \frac{\varphi}{2}. </math>

From the trigonometric identity <math> \sin 2 \varphi = 2 \sin \varphi \cos \varphi </math> this simplifies to the Euler-Rodrigues equation: 
:<math>X' = \vec{x} + 2a (\vec{\omega} \times \vec{x}) + 2 (\vec{\omega} \times (\vec{\omega} \times \vec{x})),</math>

where <math> a = \sin^2 \tfrac{\varphi}{2}</math>.

== Connection with SU(2) spin matrices ==
The [[Lie group]] [[SU(2)]] can be used to represent three-dimensional rotations in complex {{nowrap|2 × 2}} matrices. The SU(2)-matrix corresponding to a rotation, in terms of its Euler parameters, is
:<math>U = \begin{pmatrix} \ a-di & -c-bi \\ c-bi & a+di \end{pmatrix}.</math>
which can be written as the sum
:<math>\begin{align}U & = a\ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
    - ib\ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
    - ic\ \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}
    - id\ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\
&   = a\,I-ib\,\sigma_x-ic\,\sigma_y-id\,\sigma_z,\end{align}</math>
where the {{mvar|σ<sub>i</sub>}} are the [[Pauli spin matrix|Pauli spin matrices]].

Rotation is given by <math>X^\prime \equiv (x^\prime_1 \sigma_x + x^\prime_2 \sigma_y + x^\prime_3 \sigma_z ) = U \; X \; U^\dagger =  (a\,I-ib\,\sigma_x-ic\,\sigma_y-id\,\sigma_z)(x_1 \sigma_x + x_2 \sigma_y + x_3 \sigma_z )(a\,I+ib\,\sigma_x+ic\,\sigma_y+id\,\sigma_z)</math>, which it can be confirmed by multiplying out gives the Euler–Rodrigues formula as stated above.

Thus, the Euler parameters are the real and imaginary coordinates in an SU(2) matrix corresponding to an element of the [[spin group]] Spin(3), which maps by a double cover mapping to a rotation in the [[orthogonal group]] SO(3). This realizes <math>\mathbb{R}^3</math> as the unique three-dimensional irreducible [[Group representation|representation]] of the [[Lie group]] SU(2) ≈ Spin(3).

=== Cayley–Klein parameters ===
The elements of the matrix <math>U</math> are known as the '''Cayley&ndash;Klein parameters''', after the mathematicians [[Arthur Cayley]] and [[Felix Klein]],{{refn|group=lower-alpha|name=ckparams|Goldstein (1980)<ref name="Goldstein" /> considers a [[Active and passive transformation|passive]] ([[Covariance and contravariance of vectors|contravariant]], or "alias") transformation, rather than the [[Active and passive transformation|active]] (covariant, or "alibi") transformation here.  <br />His matrix <math>A</math> therefore corresponds to the transpose of the Euler–Rodrigues matrix given at the head of this article, or, equivalently, to the Euler–Rodrigues matrix for an active rotation of <math>-\varphi</math> rather than <math>\varphi</math>.  Taking this into account, it is apparent that his <math>e_1</math>, <math>e_2</math>, and <math>e_3</math> in eqn 4-67 (p.153) are equal to <math>b</math>, <math>c</math>, and <math>d</math> here.  However his <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>, and <math>\delta</math>, the elements of his matrix <math>Q</math>, correspond to the elements of matrix <math>U^\dagger</math> here, rather than the matrix <math>U</math>.  This then gives his parametrization
:<math>\begin{align} \alpha &= \;\; a+di & \beta &= c+bi \\ \gamma &= -c + bi & \delta &= a - di \end{align}</math> 
In consequence, while his formula (4-64) is identical symbol-by-symbol to the transformation matrix given here, using his definitions for <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>, and <math>\delta</math> it gives his matrix <math>A</math>, whereas the definitions based on the matrix <math>U</math> above lead to the (active) Euler–Rodrigues matrix presented here.
<br />
Pennestrì et al (2016)<ref name="Pennestrì" /> similarly define their <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>, and <math>\delta</math> in terms of the passive matrix <math>Q</math> rather than the active matrix <math>U</math>.
<br />
The parametrization here accords with that used in eg Sakurai and Napolitano (2020),<ref>{{Cite book|last1=[[J. J. Sakurai|Sakurai, J. J.]]|title=[[Modern Quantum Mechanics]]|last2=Napolitano, Jim|isbn=978-1-108-47322-4|date=2020|edition=3rd|location=Cambridge|oclc=1202949320}}</ref> p. 165,  and Altmann (1986),<ref>Altmann, S. (1986), ''Rotations, Quaternions and Double Groups''. Oxford:Clarendon Press. {{ISBN|0-19-855372-2}}</ref> eqn. 5 p. 113 / eqn. 9 p. 117. }}
:<math>\begin{align} \alpha &= a-di & \beta &= -c-bi \\ \gamma &= c - bi & \delta &= \ a + di \end{align}</math>

In terms of these parameters the Euler–Rodrigues formula can then also be written <ref name=Goldstein>[[Herbert Goldstein|Goldstein, H.]] (1980), "The Cayley-Klein Parameters and Related Quantities". §4-5 in ''[[Classical Mechanics (Goldstein)|Classical Mechanics]]'', 2nd ed. Reading, MA: Addison-Wesley. p. 153</ref><ref>[[Eric W. Weisstein|Weisstein, Eric W.]], [https://mathworld.wolfram.com/Cayley-KleinParameters.html Cayley-Klein Parameters], ''[[MathWorld]]''. Accessed 2024-05-10</ref>{{refn|group=lower-alpha|name=ckparams}}
:<math>\vec x' = \begin{pmatrix} \frac{1}{2}(\alpha^2 -\gamma^2 +\delta^2 -\beta^2) & \frac{1}{2} i (\gamma^2 -\alpha^2 +\delta^2 -\beta^2) & \gamma \delta - \alpha \beta \\
                                 \frac{1}{2}i (\alpha^2 +\gamma^2 -\beta^2 - \delta^2) & \frac{1}{2} (\alpha^2 +\gamma^2 +\beta^2 + \delta^2) & -i (\alpha \beta + \gamma \delta) \\
                                       \beta \delta - \alpha \gamma & i(\alpha \gamma + \beta \delta) & \alpha\delta + \beta\gamma \end{pmatrix}\vec x.</math>

Klein and [[Arnold Sommerfeld|Sommerfeld]] used the parameters extensively in connection with [[Möbius transformation]]s and [[cross-ratio]]s in their discussion of gyroscope dynamics.<ref name="Pennestrì">E. Pennestrì, P.P. Valentini, G. Figliolini, J. Angeles (2016), "Dual Cayley–Klein parameters and Möbius transform: Theory and applications", ''Mechanism and Machine Theory'' '''106'''(January):50-67. {{doi|10.1016/j.mechmachtheory.2016.08.008}}. [https://www.researchgate.net/publication/308037375_Dual_Cayley-Klein_parameters_and_Mobius_transform_Theory_and_applications pdf] available via [[ResearchGate]]</ref><ref>[[Felix Klein]] and [[Arnold Sommerfeld]], ''Über die Theorie des Kreisels'', vol 1. (Teubner, 1897).  Translated (2008) as: ''The Theory of the Top'', vol 1. Boston: Birkhauser. {{ISBN|0817647201}}</ref>

== See also ==
*[[Rotation formalisms in three dimensions]]
*[[Quaternions and spatial rotation]]
*[[Versor]]
*[[Spinors in three dimensions]]
*[[SO(4)]]
*[[3D rotation group]]
*[[Beltrami–Klein model]] (Cayley–Klein model)
*[[Cayley–Klein metric]]

==Notes==
{{notelist}}

==Further reading==
*{{cite book | last=Cartan |first=Élie|title=The Theory of Spinors |author-link=Élie Cartan| publisher=Dover |year=1981 |isbn=0-486-64070-1}}
*{{cite book | last=Hamilton |first=W. R.|author-link=William Rowan Hamilton |title=Elements of Quaternions | publisher=Cambridge University Press|year =1899}}
*{{cite book | last=Haug |first=E.J.|title=Computer-Aided Analysis and Optimization of Mechanical Systems Dynamics. | publisher=Springer-Verlag|year =1984}}
*{{cite journal|last=Garza |first=Eduardo |first2=M. E. |last2=Pacheco Quintanilla |title=Benjamin Olinde Rodrigues, matemático y filántropo, y su influencia en la Física Mexicana |journal=Revista Mexicana de Física |date=June 2011 |pages=109–113 |url=http://rmf.smf.mx/pdf/rmf-e/57/1/57_1_0109.pdf |format=pdf |archiveurl=https://web.archive.org/web/20120423215549/http://rmf.smf.mx/pdf/rmf-e/57/1/57_1_0109.pdf |archivedate=2012-04-23 |language=es |url-status=dead }}
*{{cite journal|last=Shuster |first1=Malcolm D. |title=A Survey of Attitude Representations |journal=Journal of the Astronautical Sciences |volume=41 |issue=4 |date=1993 |pages=439–517 |url=http://malcolmdshuster.com/Pub_1993h_J_Repsurv_scan.pdf |language=en }}
*{{cite journal|last=Dai |first1=Jian S.|title=Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections|journal=Mechanism and Machine Theory|date=October 2015|volume=92|pages=144–152|doi=10.1016/j.mechmachtheory.2015.03.004|url=https://www.sciencedirect.com/science/article/pii/S0094114X15000415|language=en|doi-access=free}}

==References==
{{reflist}}

{{Leonhard Euler}}

{{DEFAULTSORT:Euler-Rodrigues Parameters}}
[[Category:Rotation in three dimensions]]
[[Category:Euclidean symmetries]]
[[Category:Leonhard Euler]]