Euler–Rodrigues formula

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Inmathematics andmechanics, theEuler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based onRodrigues' rotation formula, but uses a different parametrization.

The rotation is described by fourEuler parameters due toLeonhard Euler. TheRodrigues' rotation formula (named afterOlinde Rodrigues), a method of calculating the position of a rotated point, is used in some software applications, such asflight simulators andcomputer games.

Definition

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A rotation about the origin is represented by four real numbers,a, b, c, d such that

a^{2}+b^{2}+c^{2}+d^{2}=1.

When the rotation is applied, a point at position ${\vec {x}}$ rotates to its new position,^[1]

{\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}}.

Vector formulation

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The parametera may be called thescalar parameter and ${\vec {\omega }}=(b,c,d)$ thevector parameter. In standard vector notation, the Rodrigues rotation formula takes the compact form^{[citation needed]}

${\vec {x}}'={\vec {x}}+2a({\vec {\omega }}\times {\vec {x}})+2\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)$

Symmetry

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The parameters(a, b, c, d) and(−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

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The composition of two rotations is itself a rotation. Let(a₁, b₁, c₁, d₁) and(a₂, b₂, c₂, d₂) be the Euler parameters of two rotations. The parameters for the compound rotation (rotation 2 after rotation 1) are as follows:

{\begin{aligned}a&=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2};\\b&=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2};\\c&=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2};\\d&=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}.\end{aligned}}

It is straightforward, though tedious, to check thata² +b² +c² +d² = 1. (This is essentiallyEuler's four-square identity.)

Rotation angle and rotation axis

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Any central rotation in three dimensions is uniquely determined by its axis of rotation (represented by aunit vectork→ = (k_x,k_y,k_z)) and the rotation angleφ. The Euler parameters for this rotation are calculated as follows:

{\begin{aligned}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{aligned}}

Note that ifφ 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,(−a, −b, −c, −d); they represent the same rotation.

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

Connection with quaternions

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The Euler parameters can be viewed as the coefficients of aquaternion; the scalar parametera is the real part, the vector parametersb,c,d are the imaginary parts.Thus we have the quaternion

q=a+bi+cj+dk,

which is a quaternion of unit length (orversor) since

\left\|q\right\|^{2}=a^{2}+b^{2}+c^{2}+d^{2}=1.

Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions $q=q_{2}\,q_{1}$ . 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 unit quaternionq:

q=\cos {\frac {\varphi }{2}}+\mathbf {k} \sin {\frac {\varphi }{2}}=a+{\vec {\omega }},

where:

$\cos {\tfrac {\varphi }{2}}$ is the scalar (real) part,
$\mathbf {k} \sin {\tfrac {\varphi }{2}}$ is the vector (imaginary) part,
$\mathbf {k}$ is a unit vector representing the axis of rotation.

For a 3D vector $\mathbf {v}$ identified as a pure quaternion, the rotated vector $\mathbf {v} '$ is given by:

\mathbf {v} '=q\mathbf {v} q^{-1},

where $q^{-1}=\cos {\tfrac {\varphi }{2}}-\mathbf {k} \sin {\tfrac {\varphi }{2}}=a-{\vec {\omega }}.$

The previous equation can be shown to be equivalent to the Euler-Rodrigues equation by exploiting thecommutation relationship

q_{2}q_{1}=q_{1}q_{2}-2\mathbf {k_{1}} \times \mathbf {k_{2}}

and the fact that $qq^{-1}=1$ as follows:

q\mathbf {v} q^{-1}=q(q^{-1}\mathbf {v} -2(-{\vec {\omega }})\times \mathbf {v} )

{\phantom {q\mathbf {v} q^{-1}}}={\cancel {qq^{-}1}}\mathbf {v} +\underbrace {(a+{\vec {\omega }})} _{q}(2{\vec {\omega }}\times \mathbf {v} )

{\phantom {q\mathbf {v} q^{-1}}}=\mathbf {v} +2a({\vec {\omega }}\times \mathbf {v} )+2({\cancel {-{\vec {\omega }}\cdot ({\vec {\omega }}\times \mathbf {v} )}}+{\vec {\omega }}\times ({\vec {\omega }}\times \mathbf {v} ))

{\phantom {q\mathbf {v} q^{-1}}}=\mathbf {v} +2a({\vec {\omega }}\times \mathbf {v} )+2({\vec {\omega }}\times ({\vec {\omega }}\times \mathbf {v} ))

where, in the second to last equality, the dot product of two orthogonal vectors is zero.

Connection with SU(2) spin matrices

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TheLie group SU(2) can be used to represent three-dimensional rotations in complex2 × 2 matrices. The SU(2)-matrix corresponding to a rotation, in terms of its Euler parameters, is

U={\begin{pmatrix}\ a-di&-c-bi\\c-bi&a+di\end{pmatrix}}.

which can be written as the sum

{\begin{aligned}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{aligned}}

where theσ_i are thePauli spin matrices.

Rotation is given by $X^{\prime }\equiv (x_{1}^{\prime }\sigma _{x}+x_{2}^{\prime }\sigma _{y}+x_{3}^{\prime }\sigma _{z})=U\;X\;U^{\dagger }=(a\,I-ib\,\sigma _{x}-ic\,\sigma _{y}-id\,\sigma _{z})(x_{1}\sigma _{x}+x_{2}\sigma _{y}+x_{3}\sigma _{z})(a\,I+ib\,\sigma _{x}+ic\,\sigma _{y}+id\,\sigma _{z})$ , 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 thespin group Spin(3), which maps by a double cover mapping to a rotation in theorthogonal group SO(3). This realizes $\mathbb {R} ^{3}$ as the unique three-dimensional irreduciblerepresentation of theLie group SU(2) ≈ Spin(3).

Cayley–Klein parameters

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The elements of the matrix $U {\displaystyle U}$ are known as theCayley–Klein parameters, after the mathematiciansArthur Cayley andFelix Klein,^[a]

{\begin{aligned}\alpha &=a-di&\beta &=-c-bi\\\gamma &=c-bi&\delta &=\ a+di\end{aligned}}

In terms of these parameters the Euler–Rodrigues formula can then also be written^[2]^[6]^[a]

{\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}}.

Klein andSommerfeld used the parameters extensively in connection withMöbius transformations andcross-ratios in their discussion of gyroscope dynamics.^[3]^[7]

Notes

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^^a ^bGoldstein (1980)^[2] considers apassive (contravariant, or "alias") transformation, rather than theactive (covariant, or "alibi") transformation here.
His matrix $A {\displaystyle A}$ 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 $-\varphi$ rather than $\varphi$ . Taking this into account, it is apparent that his $e_{1}$ , $e_{2}$ , and $e_{3}$ in eqn 4-67 (p.153) are equal to $b {\displaystyle b}$ , $c {\displaystyle c}$ , and $d {\displaystyle d}$ here. However his $\alpha$ , $\beta$ , $\gamma$ , and $\delta$ , the elements of his matrix $Q {\displaystyle Q}$ , correspond to the elements of matrix $U^{\dagger }$ here, rather than the matrix $U {\displaystyle U}$ . This then gives his parametrization
${\begin{aligned}\alpha &=\;\;a+di&\beta &=c+bi\\\gamma &=-c+bi&\delta &=a-di\end{aligned}}$
In consequence, while his formula (4-64) is identical symbol-by-symbol to the transformation matrix given here, using his definitions for $\alpha$ , $\beta$ , $\gamma$ , and $\delta$ it gives his matrix $A {\displaystyle A}$ , whereas the definitions based on the matrix $U {\displaystyle U}$ above lead to the (active) Euler–Rodrigues matrix presented here.
Pennestrì et al (2016)^[3] similarly define their $\alpha$ , $\beta$ , $\gamma$ , and $\delta$ in terms of the passive matrix $Q {\displaystyle Q}$ rather than the active matrix $U {\displaystyle U}$ .
The parametrization here accords with that used in eg Sakurai and Napolitano (2020),^[4] p. 165, and Altmann (1986),^[5] eqn. 5 p. 113 / eqn. 9 p. 117.

References

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^e.g.Felix Klein (1897),The mathematical theory of the top, New York: Scribner. p.4
^^a ^bGoldstein, H. (1980), "The Cayley-Klein Parameters and Related Quantities". §4-5 inClassical Mechanics, 2nd ed. Reading, MA: Addison-Wesley. p. 153
^^a ^bE. Pennestrì, P.P. Valentini, G. Figliolini, J. Angeles (2016), "Dual Cayley–Klein parameters and Möbius transform: Theory and applications",Mechanism and Machine Theory106(January):50-67.doi:10.1016/j.mechmachtheory.2016.08.008.pdf available viaResearchGate
^Sakurai, J. J.; Napolitano, Jim (2020).Modern Quantum Mechanics (3rd ed.). Cambridge.ISBN 978-1-108-47322-4.OCLC 1202949320.{{cite book}}: CS1 maint: location missing publisher (link)
^Altmann, S. (1986),Rotations, Quaternions and Double Groups. Oxford:Clarendon Press.ISBN 0-19-855372-2
^Weisstein, Eric W.,Cayley-Klein Parameters,MathWorld. Accessed 2024-05-10
^Felix Klein andArnold Sommerfeld,Über die Theorie des Kreisels, vol 1. (Teubner, 1897). Translated (2008) as:The Theory of the Top, vol 1. Boston: Birkhauser.ISBN 0817647201