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Euler Parameters


The four parameters e_0, e_1, e_2, and e_3 describing a finite rotation about an arbitrary axis. The Euler parameters are defined by

e_0=cos(phi/2)
(1)
e=[e_1; e_2; e_3]
(2)
=n^^sin(phi/2),
(3)

where n^^ is the unit normal vector, and are a quaternion in scalar-vector representation

 (e_0,e)=e_0+e_1i+e_2j+e_3k.
(4)

Because Euler's rotation theorem states that an arbitrary rotation may be described by only three parameters, a relationship must exist between these four quantities

e_0^2+e·e=e_0^2+e_1^2+e_2^2+e_3^2
(5)
=1
(6)

(Goldstein 1980, p. 153). The rotation angle is then related to the Euler parameters by

cosphi=2e_0^2-1
(7)
=e_0^2-e·e
(8)
=e_0^2-e_1^2-e_2^2-e_3^2
(9)

and

 n^^sinphi=2ee_0.
(10)

The Euler parameters may be given in terms of the Euler angles by

e_0=cos[1/2(phi+psi)]cos(1/2theta)
(11)
e_1=cos[1/2(phi-psi)]sin(1/2theta)
(12)
e_2=sin[1/2(phi-psi)]sin(1/2theta)
(13)
e_3=sin[1/2(phi+psi)]cos(1/2theta)
(14)

(Goldstein 1980, p. 155).

Using the Euler parameters, the rotation formula becomes

 r^'=r(e_0^2-e_1^2-e_2^2-e_3^2)+2e(e·r)+(r×n^^)sinphi,
(15)

and the rotation matrix becomes

 [x^'; y^'; z^']=A[x; y; z],
(16)

where the elements of the matrix are

 a_(ij)=delta_(ij)(e_0^2-e_ke_k)+2e_ie_j+2epsilon_(ijk)e_0e_k.
(17)

Here, Einstein summation has been used, delta_(ij) is the Kronecker delta, and epsilon_(ijk) is the permutation symbol. Written out explicitly, the matrix elements are

a_(11)=e_0^2+e_1^2-e_2^2-e_3^2
(18)
a_(12)=2(e_1e_2+e_0e_3)
(19)
a_(13)=2(e_1e_3-e_0e_2)
(20)
a_(21)=2(e_1e_2-e_0e_3)
(21)
a_(22)=e_0^2-e_1^2+e_2^2-e_3^2
(22)
a_(23)=2(e_2e_3+e_0e_1)
(23)
a_(31)=2(e_1e_3+e_0e_2)
(24)
a_(32)=2(e_2e_3-e_0e_1)
(25)
a_(33)=e_0^2-e_1^2-e_2^2+e_3^2.
(26)

See also

Euler Angles, Quaternion, Rotation Formula, Rotation Matrix

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 198-200, 1985.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.Landau, L. D. and Lifschitz, E. M. Mechanics, 3rd ed. Oxford, England: Pergamon Press, 1976.

Referenced on Wolfram|Alpha

Euler Parameters

Cite this as:

Weisstein, Eric W. "Euler Parameters." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerParameters.html

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