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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

$\displaystyle e_0$ $\textstyle \equiv$ $\displaystyle \cos\left({\phi\over 2}\right)$ (1)
$\displaystyle {\bf e}$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{c}e_1\\  e_2\\  e_3\end{array}\right] = \hat {\bf n} \sin\left({\phi\over 2}\right),$ (2)

and are a Quaternion in scalar-vector representation
(e_0, {\bf e})=e_0+e_1i+e_2j+e_3k.
\end{displaymath} (3)

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+{\bf e}\cdot{\bf e} = {e_0}^2+{e_1}^2+{e_2}^2+{e_3}^2 = 1
\end{displaymath} (4)

(Goldstein 1980, p. 153). The rotation angle is then related to the Euler parameters by
\cos\phi = 2{e_0}^2-1 = {e_0}^2-{\bf e}\cdot{\bf e} = {e_0}^2-{e_1}^2-{e_2}^2-{e_3}^2
\end{displaymath} (5)

\hat {\bf n}\sin\phi = 2{\bf e}e_0.
\end{displaymath} (6)

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

$\displaystyle e_0$ $\textstyle =$ $\displaystyle \cos[{\textstyle{1\over 2}}(\phi+\psi)]\cos({\textstyle{1\over 2}}\theta)$ (7)
$\displaystyle e_1$ $\textstyle =$ $\displaystyle \sin[{\textstyle{1\over 2}}(\phi-\psi)]\sin({\textstyle{1\over 2}}\theta)$ (8)
$\displaystyle e_2$ $\textstyle =$ $\displaystyle \cos[{\textstyle{1\over 2}}(\phi-\psi)]\sin({\textstyle{1\over 2}}\theta)$ (9)
$\displaystyle e_3$ $\textstyle =$ $\displaystyle \sin[{\textstyle{1\over 2}}(\phi+\psi)]\cos({\textstyle{1\over 2}}\theta)$ (10)

(Goldstein 1980, p. 155).

Using the Euler parameters, the Rotation Formula becomes

{\bf r}' = {\bf r}({e_0}^2-{e_1}^2-{e_2}^2-{e_3}^2)+2{\bf e}({\bf e}\cdot{\bf r})+({\bf r}\times\hat{\bf n})\sin\phi,
\end{displaymath} (11)

and the Rotation Matrix becomes
\left[{\matrix{x'\cr y'\cr z'\cr}}\right] = {\hbox{\sf A}}\left[{\matrix{x\cr y\cr z\cr}}\right],
\end{displaymath} (12)

where the elements of the matrix are
a_{ij} = \delta_{ij}({e_0}^2-e_ke_k)+2e_ie_j+2\epsilon_{ijk}e_0e_k.
\end{displaymath} (13)

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
$\displaystyle a_{11}$ $\textstyle =$ $\displaystyle {e_0}^2+{e_1}^2-{e_2}^2-{e_3}^2$ (14)
$\displaystyle a_{12}$ $\textstyle =$ $\displaystyle 2(e_1e_2+e_0e_3)$ (15)
$\displaystyle a_{13}$ $\textstyle =$ $\displaystyle 2(e_1e_3-e_0e_2)$ (16)
$\displaystyle a_{21}$ $\textstyle =$ $\displaystyle 2(e_1e_2-e_0e_3)$ (17)
$\displaystyle a_{22}$ $\textstyle =$ $\displaystyle {e_0}^2-{e_1}^2+{e_2}^2-{e_3}^2$ (18)
$\displaystyle a_{23}$ $\textstyle =$ $\displaystyle 2(e_2e_3+e_0e_1)$ (19)
$\displaystyle a_{31}$ $\textstyle =$ $\displaystyle 2(e_1e_3+e_0e_2)$ (20)
$\displaystyle a_{32}$ $\textstyle =$ $\displaystyle 2(e_2e_3-e_0e_1)$ (21)
$\displaystyle a_{33}$ $\textstyle =$ $\displaystyle {e_0}^2-{e_1}^2-{e_2}^2+{e_3}^2.$ (22)

See also Euler Angles, Quaternion, Rotation Matrix


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.

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© 1996-9 Eric W. Weisstein