Euler Parameters

The four parameters , , , and 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

$\begin{displaymath} (e_0, {\bf e})=e_0+e_1i+e_2j+e_3k. \end{displaymath}$

(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

$\begin{displaymath} {e_0}^2+{\bf e}\cdot{\bf e} = {e_0}^2+{e_1}^2+{e_2}^2+{e_3}^2 = 1 \end{displaymath}$

(4)

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

$\begin{displaymath} \cos\phi = 2{e_0}^2-1 = {e_0}^2-{\bf e}\cdot{\bf e} = {e_0}^2-{e_1}^2-{e_2}^2-{e_3}^2 \end{displaymath}$

(5)

$\begin{displaymath} \hat {\bf n}\sin\phi = 2{\bf e}e_0. \end{displaymath}$

(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

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

(11)

and the Rotation Matrix becomes

$\begin{displaymath} \left[{\matrix{x'\cr y'\cr z'\cr}}\right] = {\hbox{\sf A}}\left[{\matrix{x\cr y\cr z\cr}}\right], \end{displaymath}$

(12)

where the elements of the matrix are

$\begin{displaymath} a_{ij} = \delta_{ij}({e_0}^2-e_ke_k)+2e_ie_j+2\epsilon_{ijk}e_0e_k. \end{displaymath}$

(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)