info prev up next book cdrom email home

Orthogonal Rotation Group

Orthogonal rotation groups are Lie Groups. The orthogonal rotation group $O_3(n)$ is the set of $n\times
n$ Real Orthogonal Matrices.


The orthogonal rotation group $O_3^-(n)$ is the set of $n\times
n$ Real Orthogonal Matrices (having $n(n-1)/2$ independent parameters) with Determinant $-1$.


The orthogonal rotation group $O_3^+(n)$ is the set of $n\times
n$ Real Orthogonal Matrices, having $n(n-1)/2$ independent parameters, with Determinant $+1$. $O_3^+(n)$ is Homeomorphic with ${\it SU}(2)$. Its elements can be written using Euler Angles and Rotation Matrices as

$\displaystyle I$ $\textstyle =$ $\displaystyle \left[{\begin{array}{ccc}1 & 0 & 0 \\  0 & 1 & 0 \\  0 & 0 & 1\end{array}}\right]$ (1)
$\displaystyle R_x(\phi)$ $\textstyle =$ $\displaystyle \left[{\begin{array}{ccc}1 & 0 & 0 \\  0 & \cos\phi & \sin\phi \\  0 & -\sin\phi & \cos\phi\end{array}}\right]$ (2)
$\displaystyle R_y(\theta)$ $\textstyle =$ $\displaystyle \left[{\begin{array}{ccc}\cos\theta & 0 & -\sin\theta \\  0 & 1 & 0 \\  \sin\theta & 0 & \cos\theta\end{array}}\right]$ (3)
$\displaystyle R_z(\psi)$ $\textstyle =$ $\displaystyle \left[{\begin{array}{ccc}\cos\psi & \sin\psi & 0 \\  -\sin\psi & \cos\psi & 0\\  0 & 0 & 1\end{array}}\right].$ (4)


References

Arfken, G. ``Orthogonal Group, $O_3^+$.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 252-253, 1985.

Wilson, R. A. ``ATLAS of Finite Group Representation.'' http://for.mat.bham.ac.uk/atlas/html/contents.html#orth.




© 1996-9 Eric W. Weisstein
1999-05-26