3$\times$3 Matrices

The matrix construct allows one to specify a 3$\times$3 real matrix. There are several modes available to specify rotation matrices. An arbitrary 3$\times$3 matrix can be specified by using the MATRix construct. Note: all rotations are counterclockwise.

$<$matrix$>$:==

AXIS $<$vector$>$ $<$real$>$

specifies input through the axis vector and the rotation angle $\kappa$.

EULEr $<$vector$>$

specifies input through the Eulerian angles $\theta_1$,$\theta_2$,$\theta_3$. $\theta_1$ is the rotation around the z axis, $\theta_2$ around the new x axis, and $\theta_3$ around the new z axis.

LATTman $<$vector$>$

specifies input through Lattman's angles ( $\theta_+=\theta_1+\theta_3$, $\theta_2$, $\theta_-=\theta_1- \theta_3$).

MATRix $<$vector$>$ $<$vector$>$ $<$vector$>$

specifies direct input of the matrix by three 3d-vectors.

QUATernions

$<$real$>$ $<$real$>$ $<$real$>$ $<$real$>$ specifies quaternions $q_0$, $q_1$, $q_2$, $q_3$, which are defined as

$$q_0 = cos({\theta_2 \over 2})cos({{\theta_1+\theta_3} \over 2})$$

$$q_1 = sin({\theta_2 \over 2})cos({{\theta_1-\theta_3} \over 2})$$

$$q_2 = sin({\theta_2 \over 2})sin({{\theta_1-\theta_3} \over 2})$$

$$q_3 = cos({\theta_2 \over 2})sin({{\theta_1+\theta_3} \over 2})$$ (2.1)

with the constraint

$$q^2_{0}+q^2_{1}+q^2_{2}+q^2_{3}=1$$ (2.2)

SPHErical $<$vector$>$

specifies input through spherical polar angles $\psi$, $\phi$, $\kappa$. $\psi$ and $\phi$ specify the rotation axis. $\psi$ is the inclination versus the y-axis; $\phi$ is the azimuthal angle, i.e., the angle between the x-axis and the projection of the axis into the x,z plane; and $\kappa$ is the rotation around the rotation axis.

In the following example, a rotation matrix is specified by a rotation axis (2,3,4) and a rotation angle (40$^{\circ}$) around the axis (counterclockwise rotation):

AXIS=( 2, 3, 4 )  40.

In the next example, a matrix is specified by direct input:

MATRix=( 1. 3. 5. )
       ( 4. 2. 1. )
       ( 2. 1. 8. )

The last example shows how to specify a rotation matrix by using the Eulerian angles $\theta_1$,$\theta_2$,$\theta_3$:

EULEr=( 30. 40. 120. )