subtom_transform_motl
Apply a rotation and a shift to a MOTL file.
subtom_transform_motl(
'input_motl_fn', input_motl_fn (''),
'output_motl_fn', output_motl_fn (''),
'shift_x', shift_x (0),
'shift_y', shift_y (0),
'shift_z', shift_z (0),
'rotate_phi', rotate_phi (0),
'rotate_psi', rotate_psi (0),
'rotate_theta', rotate_theta (0),
'rand_inplane', rand_inplane (0))
Takes the motl given by input_motl_fn, and first applies the rotation
described by the Euler angles rotate_phi, rotate_psi, rotate_theta,
which correspond to an in-plane spin, azimuthal, and zenithal rotation
respectively. Then a translation specified by shift_x, shift_y,
shift_z, is applied to the existing translation. Finally the resulting
transformed motive list is written out as output_motl_fn. Keep in mind that
the motive list transforms describe the alignment of the reference to each
particle, but that the rotation and shift here describe an affine transform of
the reference to a new reference. If rand_inplane evaluates to true as a
boolean, then the final Euler angle (phi in AV3 notation, and psi/spin/inplane
in other notations) will be randomized after the given transform.
Explanation of how the transforms are derived
The alignments in the motive list describe the rotation and shift of the
reference to each particle. Since this is a rotation followed by a shift
we can describe this as an affine transform 4x4 matrix as follows:
[ R_1 T_1 ] [ V ] [ P ]
[ ] x [ ] = [ ] (1)
[ 0 1 ] [ 1 ] [ 1 ]
Where R_1 is the rotation matrix described by motl(17:19, 1) and T_1 is
the shift column vector described by motl(11:13, 1), and finally V and P
are the coordinates in the reference, and the reference in register with
the particle respectively.
Likewise the rotation and shift we apply to the reference to get a new
updated reference is also an affine transform as follows:
[ R_2 T_2 ] [ V ] [ V' ]
[ ] x [ ] = [ ] (2)
[ 0 1 ] [ 1 ] [ 1 ]
Where V' is our new reference. Therefore the affine transform we want to
find and place in our updates motive list is:
[ R_? T_? ] [ V' ] [ P ]
[ ] x [ ] = [ ] (3)
[ 0 1 ] [ 1 ] [ 1 ]
The most logical path is to go from V' to V and V to P, so we have to
invert the affine transform in (2), and then left multiply it by the
transform in (1). To find the inverse affine transform of (2) we have
that:
R_2 * V + T_2 = V'
R_2 * V = V' - T_2
V = R_2^-1 * (V' - T_2)
V = (R_2^-1 * V') - (R_2^-1 * T_2)
[ R_2^-1 -R_2^-1 * T_2 ] [ V' ] [ V ]
[ ] x [ ] = [ ] (4)
[ 0 1 ] [ 1 ] [ 1 ]
So we have that:
[ R_1 T_1 ] [ R_2^-1 -R_2^-1 * T_2 ] [ V' ] [ P ]
[ ] x [ ] x [ ] = [ ] (5)
[ 0 1 ] [ 0 1 ] [ 1 ] [ 1 ]
[ R_1 * R_2^-1 -R_1 * R_2^-1 * T_2 + T_1 ] [ V' ] = [ P ]
[ ] x [ ] = [ ] (6)
[ 0 1 ] [ 1 ] = [ 1 ]
And finally:
R_? = R_1 * R_2^-1
T_? = T_1 - R_1 * R_2^-1 * T_2
Example
subtom_transform_motl(...
'input_motl_fn', 'combinedmotl/allmotl_1.em', ...
'output_motl_fn', 'combinedmotl/allmotl_1_shifted.em', ...
'shift_x', 5, ...
'shift_y', 5, ...
'shift_z', -3, ...
'rotate_phi', 60, ...
'rotate_psi', 15, ...
'rotate_theta', 0.5, ...
'rand_inplane', 0)