Ian Tickle wrote:

Ed, the screen z axis is not the same axis in the molecule for the first and 
last
rotations, except in the special case beta = 0 or 180.  The fallacy in your 
argument is
that you're implicitly assuming that rotations commute, whereas of course they 
don't i.e.
Rz.Ry.Rz is not the same as Rz.Rz.Ry unless Ry = unit matrix or 2-fold.  The 
first and
last rotations are both indeed around the screen z axis but the orientation of 
the
molecule has changed because of the intervening y rotation, so the two z 
rotations are not
additive unless beta = 0.  Indeed if beta = 180 the net effect is the 
difference of the
two z rotations.  For other values of beta the net z rotation is a more 
complicated
function of the Eulerian angles.

OK, I need to think about this more when I have time, but at this point
I think it is a semantic difference- For me the first and last rotation are
about the same Z axis because as you say they are both around the screen Z axis
and both operators look like cos,sin,  0, -sin, cos, 0, 0, 0, 1; i.e. rotation
about "THE" z axis; and it is not helpful to consider it a different z axis just
because the atoms moved.
We come up with the same conclusions with our different ways of thinking about 
it:
for one, deriving the concatenated simple operators to represent a general 
rotation,
and the commutativity: I would say the operators do not commute as long as the 
axes
they rotate about are kept fixed, but if the axes rotate the same as the 
molecule
then the z axis will always be passing through the atoms the same way.
Then rotations would commute, because the z axis would always represent the same molecular axis. Which I am sure is NOT what you meant by saying "new z axis".

Thanks,
eab


HTH!

Cheers

-- Ian


On 29 March 2014 21:22, Edward Berry <ber...@upstate.edu 
<mailto:ber...@upstate.edu>> wrote:

    Thanks, Ian!
    I agree it may have to do with being used to computer graphics, where x,y,z 
are fixed
    and the coordinates rotate. But it still doesn't make sense:

    If the axes rotate along with the molecule, in the catenated operators of 
the polar
    angles, after the first two operators the z axis would still be passing 
through the
    molecule in the same way it did originally, so rotation about z in the 
third step
    would have the same effect as rotating about z in the original orientation.
    Or in eulerian angles, if the axes rotate along with the molecule at each 
step, the z
    axis in the third step passes through the molecule in the same way it did 
in the first
    step, so alpha and gamma would have the same effect and be additive.  In 
other words
    if the axes we are rotating about rotate themselves in lock step with the 
molecule, we
    can never rotate about any molecular axes except those that were originally 
along x,
    y, and z (because they will always be alng x,y,z) (I mean using simple 
rotations about
    principle axes: cos sin -sin cos).
    Maybe I need to think about the concept of molecular axes as opposed to lab 
axes. The
    lab axes are defined relative to the world and never change. The molecular 
axis is
    defined by how the lab axis passes through the molecule, and changes as the 
molecule
    rotates relative to the lab axis.  But then the molecular axis seems 
redundant, since
    I can understand the operator fine just in terms of the rotating 
coordinates and the
    fixed lab axes. Except the "desired rotation axis" of the polar angles 
would be a
    molecular axis, since it is defined by a line through the atoms that we 
want to rotate
    about. So it rotates along with the coordinates during the first two 
operations, which
    align it with the old lab Z axis (which is the new molecular z axis?) . . . 
  You see
    my confusion.
    Or think about the math one step at a time, and suppose we look at the 
coordinates
    after each step with a graphics program keeping the x axis horizontal, y 
axis
    vertical, and z axis coming out of the plane. For Eulerian angles, the 
first rotation
    will be about Z. This will leave the z coordinate of each atom unchanged 
and change
    the x,y coordinates.  If we give the new coordnates to the graphics 
program, it will
    display the atoms rotated in the plane of the screen (about the z axis 
perpendicular
    to the screen).  The next rotation will be about y, will leave the y 
coordinates
    unchanged, and we see rotation about the vertical axis. Final rotation 
about z is in
    the plane of the screen again, although this represents rotation about a 
different
    axis of the molecule.  My view would be to say the first and final rotation 
are
    rotating about the perpendicular to the screen which we have kept equal to 
the z axis,
    and it is the same z axis.

    Ed

     >>> Ian Tickle __ 03/29/14 1:39 PM >>>

    Hi Edward

    As far as Eulerian rotations go, in the 'Crowther' description the 2nd 
rotation can
    occur either about the new (rotated) Y axis or about the old (unrotated) Y 
axis, and
    similarly for the 3rd rotation about the new or old Z.  Obviously the same 
thing
    applies to polar angles since they can also be described in terms of a 
concatenation
    of rotations (5 instead of 3).  So in the 'new' description the rotation 
axes do
    change: they are rotating with the molecule.

    For reasons I find hard to fathom virtually all program documentation seems 
to
    describe it in terms of rotations about already-rotated angles.  If as you 
say you
    find this confusing then you are not alone!  However it's very easy to 
change from a
    description involving 'new' axes to one involving 'old' axes: you just 
reverse the
    order of the angles.  So in the Eulerian case a rotation of alpha around Z, 
then beta
    around new Y, then gamma around new Z (i.e. 'Crowther' convention) is 
completely
    equivalent to a rotation of gamma around Z, then beta around _old_ Y, then 
alpha
    around _old_ Z.

    So if you're used to computer graphics where the molecules rotate around 
the fixed
    screen axes (rotation around the rotating molecular axes would be very 
confusing!)
    then it seems to me that the 'old' description is much more intuitive.

    Cheers

    -- Ian


    On 27 March 2014 22:18, Edward A. Berry <ber...@upstate.edu
    <mailto:ber...@upstate.edu>> wrote:

                According to the html-side the 'visualisation' includes two
                back-rotations in addition to what you copied here, so there is 
at
                least one difference to the visualisation of the Eulerian 
angles.


        Right- it says:
        "This can also be visualised as

        rotation ϕ about Z,
        rotation ω about the new Y,

        rotation κ about the new Z,

        rotation (-ω) about the new Y,
        rotation (-ϕ) about the new Z."

        The first two and the last two rotations can be seen as a "wrapper" 
which
        first transforms the coordinates so the rotation axis lies along z, 
then after
        the actual kappa rotation is carried out (by rotation about z), 
transforms the
        rotated molecule back to the otherwise original position.
        Or which transforms the coordinate system to put Z along the rotation 
axis, then after
        the rotation by kappa about z transforms back to the original 
coordinate system.

        Specifically,
           rotation ϕ about Z brings the axis into the x-z plane so that

           rotation ω about the Y brings the axis onto the z axis, so that

           rotation κ about Z is doing the desired rotation about a line that 
passes through
             the  atoms in the same way the desired lmn axis did in the 
original orientation;

           Then the 4'th and 5'th operations are the inverse of the 2nd and 
first,
            bringing the rotated molecule back to its otherwise original 
position

        I think all the emphasis on "new" y and "new" z is confusing. If we are 
rotating
        the molecule (coordinates), then the axes don't change. They pass 
through the molecule
        in a different way because the molecule is rotated, but the axes are 
the same.
        After the first two rotations the Z axis passes along the desired 
rotation axis,
        but the Z axis has not moved, the coordinates (molecules) have.
        Of course there is the alternate interpretation that we are doing a 
change of
        coordinates and expressing the unmoved molecular coordinates relative 
to new
        principle axes. but if we are rotating the coordinates about the axes 
then the
        axes should remain the same, shouldn't they? Or maybe there is yet 
another way of
        looking at it.



        Tim Gruene wrote:

            -----BEGIN PGP SIGNED MESSAGE-----
            Hash: SHA1

            Dear Qixu Cai,

            maybe the confusion is due to that your quote seems incomplete.
            According to the html-side the 'visualisation' includes two
            back-rotations in addition to what you copied here, so there is at
            least one difference to the visualisation of the Eulerian angles.

            Best,
            Tim

            On 03/27/2014 07:11 AM, Qixu Cai wrote:

                Dear all,

                  From the definition of CCP4
                (http://www.ccp4.ac.uk/html/__rotationmatrices.html
                <http://www.ccp4.ac.uk/html/rotationmatrices.html>), the polar 
angle
                (ϕ, ω, κ) can be visualised as rotation ϕ about Z, rotation ω 
about
                the new Y, rotation κ about the new Z. It seems the same as the 
ZXZ
                convention of eulerian angle definition. What's the difference
                between the CCP4 polar angle definition and eulerian angle ZXZ
                definition?

                And what's the definition of polar angle XYK convention in GLRF
                program?

                Thank you very much!

                Best wishes,


            - --
            - --
            Dr Tim Gruene
            Institut fuer anorganische Chemie
            Tammannstr. 4
            D-37077 Goettingen

            GPG Key ID = A46BEE1A

            -----BEGIN PGP SIGNATURE-----
            Version: GnuPG v1.4.12 (GNU/Linux)
            Comment: Using GnuPG with Icedove - http://www.enigmail.net/

            iD8DBQFTNAz0UxlJ7aRr7hoRAj7IAK__Ds/J0L/__XCYPpQSyB2BPJ2uWV2lVgCeKD72
            0DemwU57v6fekF6iOC4/5IA=
            =PeT9
            -----END PGP SIGNATURE-----


    __


Reply via email to