There are good arguiments for using quaternions rather than Eulerian
(or other) angles anyway, this is very well explained in the paper
"*Quaternions *in *molecular modeling*
<http://scholar.google.de/scholar_url?hl=en&q=http://arxiv.org/pdf/physics/0506177&sa=X&scisig=AAGBfm13yuMgR9JJ3LvihnDJIoFFejNTrg&oi=scholarr&ei=Zzs3U__QEYKw7AaL4IDYAw&ved=0CCoQgAMoADAA>'
by Karney.
George
On 03/29/2014 10:22 PM, Edward Berry 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 <[email protected]
<mailto:[email protected]>> 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:
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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), 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
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--
Prof. George M. Sheldrick FRS
Dept. Structural Chemistry,
University of Goettingen,
Tammannstr. 4,
D37077 Goettingen, Germany
Tel. +49-551-39-33021 or -33068
Fax. +49-551-39-22582
