JISCMail - CCP4BB Archives

"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."

Maybe in my thinking I am going in reverse order- didn't pay attention to sign of the angles.
If you think of the Eulerian navigator at http://sb20.lbl.gov/berry/Euler2.gif
it is obvious that the same setting on each of the three angles will give the same orientation. Now assuming the outside frame is fixed (bolted to the bench) and you adjust the angles starting with the inside ring, you will be using lab axes all the way. If you first adjust the outside ring, then the next two rotation will be about "new" axes.
Computationally it must be much easier to use "old" or "Lab" axes. In the case of polar coordinates, the whole problem involves rotation by kappa about an axis at odd angles to x,y,z. If in order to do that, we introduce 3 more rotations about non-standard axes, and the same for each of them, we will never get there!

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 <[log in to unmask]> 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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