Miguel, we have this book at St. Olaf. I'll take a look.
Anthony, do you see any practical problems with delivering RxRyRz information to the user, or is this more of a concern in principle? Does it make any difference, since Jmol doesn't use RxRyRz for anything right now and wouldn't except for informing the user of a simple set of rotations that will generate one specific structure? Probably when output, these would be rounded to the nearest integer anyway. (Though I don't have Miguel's thoughts on that.)
No one is suggesting that Jmol use RxRyRz for representing arbitrary rotations internally.
Bob
Anthony Stone wrote:
There are several further references at the end of the Web page that Miguel mentions. A comprehensive book on the subject is S. L. Altmann, "Rotations, Quaternions and Double Groups", Oxford University Press. Altmann explains why the RxRyRz or yaw-pitch-roll form is unsatisfactory for arbitrary rotations. (It's fine for small rotations, and natural for controlling aeroplanes.)
There are numerous pitfalls for the unwary in this area. For Euler angles, the rotations are about z,y,z (Whitaker convention) or z,x,z (Goldstein convention). An Euler rotation (phi,theta,psi) may be achieved by rotating first through phi about z, then through theta about the _rotated_ y (or x), and then through psi about the _rotated_ z. Alternatively, and with exactly the same result, one may rotate through psi about z, then through theta about the _original_ y axis, then through phi about the _original_ z axis.
For many purposes, the angle-axis formulation, which is closely related to the quaternion approach, is the simplest and most convenient. The matrix describing a rotation through psi about an axis described by the unit vector (nx,ny,nz) is
1-2*(ny^2+nz^2)*s^2 -nz*S+2*nx*ny*s^2 ny*S+2*nz*nx*s^2
nz*S+2nx*ny*s^2 1-2*(nz^2+nx^2)*s^2 -nx*S+2+ny*nz*s^2
-ny*S+2nz*nx*s^2 nx*S+2*ny*nz*s^2 1-2*(nx^2+ny^2)*s^2
where S=sin psi and s=sin(psi/2).
Anthony
At 07:19 on 5 October, Miguel wrote:
> Egon wrote:
> > > >> I suggest that Jmol should use one of these standard, well-documented,
> >> approaches rather than introducing something different.
> >
> > Can you send us some pointers to such documentation?
> > Egon,
> > I found the following reference by using goole for 'euler angle rotation'
> > http://mathworld.wolfram.com/EulerAngles.html
> > other googling for 'euler angle rotation x convention' generated other
> references.
> > > Miguel
> >
-- Robert M. Hanson, [EMAIL PROTECTED], 507-646-3107 Professor of Chemistry, St. Olaf College 1520 St. Olaf Ave., Northfield, MN 55057 mailto:[EMAIL PROTECTED] http://www.stolaf.edu/people/hansonr
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