The main reason for using Eulerian (or polar) angles is speed (not for nothing is Crowther's implementation called the "Fast Rotation Function"). Expression of the rotation in terms of Eulerian or polar angles makes it possible to express the Patterson functions in terms of orthogonal spherical harmonics and thus decompose the problem into a set of simpler problems involving spherical Bessel functions for the radial variable and Fourier summations for the angular variables of the spherical harmonics, which can then take advantage of the FFT algorithm. I'm not aware of any algorithm that makes use of quaternions which can decompose the problem similarly in order to take advantage of the speed of FFT to do most of the work.
Cheers -- Ian On 29 March 2014 21:41, George Sheldrick <gshe...@shelx.uni-ac.gwdg.de>wrote: > 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 <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), 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/ >>> >>> iD8DBQFTNAz0UxlJ7aRr7hoRAj7IAKDs/J0L/XCYPpQSyB2BPJ2uWV2lVgCeKD72 >>> 0DemwU57v6fekF6iOC4/5IA= >>> =PeT9 >>> -----END PGP SIGNATURE----- >>> >>> > > > -- > 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 > >
