Thanks, lan. For quaternions, it needs w^2+x^2+y^2+z^2=1, thus reduces variables to 3. But fortunately,(x,y,z) here only represents a direction of the rotation angle, and the absolute value of them are not that important. I think a rotation vector and the rotation angle could be a very good representation of any rotation in 3D space, and that's more directviewing than the rotation Matrix which is a Matrix, or Euler Angles, which need three rotation angles. The Exponential Map, it's too complicated, so many mathematics....But, from the definition of q = e^v, I think they are identical. and the direction of v is identical to (x,y,z) in the quaternion.
Regards, Yuan SHANG On Tue, Sep 14, 2010 at 12:09 AM, Ian Tickle <ianj...@gmail.com> wrote: > Hi Yuan > > You might want to look at the 'exponential map' as an alternative to > quaternions. This article evaluates all the various representations > of rotations, including Eulerian angles, polar angles, rotation > matrices, quaternions etc: > > http://webhome.cs.uvic.ca/~blob/courses/485c/notes/pdf/expmap.pdf<http://webhome.cs.uvic.ca/%7Eblob/courses/485c/notes/pdf/expmap.pdf> > > The advantage of the exponential map representation is that only 3 > variables are used, which, as for the Eulerian & polar angle > representations, is exactly the number that are needed to represent an > arbitrary rotation in 3-D so it's an optimally parsimonious > representation. However it doesn't suffer from the well-known > singularities as Eulerian & polar angles (it suffers from different > singularities but it's possible to "sweep them under the carpet"). > Quaternions, as you say, require 4 variables but a constraint is > needed (i.e. normalisation of the vector) to reduce it back to 3. > > Cheers > > -- Ian > > On Mon, Sep 13, 2010 at 4:21 PM, 商元 <shangyuan5...@gmail.com> wrote: > > Dear CCP4 members, > > I have finally used Eleanor's idea, and it works very well. After > > applying SSM, we can get a rotation matrix(M), and a displacement vector > N: > > (Xnew,Ynew,Znew)'=M*(X,Y,Z)'+N > > Then, from the rotation matrix M, we can get its another representative > > format-----quaternion number. The quaternion number has 4 elements, the > > first element(w) of which represents the cosine value of half of the > > rotation anger, and the next 3 elements(x0,y0,z0) represent the rotation > > vector. That means the molecule rotates an angle of acos(w)*2*180/pi > degrees > > around the vector (x0,y0,z0). > > The attached file is the matlab program i wrote to translate M into a > > quaternion number. That should be very easy to be translated into other > > language formats. > > > > Best & regards, > > Yuan SHANG > > > > > > > > On Thu, Aug 19, 2010 at 7:34 PM, Frances C. Bernstein > > <f...@bernstein-plus-sons.com> wrote: > >> > >> Pete Artymiuk wrote: > >> ----------------------- > >> I have an old badly-written Fortran program (I wrote it for a Vax, but > it > >> still compiles and runs in g95 - isn't Fortran wonderful?) that takes > >> Arnott & Dover's polar coordinates and converts them to a helix of any > >> required length* in PDB (or Diamond!) format. > >> ----------------------- > >> > >> Wow, that is an old program! > >> > >> For everyone under the ago of 60 reading this list, Diamond > >> format was the very first PDB format, used for the first 100 > >> or so entries that we had. It was based on the output format > >> of the Diamond real-space refinement program and each line was > >> 132 characters long. Long lines were awkward, in some ways, > >> to handle on computers of that time so we designed what is > >> now known as PDB format. If you want to know more, you can > >> look at page 9 of the September 1974 PDB Newsletter (available > >> on the RCSB web site if you start at > >> > >> > http://www.rcsb.org/pdb/static.do?p=general_information/news_publications/newsletters/newsletter.html#pre1999 > ) > >> for the format of coordinate records in the original format. > >> > >> The reason that I know that there were about 100 entries > >> released in the original format is that I was the one who had > >> to convert them all into the new PDB format in 1976. > >> > >> Frances Bernstein > >> > >> ===================================================== > >> **** Bernstein + Sons > >> * * Information Systems Consultants > >> **** 5 Brewster Lane, Bellport, NY 11713-2803 > >> * * *** > >> **** * Frances C. Bernstein > >> * *** f...@bernstein-plus-sons.com > >> *** * > >> * *** 1-631-286-1339 FAX: 1-631-286-1999 > >> ===================================================== > > > > >
