Re: [PyMOL] Algorithm to Rotate One Set of Vectors onto Another

Buz,

Tsjerk's answer is right on.  This has already been implemented for PyMOL a=
s a=20
plugin.  See Kabsch/optAlign from "cealign," here:  =20
  http://www.pymolwiki.org/index.php/Kabsch
The code is open-source and so can be applied elsewhere.

Another simple method is to simply calculate the SVD of the correlation=20
matrix.  Then multiply the right and left singular vectors by each other --=
=20
that will yield the DxD rotation matrix (where D is the dimension of your=20
vector sets).  (This is how Kabsch/optAlign works.)

=2D- Jason

On Monday 28 January 2008 10:48:29 pm=20
pym...@li... wrote:
> ------------------------------
>
> Message: 3
> Date: Sat, 26 Jan 2008 10:04:06 +0100
> From: "Tsjerk Wassenaar" <ts...@gm...>
> Subject: Re: [PyMOL] Algorithm to Rotate One Set of Vectors onto
> =A0=A0=A0=A0=A0=A0=A0=A0Another
> To: "Buz Barstow" <bu...@ma...>
> Cc: pym...@li...
> Message-ID:
> =A0=A0=A0=A0=A0=A0=A0=A0<8ff898150801260104k4902e5f0tcb94e8ebbbf1e027@mai=
l.gmail.com>
> Content-Type: text/plain; charset=3DISO-8859-1
>
> Hi Buz,
>
> To my opinion, this is not the best place for your question. Pymol is
> a molecular viewer...
> But the question itself is basically trivial from the linear algebra
> point of view.
>
> If X is your source set of orthogonal vectors and Y is the target,
> then you should have some sort of matrix R to satisfy
>
> Y =3D RX
>
> But, since it should only be a rotation, you'll first have to
> transform X and Y to their orthonormal counterparts N and M:
>
> M =3D RN
>
> Then
>
> MN^-1=3DRNN^-1
>
> such that
>
> R =3D MN^-1
>
> If both sets are of equal dimensions (and full rank), there's an exact
> solution. Otherwise, there's a bit more trouble...
>
> So, taking your favourite language with the proper linear algebra
> package, it comes down to:
>
> normalize X -> N
> normalize Y -> M
> invert N
> multiply M with the inverse of N
>
> By the way, you're probably dealing with 3x3 matrices here (molecules
> in cartesian space), in which case the routines are simple enough to
> write down yourself (I believe these were even in the array.py I
> posted like two days ago).
>
> Hope it helps,
>
> Tsjerk
>
> On Jan 25, 2008 10:55 PM, Buz Barstow <bu...@ma...> wrote:
> > Dear All,
> >
> > I'm looking for an algorithm that will allow me to derive a
> > transformation matrix that superimposes one set of orthogonal vectors
> > onto another set of orthogonal vectors, that I can then use to
> > transform another set of orthogonal vectors.
> >
> > Thanks! and all the best,
> >
> > --Buz



=2D-=20

Jason Vertrees (jav...@ut...)
Doctoral Candidate
Biophysical, Structural & Computational Biology Program
University of Texas Medical Branch=20
Galveston, Texas

http://www.best.utmb.edu/
http://www.pymolwiki.org/