Hi Jason,
Thanks a lot for this! I was able to code up a very simple solution to
my problem using Tsjerk's suggestion.
In my problem I have a series of very similar structures, and I wanted
to calculate the principal axes of a section of these structures and
then align the principal axes of another section of the structure with
those principal axes:
while i < len(eigVectors0001_1):
# the m matrix is a matrix of the principal axes of segment 1 of
structure 1
m = array([eigVectors0001_1[1], eigVectors0001_2[1],
eigVectors0001_3[1]])
m = transpose(m)
# The n matrix is a matrix of the principal axes of segment of
structure i
n = array([eigVectors0001_1[i], eigVectors0001_2[i],
eigVectors0001_3[i]])
n = transpose(n)
# r is the rotation matrix to rotate the n axes onto the m axes
r = dot(m,inverse(n))
# p is a matrix of the principal axes of segment 2 in structure i
p = array([eigVectors0002_1[i], eigVectors0002_2[i],
eigVectors0002_3[i]])
p = transpose(p)
# q are the rotated principal axes of segment 2 in structure i
q = dot(r,p)
i+=1
Thanks again! and all the best,
--Buz
On Jan 29, 2008, at 8:31 AM, Jason Vertrees wrote:
Buz,
Tsjerk's answer is right on. This has already been implemented for
PyMOL as a
plugin. See Kabsch/optAlign from "cealign," here:
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
matrix. Then multiply the right and left singular vectors by each
other --
that will yield the DxD rotation matrix (where D is the dimension of
your
vector sets). (This is how Kabsch/optAlign works.)
-- Jason
On Monday 28 January 2008 10:48:29 pm
[email protected] wrote:
------------------------------
Message: 3
Date: Sat, 26 Jan 2008 10:04:06 +0100
From: "Tsjerk Wassenaar" <[email protected]>
Subject: Re: [PyMOL] Algorithm to Rotate One Set of Vectors onto
Another
To: "Buz Barstow" <[email protected]>
Cc: [email protected]
Message-ID:
<[email protected]>
Content-Type: text/plain; charset=ISO-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 = 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 = RN
Then
MN^-1=RNN^-1
such that
R = 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 <[email protected]> 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
--
Jason Vertrees ([email protected])
Doctoral Candidate
Biophysical, Structural & Computational Biology Program
University of Texas Medical Branch
Galveston, Texas
http://www.best.utmb.edu/
http://www.pymolwiki.org/
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