Extracting a rotation axis from a rotation matrix

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William
William 2011-7-23
I am working with 3 x 3 rotation matrices, R(ij), of cubic bicrystal misorientations, so there are up to 24 symmetry related descriptions (i.e. different permutations of the rotation matrix) for any starting misorientation. Usually I extract the axis/angle description from each one by standard methods - acos (theta) = acos(((traceR)-1)/2), and the axis as [R32-R23, R13-R31, R21-R12]. This works fine, except when theta = 180 degrees, for which this formula gives the axis as [0 0 0], as we know. So to get the axes for the 180 case, it seems to me I can do the following:
1) Use [V,D] = eig® to get a matrix D whose columns are the eigenvalues of R. One of these will be equal to +1, the other two will be -1.
2) Identify which column of D contains the +1, and then take the equivalent column of V as the eigenvector, i.e. the rotation axis. So far so good.
I am trying to use a for-(else)-end loop to do this, by using a counter i to examine each column of D in turn, and then summing the three elements of that column. If the sum comes to +1, I want to use that corresponding i to obtain the eigenvector as:
[V(1,i), V(2,1), V(3,1)], and then come out of the loop. I am relatively new to MATLAB, and I cannot get a loop to work. Can anyone help point me to a working code to do this loop?
Thanks.

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采纳的回答

Andrew Newell
Andrew Newell 2011-7-23
tol = 100*eps;
i = find(abs(diag(D)-1)<tol);
rotAxis = V(:,i);
(Edited to take rounding error into account. Increase tol if necessary.)

更多回答（2 个）

William
William 2011-7-24
Andrew, I tried this, but it comes back with the response i = Empty matrix 0-by-1 for some of the 24 symmetry operations, and then these return "Empty matrix 3-by-0" for the rotation axis, and produces no axis.
This is the loop for getting the degenerate descriptions:
for i=1:1:24
RotMat2 = RotMat*e(:, :, i);
[V,D] = eig(RotMat2);
RotAng2 = acosd((trace(RotMat2)-1)/2);
j = find(diag(D)==1)
RotAxis2 = V(:,j)
"Then come some printing commands, to print RotAng2, RotAxis2, and RotMat2"
end
Where RotMat is the starting rotation matrix. I am sure I am missing the obvious, but I can't get this to work. Hoping a more experienced user can :-)
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Walter Roberson
Walter Roberson 2011-7-24
Take one of the cases where you are expecting 1, and subtract 1 from it and look at the result. I suspect that what you will see a small residue, indicating that the value is not exactly 1 due to round-off error. It is seldom a good idea to use "==" for a floating point calculation.
Andrew Newell
Andrew Newell 2011-7-24
Right. That was careless of me. I have corrected my answer.

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William
William 2011-7-24
Gentlemen, thank you both for the help. So far, this seems to work.
Much appreciated, from a newbie :-)

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