Take a rubber gasket. Make a slice from the middle to the outside, like the first cut in a pie. Because there was some strain in the rubber, the gasket doesn't close into a ring now, it's more like the letter C. Worse still, there is some rotational strain, so it's not even flat; it's more like a lock washer. Now, cut the gasket into six identical pieces, like a pizza.
Here's my question: How do I calculate the number of slices that would be required to form a complete ring, given the geometry that the relaxed slices assume?
Here's the data I have: I have two sets of points representing two of the slices of the gasket ($PS_1$ for the first slice and $PS_2$ for the second). I know the affine transformation matrix (rotation and translation only) that gives the least-squares mapping from points in the first slice to those in the second slice. Call this matrix $M_{12}$. Assuming the relation between all the slices is the same, I can reproduce slice n with $PS_n=PS_1*M_{12}^n$
If we ignore the z axis for a moment (the washer is in the xy plane), then I should be able to reproduce $PS_1$ by $PS_1=PS_1*M_{12}^N$ where N is the number of slices required to form the ring. N will almost never be an integer - the ring relaxed a bit when I made the first slice, so I need a non-integer number of slices to close it. How do I solve for N? (obviously, the trivial solution is 0, what's the first non-trivial solution?) Given that there might not be exactly $2\pi{}/N$ xy rotation between $PS_1$ and $PS_2$, how do I find a least-squares solution for N?
Adding back in the z axis, there are three problematic degrees of freedom: xz rotation, yz rotation, and z translation. If I tile out the slices of my gasket now, they could corkscrew off into space. With that in mind, what value of N will give the second least-squares minimum of $PS_1 - PS_1*M_{12}^N$? (The first minimum being, of course, N=0.)
You can assume that the ring doesn't relax to a crazy degree, it's only a minor change. In the corkscrew case, you can assume that a second minimum exists; the pitch of the corkscrew is pretty shallow.
Also, if anyone can explain what it means to raise a matrix to an irrational power, I will love you forever.
Thanks!
Suppose you want to raise the nonsingular matrix $M$ to the power $p$. There are branches of the function $z \mapsto z^p$ that are analytic on a neighbourhood of the set $\sigma(M)$ of eigenvalues of $M$. Then $M^p$ can be computed using the holomorphic functional calculus for such functions.
In particular, suppose $M$ is diagonalizable, so that $M = S D S^{-1}$ for some nonsingular matrix $S$ where $D$ is the diagonal matrix with diagonal entries $\lambda_j$, the eigenvalues of $M$. Then we have $M^p = S D^p S^{-1}$, where $D^p$ is diagonal with diagonal entries $\lambda_j^p$.
For irrational $p$ there are infinitely many branches of $z^p$, and thus infinitely many possibilities for $M^p$. A popular choice, which will work if $M$ has no negative real eigenvalues, is the principal branch, which has branch cut on the negative real axis and coincides with the usual $z^p$ on the positive reals.