The basic thought is that t is time and rotations proceed as a function of time. If matrix A is a rotation of x degrees, we can arrange that the matrix valued function A(t) = matrix A when t=1 and Identity when t=0. ( I probably should have chosen a different letter than "A" for the function since I don't have multiple fonts...) When t=x/n, A(t) = A(x/n). When this is raised to the power n, it means the rotation of x/n degrees is repeated n times, hence is a rotation of x degrees. I could imagine A(t) for a variable rotation rate, but for the constant rotation rate the parameter t linearly scales the rotation angle. The two dimensional case of A(t) would be [ cos(xt) sin(xt) ] [ -sin(xt) cos(xt) ] Apropos your response to math-fun, what relationship does Log(A) have to the eigenvector of A, if any? The other way to look at this sort of rotation combination is as a linear combination of the axes of rotation, and the axes are the eigenvectors of the rotation matrices. When I fiddled around with this stuff last time it came up, I could only make it work using the Log hack when A and B were rotations with the same axis. One of these days I'll "get it" for the more general case Shel At 06:18 PM 11/8/2002 -0500, you wrote:

Hi, Shel.

I'm not clear on how A(t) and B(t) depend on t in what you wrote below.

I have posted a response to math-fun that should appear any minute.

--Dan --------------------------------------------------------------------------------------------------------------

<< If A and B are thought of as matrix-valued functions of a parameter t where, e.g., A(1/n) ^ n = A(1), B(1/n) ^ n = B(1), which makes easy to see geometric sense for constant rate rotations, then the product integral P (A(t)B(t))^dt seems to be the continuously combined rotation (at least according to one way of defining it), and seems to be equal to P (B(t)A(t))^dt.

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