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.
>>