Dear Tim, Ian, Tim Gruene wrote:
[...] Actually it does not depend - the rotation matrices are a representation of the group SO(3) and hence the matrix multiplication is associative. It does not matter whether you start left or right or in the middle - that is not the same as not being commutative.
Yes of course, but if you start from the left you are not (at least not obviously) "first rotating the coordinates by alpha about z"; you are rotating the columns of the second matrix. And in the second multiplication you are not rotating the already-rotated coordinates by beta about new y: you haven't touched the coordinates x yet! Apparently from Ian's derivation this can be expanded into something that would be rotating the coordinates in the first step, but not simply by using the associative property. And think how ridiculous this would be in the case of the polar angles: the whole purpose of the first two and last two rotations is to bring the desired general rotation axis onto the z axis in order to rotate about it with a simple matrix. However if each of these rotations is a general rotation and has to be brought back to standard orientation in order to perform it, the problem grows without bounds. I need to work on something else now, but when i get time I will go through Ian's derivation and see how it is in fact tractable. Fortunately, as described in the write-up that started this thread, the math just involves multiplying the three matrices and operating on the coordinates with that, and this can perfectly well be visualized by simple rotations about fixed axes if you will do them in the order first gamma, then beta then alpha. So if in fact it is also possible to visualize them in terms of moving axes with the rotations in the the opposite order, that is irrelevant and not helpful to my understanding. Tim Gruene wrote:
Dear Ed, On 03/31/2014 08:55 PM, Edward A. Berry wrote: [...]Looking at the math, it depends whether you multiply from right to left or left to right x' = Rz(a) Ry(b) Rz(g) x or x' = Rz(a) (Ry(b) (Rz(g) x))[...] Actually it does not depend - the rotation matrices are a representation of the group SO(3) and hence the matrix multiplication is associative. It does not matter whether you start left or right or in the middle - that is not the same as not being commutative. Cheers, Tim
