On 1 April 2014 00:12, Edward A. Berry <[email protected]> wrote:
> 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. > Ed, you are not "starting from the left", you are starting the matrix multiplication from the right in the usual way. From my hand-out the equation for the rotating axes interpretation is: *R'*(*a**,**b*,*g*) = *R*(*Z*2,*g*;*a*,*b*) *R*(*Y*1,*b*;*a*) *R*(*Z*,*a*) The first rotation is the alpha one on the right consistent with alpha around Z being the first rotation: note that we write Y1 and Z2 for the other two axes because they are not the same as Y and Z (they are 'new Y' and 'new Z' resp.). Compare this with the equation for the fixed-axes interpretation: *R*(*a**,**b*,*g*) = *R*(*Z*,*a*) *R*(*Y*,*b*) *R*(*Z*,*g*) Now gamma on the right is the first rotation and Y and Z remain the same throughout (since they are the fixed 'screen' axes). The whole point of the exercise is that, as proved: *R'*(*a**,**b*,*g*) = *R*(*a**,**b*,*g*) So you can write the equation either way, but you can interpret it in two ways. My complaint is that many of the papers describing the rotation function go from the "alpha around Z, then beta around new Y, then gamma around new Z" description to the equation for the fixed-axes version of *R*( *a**,**b*,*g*) with no explanation, as though it were obvious how you get from A to B, when as is clear from the proof, it is far from obvious. I believe this is the source of much of the confusion surrounding the implementation of Eulerian angles in the RF programs. Cheers -- Ian
