Rotation as a vector
The situation is that I have x-rays of implanted hemispherical surgical prostheses. Most of the prosthesis is invisible as it is made of polythene but there are two circular wires attached which are visible on the xray. The circles appear in projection as ellipses - the ellipticity and major axis of these permit me to deduce the angle of these wires to the plane of the x-ray along 3 orthogonal axes and from this I can deduce the orientation of the prosthesis itself. The transformation I need to do is to figure out at what angles the prosthesis lies to the body - as the orientation may affect the surgical outcome. I also need to handle a spatial distortion due to divergence of the x-ray beam.
So, if I expressed myself properly - I have 3 orthogonal angular positions in one frame of reference. I don't care much about the linear position but I need to understand the mathematics necessary to transform the angular coordinates. I had thought that simple vector arithmetic would do the trick but it is clear (I think) from this thread that even though the angular positions have magnitude and direction, they are not vectors (as they don't obey the laws of vector algebra).
It seems intutively obvious that these transformations are deterministic and thus that there must be an algebra that will allow me to calculate - but it is not untuitively obvious what the laws are. I am hoping for some guidance as to how angular positions should be handled mathematically.