Coordinate System Transformation

[dhume878]

Hey everyone,

I'm working on my degree and have started getting into some deeper lin alg than I took previously regarding coordinate system transformations. I was hoping someone might be able to shed some light on it for me. I'll do my best to explain the problem ..

I have a global coordinate system for a volume in space created by a motion capture device. Thus three unit vectors representing the x, y and z vectors of the global space are
[1 0 0
0 1 0
0 0 1]

I then have a person standing in space, with markers on their hips in such a way I can determine a local system for the person's pelvis. The unit vectors representing this local system are as follows

[0.9625 -0.0326 -0.266
0.0268 0.9999 -0.0256
0.2671 0.6175 0.9627]

So the local system is oriented very close to the global system.

I then calculate two points in space, but in the global space. I in essence need to rotate them about the origin of my local system as much as my local system is rotated from my global system.

I'm sure I sound like a bumbling goon, but I hope you guys can make heads or tails of this. I'm guessing there's a way to come up with a rotation matrix from system 1 to system 2, and from there .. hmm.. somehow translate my points about the origin of my local system.

I can clarify anything if need be.

 

[fresh_42]

Maybe this can help: https://en.wikipedia.org/wiki/Orthogonal_matrix
I would search for 'normal forms in orthogonal groups' or 'word problem in O(n)'.

 

[dhume878]

I appreciate the response after 8 years. This was the question of a young academic, which has since been solved, published, and laid to rest. However I would point people toward the wiki article on rotation matrices as opposed to orthogonality wrt to the relevance of the question. https://en.wikipedia.org/wiki/Rotation_matrix

 

[fresh_42]

We currently try to avoid any empty threads, which implies to work through old ones, such that anyone who stops by has at least a hint on how to proceed.