0
$\begingroup$

I have three linearly independent vectors ($\vec{a}$, $\vec{b}$, and $\vec{c}$) which have been rotated to three other linearly independent vectors $\vec{a}'$, $\vec{b}'$, and $\vec{c}'$. I would like to find this rotation.

I've looked at quaternions but each pair of vectors $\vec{a}$ and $\vec{a}'$ yields different quaternions, non of which are correct.

For example, let $\vec{a} = (1, 1, 4)$, $\vec{b} = (4, 1, -1)$, $\vec{c} = (-5, 17, -3)$, $\vec{a}' = (2.760834, 1, 3.062319)$, $\vec{b}' = (3.062319, 1, -2.760834)$, and $\vec{c}' = (-5.823153, -17, -0.301485)$.

Using this method, I get $q_{a} = 2.91634 + -0.160763i + 1.36833j + -0.301891k$, $q_{b} = 2.91634 + -0.301891i + 1.36833j + 0.160763k$, and $q_{c} = 12.6495 + 1.8133i + 0.630935j + 0.553128k$.

The real rotation is 28 degree in the y axis.

My question is, is there a way of finding the angle of rotation and the axis of rotation, preferably in the form of quaternions but I don't mind, of these three vectors?

$\endgroup$
1
$\begingroup$

You can obtain the rotation matrix (and from this, the axis rotation and the angle of rotation), by solving the sistem

$R\cdot(\vec{a},\vec{b},\vec{c})=(\vec{a}',\vec{b}',\vec{c}')$

That is:

$R=(\vec{a}',\vec{b}',\vec{c}')(\vec{a},\vec{b},\vec{c})^{-1}$

Here $R$ is the rotation matrix, and $(\vec{a},\vec{b},\vec{c})$ is the matrix with the three vectors as columns

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.