I have a 3D rotation matrix, R which is a combination of rotations around x-axis , y-axis and z-axis. I know how to calculate n(the arbitrary axis around which a point rotated about theta angle and this rotation is equal to rotating that point using the 3D matrix above) i.e. by finding the eigenvector of R corresponding to the eigenvalue equal to 1. How can I calculate theta, the angle to rotate about the arbitrary axis?
[Math] 3D rotation around arbitrary axis
linear algebrarotations
Related Solutions
If $R_x$ rotates around the $x$-axis, and $R_y$ rotates around the $y$-axis, and you want to rotate first around $x$, and then around $y$, simply apply $R_y R_x$ to your vector, let's call it $v$.
This is because $R_x v$ rotates $v$ around the $x$-axis, then $R_y(R_x v)$ rotates $R_x v$ around the $y$-axis.
Stick your thumb of your right hand along the direction of the rotation axis. A positive angle is along the way your fingers point.
Like this
http://electron9.phys.utk.edu/Collisions/rotational_motiondetails.htm
Best Answer
I'll describe a computational method, assuming the rotation has been translated to be around an axis through the origin.
If you know the axis of rotation $A=(a,b,c)$, then you can find a vector orthogonal to this one. For example, if $a \ne 0$, $b \ne 0$, then $V=(b,-a,0)$ is such a vector. Or, if one component is $0$--say, for example $c=0$, then $V=(0,0,1)$ is orthogonal. And that covers all cases.
A third vector in a right-handed triple is found using the vector cross product: $W=A\times V$. Normalize $V$ and $W$ to unit vectors $\hat{V}$, $\hat{W}$.
Now, feed $\hat{V}$ into your matrix and determine the output in terms of $\hat{V}$, $\hat{W}$ using dot products $$ R\hat{V} = \{(R\hat{V})\cdot\hat{V}\}\hat{V}+\{(R\hat{V})\cdot\hat{W}\}\hat{W}=\alpha\hat{V}+\beta\hat{W}, $$ and compare to the action of a right-handed rotation about $A$ through an angle $\theta$: $$ \hat{V} \mapsto \cos\theta \hat{V}+\sin\theta \hat{W}. $$