Finding Plane of Rotation and Angles of Rotation from Rotation Matrix

1. Any given Rotation Matrix \(R\) in Any Dimension can be resolved into 2 Non-Parallel Vectors Representing the Plane of Rotation and 2 Rotation Angles (\(\phi\) and \(2\pi - \phi\) where \(0 \leq \phi < 2\pi\)) corresponding to the Rotation as follows
   1. Calculate \(\cos \phi\) and \(\sin \phi\) corresponding to Angle of Rotation \(\phi\) as follows
      
      \(\cos (\phi)= {\Large \frac{Trace(R) - (R_n-2)}{2}}\)    (where \(R_n\) is the Number of Rows/Columns in the Rotation Matrix)
      
      \(\sin (\phi)= \sqrt{1-\cos^2(\phi)}\)
   2. Calculate Rotation Angles \(\phi\) and \(2\pi-\phi\) using the values of \(\cos \phi\) and \(\sin \phi\).
   3. Calculate Matrix \(M\) as follows
      
      \(M=R-I\)
      
      Any 2 Columns of Matrix \(M\) give the 2 Non-Parallel Vectors corresponding to the Plane of Rotation of the given Rotation Matrix.