The difference between two coordinates immediately yields the single axis of rotation and angle between the two orientations. The natural log of a quaternion represents curving space by 3 angles around 3 axles of rotation, and is expressed in arc-length; similar to Euler angles, but order independent. There is a Lie product formula definition of the addition of rotations, which is that they are sum of infinitesimal steps of each rotation applied in series; this would imply that rotations are the result of all rotations in the same instant are applied, rather than a series of rotations applied subsequently. The axes of rotation are aligned to the standard Cartesian axes. These rotations may be simply added and subtracted, especially when the frames being rotated are fixed to each other as in IK chains. Differences between two objects that are in the same reference frame are found by simply subtracting their orientations. Rotations that are applied from external sources, or are from sources relative to the current rotation still require multiplications, application of the Rodriguez Formula is provided. The rotation from each axle coordinate represent rotating the plane perpendicular to the specified axis simultaneously with all other axles. Although the measures can be considered in angles, the representation is actually the arc-length of the curve; an angle implies a rotation around a point, where a curvature is a delta applied to the current point in an inertial direction. Just an observational note: log quaternions have rings, or octaves of rotations; that is for rotations greater than 4 have related curves. Curvatures of things that approach this boundary appear to chaotically jump orbits. For 'human readable' angles the 1-norm can be used to rescale the angles to look more 'appropriate'; much like Celsius might be considered more correct than Fahrenheit. Other related values are immediately derivable: The total angle of rotation.... The axis of rotation... Quaternion Representation Basis Matrix Computation This was built from rotating the vectors (1,0,0), (0,1,0), (0,0,1), and reducing constants. Given an input , Which are used to compute the resulting matrix... Alternate Basis Calculation Alternatively this can be used given: convert to angle-axis , and Compute some partial expressions: Compute the resulting matrix: Expanded: Vector Rotation Rotate the vector around the rotation vector . The angle of rotation will be . Calculate the cosine of the angle times the vector to rotate, plus sine of the angle times