text: In materials science, misorientation is the difference in crystallographic orientation between two crystallites in a polycrystalline material.
  In crystalline materials, the orientation of a crystallite is defined by a transformation from a sample reference frame (i.e. defined by the direction of a rolling or extrusion process and two orthogonal directions) to the local reference frame of the crystalline lattice, as defined by the basis of the unit cell.  In the same way, misorientation is the transformation necessary to move from one local crystal frame to some other crystal frame.  That is, it is the distance in orientation space between two distinct orientations.  If the orientations are specified in terms of matrices of direction cosines  and , then the misorientation operator  going from  to  can be defined as follows:
  where the term  is the reverse operation of , that is, transformation from crystal frame  back to the sample frame.  This provides an alternate description of misorientation as the successive operation of transforming from the first crystal frame () back to the sample frame and subsequently to the new crystal frame ().
  Various methods can be used to represent this transformation operation, such as: Euler angles, Rodrigues vectors, axis/angle (where the axis is specified as a crystallographic direction), or unit quaternions.