This detailed monograph treats finite point groups as subgroups of the full rotation group, providing geometrical and topological methods which allow a unique definition of the quaternion parameters for all operations. An important feature is an elementary but comprehensive discussion of projective representations and their application to the spinor representations, which yield great advantages in precision and accuracy over the more classical double group method. A self-contained treatment, with many solved problems to clarify key points, this monograph provides a powerful tool for handling rotations and double groups.