Before revision:  While our main contribution is the theory, we also demonstrate three different applications of the proposed pose mappings: (i) pose interpolation, (ii) pose servoing control, and (iii) pose estimation in a pointcloud alignment problem.
After revision:  We extend what is known for these pose mappings to the 4 x 4 representation common in robotics, and also demonstrate three different applications of the proposed pose mappings: (i) pose interpolation, (ii) pose servoing control, and (iii) pose estimation in a pointcloud alignment problem.