Another Map from P^7 to the Study Quadric

In [5] Pfurner, Schröcker and Husty introduced a mapping from P^7 to the Study quadric. In [9], it was shown that this map could be thought of as the composition of an extended version of the inverse Cayley map based on the 6x6 adjoint representation of the group, and the Cayley map itself. Here, the analogous map using the Cayley map based on the standard 4x4 representation of SE(3) is studied. It is shown that mapping a general line in P^7 results in a motion with cubic trajectories. A different view of the map is then studied. A birational map between the Study quadric and the variety defined by the adjoint representation of the group is given. The new map is then the composition of the map from the Study quadric, extended to all P^7, with the map from the P^17 back to the Study quadric. The effect of the new map on symmetric subspaces of SE(3) is also considered. Lastly, an example is given showing how the interpolation techniques can be extended to point constraints. That is, where a point on the body is required to pass through a sequence of successive points in space.