  We use classical (Penrose) two-component spinors to set up the differential
geometry of two parabolic contact structures in five dimensions, namely $G_2$
contact geometry and Legendrean contact geometry. The key players in these two
geometries are invariantly defined directional derivatives defined only in the
contact directions. We explain how to define them and their usage in
constructing basic invariants such as the harmonic curvature, the obstruction
to being locally flat from the parabolic viewpoint. As an application, we
calculate the invariant torsion of the $G_2$ contact structure on the
configuration space of a flying saucer (always a five-dimensional contact
manifold).
