For any point A in R 3 which is not on the tangent plane to X at P , the point P is a swallowtail point of the pedal surface of X from A . P is a parabolic point of X , and a line in R 3 has order of contact greater than two with X at P . Our final two characterizations of Gaussian cusps involve notions which aren't defined for an arbitrary immersion. The osculating plane at P of the parabolic image curve of X is the tangent plane of X at P . Furthermore, the set of immersions such that the Gauss map is stable and the parabolic image curve has nonzero curvature at each Gaussian cusp is open and dense in the space of all immersions of U in R 3 .
P is an inflection point of the asymptotic direction map of X .
Furthermore, the set of immersions such that the Gauss map is stable and the asymptotic direction map is regular at each Gaussian cusp is open and dense in the space of all immersions of U in R 3 . If the Gauss mapping is good then its Jacobian has rank one at each parabolic point P . The kernel of the Jacobian map is the zero principal curvature direction at P . If P is a cusp of the Gauss map of X , let l be the line of curvature of X which is tangent to the parabolic curve at P . We will show in chapter 5 that these three critical points are nondegenerate if P is a cusp of the Gauss map N .