Parallel Surfaces of Cuspidal Cross Caps

This paper investigates the geometry and singularities of parallel surfaces of cuspidal cross caps, the fundamental non-front frontal singularities. We establish a criterion for the degeneracy of the distance squared function in terms of known geometric invariants and describe the resulting configuration of singularities. Our main result demonstrates that while the parallel surface is generically $\mathcal{A}$-equivalent to a cuspidal cross cap, it degenerates into a degenerated cuspidal $S_1$ singularity at specific distances characterized by the equation $C_2(\varepsilon)=0$. These distances act as a novel analogue of the principal radii of curvature. Indeed, although the Gaussian and mean curvatures diverge at the singularity, their asymptotic expansions reveal that their constant terms correspond to the product and average of the reciprocals of these distances, respectively.