Parallel Surfaces of Cuspidal Cross Caps

TL;DR

This paper studies the geometry and singularities of parallel surfaces of cuspidal cross caps, establishing criteria for degeneracy and describing how these surfaces typically resemble cuspidal cross caps but degenerate at specific distances.

Contribution

It introduces a criterion for degeneracy of parallel cuspidal cross caps and characterizes the singularity configurations and their relation to geometric invariants.

Findings

Parallel surfaces are generally $\\mathcal{A}$-equivalent to cuspidal cross caps.

Degeneration occurs at specific distances where $C_2(\varepsilon)=0$.

Distances serve as an analogue of principal radii of curvature.

Abstract

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.