Singularities of parallel surfaces

TL;DR

This paper classifies and provides criteria for singularities of parallel surfaces in differential geometry, including cuspidal edges, swallowtails, and other complex singularities, enhancing understanding of surface behavior.

Contribution

It offers a comprehensive classification and differential geometric criteria for singularities of parallel surfaces, extending prior work to a broader set of singularities.

Findings

Identifies key types of singularities in parallel surfaces.

Provides differential geometric criteria for each singularity type.

Enhances understanding of surface singularity structures.

Abstract

We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional $D_4^\pm$ singularities. We give criteria for these singularities types in terms of differential geometry (Theorem 3.4 and 3.5).