Parallel and dual surfaces of cuspidal edges

TL;DR

This paper investigates the geometric properties of cuspidal edges, focusing on their parallel and dual surfaces, and introduces concepts like principal curvatures, ridge points, and relations between singularities and differential geometry.

Contribution

It provides explicit formulas for principal curvatures and directions, and defines ridge points for cuspidal edges, linking singularities with geometric properties.

Findings

Explicit forms of principal curvature and direction for cuspidal edges

Definition of ridge points based on principal curvatures

Relations between singularities of parallel/dual surfaces and initial cuspidal edges

Abstract

We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometric properties of initial cuspidal edges.