Duality on generalized cuspidal edges preserving singular set images and first fundamental forms

TL;DR

This paper extends a previously known isometric duality for cuspidal edges to more general singularities like cuspidal cross caps and 5/2-cuspidal edges, providing new geometric insights.

Contribution

It generalizes the duality concept to broader classes of singularities and offers new geometric perspectives on this duality.

Findings

Duality extends to generalized cuspidal edges including cuspidal cross caps and 5/2-cuspidal edges.

Dual cuspidal edges are not congruent to the original when the singular set image lacks symmetries.

New geometric insights are provided on the nature of this duality.

Abstract

In the second, fourth and fifth authors' previous work, a duality on generic real analytic cuspidal edges in the Euclidean 3-space $\boldsymbol R^3$ preserving their singular set images and first fundamental forms, was given. Here, we call this an `isometric duality'. When the singular set image has no symmetries and does not lie in a plane, the dual cuspidal edge is not congruent to the original one. In this paper, we show that this duality extends to generalized cuspidal edges in $\boldsymbol R^3$, including cuspidal cross caps, and $5/2$-cuspidal edges. Moreover, we give several new geometric insights on this duality.