Cuspidal edges on focal surfaces of regular surfaces

We investigate geometric invariants of cuspidal edges on focal surfaces of regular surface. In particular, we shall clarify the sign of the singular curvature at a cuspidal edge on a focal surface using singularities of parallel surface of a given surface satisfying certain conditions.

💡 Research Summary

The paper investigates the geometric invariants of cuspidal edges that appear on the focal surfaces of regular (immersed) surfaces in Euclidean 3‑space. Starting from a regular surface (f:U\subset\mathbb R^{2}\to\mathbb R^{3}) with unit normal (\nu) and principal curvatures (\kappa_{1},\kappa_{2}), the authors consider the parallel surfaces (f_{t}=f+t\nu). It is well‑known that a parallel surface is a front and becomes singular precisely along the curves where (t=1/\kappa_{i}). By assuming the surface has no umbilic points, a curvature‑line coordinate system ((u,v)) can be chosen, which yields the simple expressions (\partial_{u}f_{t}=(1-t\kappa_{1})f_{u}) and (\partial_{v}f_{t}=(1-t\kappa_{2})f_{v}).

The authors recall the classification of rank‑one singularities of fronts (cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, and cuspidal butterfly) via (\mathcal A)-equivalence and present the known criteria (Fukui–Hasegawa) in terms of a signed area density function (\tilde\lambda) and a null vector field (\eta). In the curvature‑line coordinates the identifier of singularities for the parallel surface (f_{t}) is simply (\tilde\lambda=\kappa_{i}-\kappa_{i}(p)); the null direction is (\partial_{u}) (or (\partial_{v}) when (t=1/\kappa_{2}(p))). Proposition 3.1 (Theorem 3.1) translates the vanishing of first and second derivatives of (\kappa_{i}) at a point (p) into precise statements about which singularity type occurs on (f_{t}). For example, a cuspidal lips appears exactly when (\kappa_{i,u}=\kappa_{i,v}=0) and the Hessian determinant of (\kappa_{i}) is positive, while a cuspidal beaks corresponds to a negative Hessian determinant.

A central technical result is Proposition 1.1, which gives an explicit formula for the limiting normal curvature (\kappa_{t\nu}) of the parallel surface at any rank‑one singular point, expressed solely in terms of the original principal curvatures: \