Cuspidal edges on focal surfaces of regular surfaces

TL;DR

The paper addresses the problem of determining the sign of the singular curvature $\kappa_s$ at cuspidal-edge singularities on focal surfaces arising from a regular surface $f:U\to \mathbb{R}^3$. It combines the study of parallel surfaces $f^t(u,v)=f(u,v)+t\,\nu(u,v)$ with the theory of fronts and rank-one singularities to derive a geometric criterion: when the parallel surface $f^t$ is a cuspidal lips or cuspidal beaks at a point $p$ with $t=1/\kappa_i(p)$ and the limiting normal curvature $\kappa_\nu^t(p)=0$, the sign of the focal-surface singular curvature $\kappa_s^{C_i}(p)$ is determined (positive for lips, negative for beaks). The authors provide explicit formulas for $\kappa_\nu^t$ and for $\kappa_\nu^{C_i},\kappa_s^{C_i}$ in curvature-line coordinates, relate the sign of $\kappa_s^{C_i}$ to the Hessian of $\kappa_1$ (or $\kappa_2$), and illustrate the necessity of the hypotheses via examples. Overall, the work clarifies how focal-surface convexity/concavity at cuspidal edges is controlled by the ambient surface geometry and its parallel-surface singularities, enhancing geometric understanding of caustics and their invariants. The results have implications for the qualitative study of focal surfaces and their singularities in differential geometry.

Abstract

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.

Figures (7)

• Figure 1: $\kappa_\nu=0$
• Figure 2: $\kappa_\nu\neq0$
• Figure 4: $\kappa_s>0$
• Figure 5: $\kappa_s<0$
• Figure 7: The images of $f$ (blue), $f^1$ (purple) and $C_1$ (gray).

Theorems & Definitions (19)

• Proposition 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Definition 2.2
• Definition 2.3
• Proposition 3.1: fh-para
• proof : Proof of Proposition \ref{['prop:kn-para']}
• Proposition 4.1: cf. usy-book
• Proposition 4.2: cf. tera2