On surfaces obtained as singular loci of normal congruences of frontals with pure-frontal singular points

TL;DR

This paper investigates the singularities and geometric features of surfaces derived from the singular loci of normal congruences of frontals with pure-frontal singular points, linking singularities to geometric invariants and curvature behavior.

Contribution

It provides new characterizations of singularities on normal ruled surfaces and focal surfaces of frontals, connecting them to geometric invariants and curvature properties.

Findings

Characterization of singularities via geometric invariants.

Relation between focal surface singularities and frontal properties.

Analysis of Gaussian curvature behavior at specific cuspidal edges.

Abstract

We study singularities and geometric properties of surfaces given by the singular loci of normal congruence of frontals with pure-frontal singular points. These surfaces consist of the normal ruled surface and focal surfaces of the initial frontal. For the normal ruled surface, we give characterizations of singularities in terms of geometric invariants of the initial frontal defined on the set of singular points. For focal surfaces, we show relation between certain singularities of them and geometric property of the given frontal. Moreover, we consider behavior of Gaussian curvature of focal surfaces of frontal with a $5/2$-cuspidal edge.