Focal surfaces of fronts associated to unbounded principal curvatures

TL;DR

This paper investigates the properties and singularities of focal surfaces linked to wave fronts with unbounded principal curvatures, providing characterizations and analyzing Gaussian curvature behavior near singular points.

Contribution

It offers new characterizations of focal surface singularities and explores Gaussian curvature behavior in the context of unbounded principal curvatures.

Findings

Characterization of focal surface singularities in terms of initial front properties

Analysis of Gaussian curvature behavior near non-degenerate singular points

Descriptions of geometrical properties influencing focal surface singularities

Abstract

We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of singularities and geometrical properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature of the focal surface.