Distance squared functions on singular surfaces parameterized by smooth maps $\mathcal{A}$-equivalent to $S_k$, $B_k$, $C_k$ and $F_4$

TL;DR

This paper classifies and analyzes the singularities of distance squared functions on certain singular surfaces in three-dimensional space, using geometric methods and blow-up techniques, with applications to wave-fronts and caustics.

Contribution

It provides a detailed description of singularities on surfaces parameterized by specific $ ext{A}$-equivalent singularities, extending geometric understanding of these complex structures.

Findings

Classification of singularities for distance squared functions on these surfaces

Description of wave-fronts and caustics related to the singular surfaces

Application of blow-up techniques to analyze singularities

Abstract

We describe singularities of distance squared functions on singular surfaces in $\mathbb{R}^3$ parameterized by smooth map-germs $\mathcal{A}$-equivalent to one of $S_k$, $B_k$, $C_k$ and $F_4$ singularities in terms of extended geometric language via finite succession of blowing-ups. We investigate singularities of wave-fronts and caustics of such singular surfaces.