# Characterizing singularities of a surface in Lie sphere geometry

## TL;DR

This paper explores the conditions for fundamental surface singularities within Lie sphere geometry and analyzes how these singularities behave under Lie sphere transformations.

## Contribution

It provides new criteria for identifying surface singularities in Lie sphere geometry and examines their transformation properties.

## Key findings

- Conditions for cuspidal edges and swallowtails in Lie sphere geometry
- Behavior of surface singularities under Lie sphere transformations
- New geometric criteria for surface singularities

## Abstract

The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.