Singularities of translation surfaces under the linearly dependent condition in Euclidean 3-space

TL;DR

This paper studies the singular points of translation surfaces in Euclidean 3-space, focusing on the linearly dependent condition, and introduces generalized framed surfaces to classify these singularities.

Contribution

It extends the theory of translation surfaces by defining generalized framed surfaces and analyzing their singular points under linear dependence.

Findings

Classification of singular points on translation generalized framed surfaces

Introduction of the notion of translation generalized framed surfaces

Analysis of singularities under linearly dependent conditions

Abstract

We investigated singular points of translation surfaces under the linearly independent condition. In this paper, as completion, we investigate singular points of translation surfaces under the linearly dependent condition, using the theories of generalised framed surfaces and framed surfaces. We introduce the notion of translation generalised framed surfaces and investigate types of singular points that appear on translation generalised framed base surfaces.