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Singularities of helicoidal surfaces of frontals in the Euclidean space

N. Nakatsuyama, K. Saji, R. Shimada, M. Takahashi

Abstract

We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a frontal can naturally be considered as a generalised framed base surface. Moreover, we show that it is also a framed base surface under a mild condition. We give basic invariants and curvatures for helicoidal surfaces of frontals by using the curvatures of Legendre curves. Moreover, we also give criteria for singularities of helicoidal surfaces.

Singularities of helicoidal surfaces of frontals in the Euclidean space

Abstract

We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a frontal can naturally be considered as a generalised framed base surface. Moreover, we show that it is also a framed base surface under a mild condition. We give basic invariants and curvatures for helicoidal surfaces of frontals by using the curvatures of Legendre curves. Moreover, we also give criteria for singularities of helicoidal surfaces.

Paper Structure

This paper contains 6 sections, 9 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Legendre curves
  4. Framed surfaces
  5. Generalised framed surfaces
  6. Helicoidal surfaces of frontals

Key Result

Theorem 2.1

Let $(\ell,\beta):I \to \mathbb{R}^2$ be a smooth mapping. There exists a Legendre curve $(\gamma,\nu):I \to \mathbb{R}^2 \times S^1$ whose curvature of the Legendre curve is $(\ell, \beta)$.

Theorems & Definitions (16)

  • Theorem 2.1: Existence Theorem for Legendre curves Fukunaga-Takahashi2013
  • Definition 2.2
  • Theorem 2.3: Uniqueness Theorem for Legendre curves Fukunaga-Takahashi2013
  • Theorem 2.4: Existence Theorem for framed surfaces Fukunaga-Takahashi2019
  • Definition 2.5
  • Theorem 2.6: Uniqueness Theorem for framed surfaces Fukunaga-Takahashi2019
  • Definition 2.7
  • Remark 2.8
  • Proposition 2.9: Fukunaga-Takahashi2019
  • Definition 2.10
  • ...and 6 more