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Helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space

Kaixin Yao, Wei Zhang

Abstract

In this paper, we define two types of helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space and investigate when they become lightcone framed base surfaces. Moreover, by constructing appropriate diffeomorphic transformations and using the criteria of $(i,j)$-cusps and $(i,j)$-cuspidal edges, we establish identification theorems for the singular types of both 1-type and 2-type helicoidal surfaces on their singular loci.

Helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space

Abstract

In this paper, we define two types of helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space and investigate when they become lightcone framed base surfaces. Moreover, by constructing appropriate diffeomorphic transformations and using the criteria of -cusps and -cuspidal edges, we establish identification theorems for the singular types of both 1-type and 2-type helicoidal surfaces on their singular loci.

Paper Structure

This paper contains 9 sections, 9 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Priliminaries
  3. Helicoidal surfaces of non-lightlike frontals
  4. 1-type helicoidal surfaces
  5. 2-type helicoidal surfaces
  6. Singularities of helicoidal surfaces
  7. Singularities of 1-type helicoidal surfaces
  8. Singularities of 2-type helicoidal surfaces
  9. Examples

Key Result

Proposition 3.2

The 1-type helicoidal surface $\bm{r}_1$ is singular at $(u_0, v_0) \in I \times \mathbb{R}$ if and only if $\beta(u_0) = 0$ or $a(u_0) = x_2(u_0) = 0.$$\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Left: Intersection between a helicoidal string world sheet and a plane is a straight line. Right: Same situation, but now the intersection is curved (cf. BMS).
  • Figure 2: The 1-type helicoidal surface (mesh) and its singular locus (red curve).
  • Figure 3: The 2-type helicoidal surface (mesh) and its singular locus (red curve).

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • ...and 7 more