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

Tomonori Fukunaga, Nozomi Nakatsuyama, Masatomo Takahashi

TL;DR

This work extends the study of translation surfaces to the linearly dependent regime by developing translation generalised framed surfaces and a frame-matrix formalism between framed curves. It provides a comprehensive set of criteria for classical and higher-order singularities, including cross caps (S0), S1^± points, cuspidal edges, swallowtails, cuspidal cross caps, lips, beaks, and D4-type points, across four structural cases determined by the tangency data and the linearly dependent condition. The results unify the treatment of singularities for translation surfaces and their self-translation counterparts, with explicit discriminants and derivative conditions that are amenable to computation and geometric interpretation. The findings advance the differential-geometric understanding of translation surfaces in R^3 and offer a framework for further exploration of frontality and singularity types in generalized framed surface theory.

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.

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

TL;DR

This work extends the study of translation surfaces to the linearly dependent regime by developing translation generalised framed surfaces and a frame-matrix formalism between framed curves. It provides a comprehensive set of criteria for classical and higher-order singularities, including cross caps (S0), S1^± points, cuspidal edges, swallowtails, cuspidal cross caps, lips, beaks, and D4-type points, across four structural cases determined by the tangency data and the linearly dependent condition. The results unify the treatment of singularities for translation surfaces and their self-translation counterparts, with explicit discriminants and derivative conditions that are amenable to computation and geometric interpretation. The findings advance the differential-geometric understanding of translation surfaces in R^3 and offer a framework for further exploration of frontality and singularity types in generalized framed surface theory.

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.

Paper Structure

This paper contains 13 sections, 27 theorems, 128 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Translation surfaces as generalised framed surfaces
  3. A criteria for the cross cap ($S_0$ singularity)
  4. Criterion for the $S^\pm_1$ singularities
  5. Self translation surfaces
  6. Translation surfaces as framed surfaces
  7. $\rm(I)$ The case of $\alpha(u_0) \neq 0, \widetilde{\alpha}(v_0) \neq 0, \mu(u_0) \times \widetilde{\mu}(v_0) = 0$
  8. $\rm(II)$ The case of $\alpha(u_0) = 0, \widetilde{\alpha}(v_0) \neq 0, \mu(u_0) \times \widetilde{\mu}(v_0) = 0$
  9. $\rm(III)$ The case of $\alpha(u_0) \neq 0, \widetilde{\alpha}(v_0) = 0, \mu(u_0) \times \widetilde{\mu}(v_0) = 0$
  10. $\rm(IV)$ The case of $\alpha(u_0) = 0, \widetilde{\alpha}(v_0) = 0, \mu(u_0) \times \widetilde{\mu}(v_0) = 0$
  11. Framed curves and regular space curves
  12. Generalised framed surfaces and framed surfaces
  13. Criterion for singularities

Key Result

Proposition 2.2

Under the above notations, we have It follows that we have for $i,j=1,2,3$.

Figures (5)

  • Figure 1: The translation surface $\hbox{\boldmath $x$}$ in Example \ref{['example-S0']}. $(0,0)$ is a cross cap singular point.
  • Figure 2: The translation surface $\hbox{\boldmath $x$}$ in Example \ref{['example-S1+']}. $(0,0)$ is an $S_1^+$ singular point and the dot in the figure is the point $\hbox{\boldmath $x$}(0,0)$.
  • Figure 3: The translation surface $\hbox{\boldmath $x$}$ in Example \ref{['example-S1-']}. $(0,0)$ is an $S_1^-$ singular point and the dot in the figure is the point $\hbox{\boldmath $x$} (0,0)$.
  • Figure 4: The curve $\gamma$ and the self translation surfaces $\hbox{\boldmath $x$}^+$ and $\hbox{\boldmath $x$}^-$ in Example \ref{['sin']}.
  • Figure 5: The curve $\gamma$ and self translation surface $\hbox{\boldmath $x$}^+$ in Example \ref{['selfS1+exam']}. $(0,\pi)$ is an $S_1^+$ singular point of $\hbox{\boldmath $x$}^+$ and the dot in the figure on the right is the point $\hbox{\boldmath $x$}^+ (0,\pi)$.

Theorems & Definitions (43)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Example 2.7: Cross cap
  • Remark 2.8
  • Corollary 2.9
  • Theorem 2.10
  • ...and 33 more