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Bour's theorem for helicoidal surfaces with singularities

Yuki Hattori, Atsufumi Honda, Tatsuya Morimoto

Abstract

In this paper, generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge, more generally, generic $n$-type edge, which is invariant under a helicoidal motion in Euclidean $3$-space admits non-trivial isometric deformations. As a corollary, several geometric invariants, such as the limiting normal curvature, the cusp-directional torsion, the higher order cuspidal curvature and the bias, are proved to be extrinsic invariants.

Bour's theorem for helicoidal surfaces with singularities

Abstract

In this paper, generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge, more generally, generic -type edge, which is invariant under a helicoidal motion in Euclidean -space admits non-trivial isometric deformations. As a corollary, several geometric invariants, such as the limiting normal curvature, the cusp-directional torsion, the higher order cuspidal curvature and the bias, are proved to be extrinsic invariants.
Paper Structure (11 sections, 18 theorems, 87 equations, 4 figures)

This paper contains 11 sections, 18 theorems, 87 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wave fronts
  4. Geometric invariants
  5. Singular points of curves with finite multiplicities
  6. Finite type edge singular points
  7. Generalized Bour's theorem
  8. Applications of Bour-type theorem
  9. Isometric deformations of $r/n$-cuspidal edges
  10. Extrinsicity of invariants
  11. Criterion for $7/2$-cusp

Key Result

Lemma 3.2

Let $f_{\gamma,h}:I\times \boldsymbol{R}\to \boldsymbol{R}^3$ be a helicoidal frontal given by cuspsur, and $k$ be a positive integer. Set $n=k+1$. If $p=(u_0,v_0)\in I\times \boldsymbol{R}$ is a generic $n$-type edge singular point of $f_{\gamma,h}$, then hold, where the $x^{(i)}$ means $d^{i}x/du^{i}$.

Figures (4)

  • Figure 1: Left: the image of the standard cuspidal edge $f_C$. Right: the image of a generic cuspidal edge (cf. Example \ref{['ex:GHF']}).
  • Figure 2: The generic helicoidal cuspidal edges $\Psi_h$ in Example \ref{['ex:GHF']}. The left (resp. center, right) figure shows the graphic of $h=0$ (resp. $h=0.1$, $0.2$).
  • Figure 3: The generic helicoidal $4/3$-cuspidal edges $\Psi_h$ in Example \ref{['ex:GHF']}. The left (resp. center, right) figure shows the graphic of $h=0$ (resp. $h=0.1$, $0.2$).
  • Figure 4: The generic helicoidal cuspidal edge $\Psi_h$ in Example \ref{['ex:GHF']} (meshed one), and the isomer of $\Psi_h$ (the one without mesh) in the case of $h=0.2$.

Theorems & Definitions (35)

  • Definition 2.1: SUY-annMSUY
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4: Bour-type representation formula for generic helicoidal $n$-type edges
  • proof
  • Lemma 3.5
  • Lemma 3.6
  • ...and 25 more