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Geometry on ruled surfaces with finite multiplicity

Hiroyuki Hayashi

TL;DR

The paper addresses geometry of ruled surfaces with finite multiplicity by introducing and analyzing pseudo-cylindrical ruled surfaces, where $\xi'(x)=\tilde{\xi}(x)x^k$, and develops a framework to study striction curves, developability, and singularities. It constructs a Frenet-like frame $\{\xi,\xi_d,\xi\times\xi_d\}$ and derives explicit expressions for the first and second fundamental forms, curvature, and the signed area density $\lambda$, linking developability to $r(x)$ and frontal/wave-front properties to $\psi$ and related invariants $\delta$, $\rho$, and $\sigma$. The work provides detailed criteria for a surface to be a front or wave front, classifies singularities on the surface and along rulings (including cuspidal edges, swallowtails, cuspidal cross caps, cuspidal beaks, and Scherbak-type surfaces), and reveals how these singularities are governed by the triple of invariants and their derivatives, with concrete examples illustrating the phenomena. These results extend classical developable and cylindrical ruled surface theory to the finite-multiplicity setting and clarify the geometric role of striction and invariants in controlling singular behavior.

Abstract

We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.

Geometry on ruled surfaces with finite multiplicity

TL;DR

The paper addresses geometry of ruled surfaces with finite multiplicity by introducing and analyzing pseudo-cylindrical ruled surfaces, where , and develops a framework to study striction curves, developability, and singularities. It constructs a Frenet-like frame and derives explicit expressions for the first and second fundamental forms, curvature, and the signed area density , linking developability to and frontal/wave-front properties to and related invariants , , and . The work provides detailed criteria for a surface to be a front or wave front, classifies singularities on the surface and along rulings (including cuspidal edges, swallowtails, cuspidal cross caps, cuspidal beaks, and Scherbak-type surfaces), and reveals how these singularities are governed by the triple of invariants and their derivatives, with concrete examples illustrating the phenomena. These results extend classical developable and cylindrical ruled surface theory to the finite-multiplicity setting and clarify the geometric role of striction and invariants in controlling singular behavior.

Abstract

We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.

Paper Structure

This paper contains 9 sections, 15 theorems, 58 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singularities of surface with a frontal and a wave front
  4. Ruled surfaces and developable surfaces
  5. Pseudo cylindrical ruled surfaces
  6. Fundamental properties of pseudo cylindrical ruled surfaces
  7. Singularities of pseudo cylindrical ruled surfaces
  8. Singularities of pseudo cylindrical developable surfaces
  9. Geometric meanings of invariants $\delta$, $\rho$, and $\sigma$

Key Result

Lemma 2.1

Let $f:(\mathbb{R}^2,0)\to(\mathbb{R}^3,0)$ be a wave front and $p_0\in(\mathbb{R}^2,0)$ be a non-degenerate point of $f$. Then the singular point $p_0$ is Let $f:(\mathbb{R}^2,0)\to(\mathbb{R}^3,0)$ be a frontal and $p_0\in(\mathbb{R}^2,0)$ be a non-degenerate point of $f$. The singular point $p_0$ is a cuspidal cross cap if and only if $\eta\lambda(p_0)\ne0$, $\phi_f(p_0)=0$, and $\phi_f'(p_0)\

Figures (7)

  • Figure 3.1: Ruled surfaces $g_2$
  • Figure 3.2: Ruled surfaces $g_3$
  • Figure 3.3: Case $\rm(I\space I)$ (cuspidal edge (left) and swallowtail (right))
  • Figure 3.4: Case $\rm(I\space V)$ (cuspidal beaks)
  • Figure 3.5: Case $\rm(I\space V)$ (Scherbak surface)
  • ...and 2 more figures

Theorems & Definitions (27)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3.1: H2024
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Example
  • Lemma 3.4
  • Proposition 3.5
  • ...and 17 more