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Geometry of curves passing through Whitney umbrella

Hiroyuki Hayashi

TL;DR

The paper analyzes curves that pass through a Whitney umbrella singularity by employing a smoothly extendable Darboux frame along the curve. It defines three Frenet-Serret-type invariants tied to the geodesic curvature, normal curvature, and geodesic torsion, and studies their degrees and top-terms to extract geometric meaning, including cases where these terms vanish. It then constructs developable surfaces along the curve, introducing osculating developables and pseudo-cylindrical/pseudo-conical generalizations, and derives invariant quantities $\delta$ and $\sigma$ that govern these surfaces. The results clarify how singularity structure and curve direction interact, offering criteria for alignment with tangent, principal plane, and self-intersecting curves of the Whitney umbrella and enabling a structured view of developable geometry near singular points.

Abstract

We study geometry of curves passing through a Whitney umbrella by using a Darboux frame along it. We define three invariants by using Frenet-Serre type formula relating to the geodesic curvature, the normal curvature, and the geodesic torsion. We investigate the degrees of divergence and the top-terms of these invariants and their geometric meanings. We also consider a developable surface along the curve.

Geometry of curves passing through Whitney umbrella

TL;DR

The paper analyzes curves that pass through a Whitney umbrella singularity by employing a smoothly extendable Darboux frame along the curve. It defines three Frenet-Serret-type invariants tied to the geodesic curvature, normal curvature, and geodesic torsion, and studies their degrees and top-terms to extract geometric meaning, including cases where these terms vanish. It then constructs developable surfaces along the curve, introducing osculating developables and pseudo-cylindrical/pseudo-conical generalizations, and derives invariant quantities and that govern these surfaces. The results clarify how singularity structure and curve direction interact, offering criteria for alignment with tangent, principal plane, and self-intersecting curves of the Whitney umbrella and enabling a structured view of developable geometry near singular points.

Abstract

We study geometry of curves passing through a Whitney umbrella by using a Darboux frame along it. We define three invariants by using Frenet-Serre type formula relating to the geodesic curvature, the normal curvature, and the geodesic torsion. We investigate the degrees of divergence and the top-terms of these invariants and their geometric meanings. We also consider a developable surface along the curve.
Paper Structure (10 sections, 8 theorems, 94 equations, 4 figures)

This paper contains 10 sections, 8 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of curves passing through Whitney umbrella
  4. Darboux frame and Curvatures
  5. Degrees and top-terms of curvatures
  6. Differential geometric meanings of top-terms of curvatures
  7. Developable surfaces along a curve passing through Whitney umbrella
  8. Ruled surfaces and developable surfaces
  9. Developable surfaces along a curve passing through Whitney umbrellas
  10. Degrees and top-terms of $\delta$ and $\sigma$

Key Result

Proposition 2.2

Let $c_w : (\mathbb{R},0) \to (\mathbb{R}^2 ,0)$ be a curve of finite multiplicity, and let $W:(\mathbb{R}^2 ,0) \to (\mathbb{R}^3 ,0)$ be a Whitney umbrella. Then a unit normal vector field $\bm{n}$ of $W$ along $\gamma$ can be smoothly extended across the origin

Figures (4)

  • Figure 2.1: The tangent line, principal plane, normal plane, and principal intersection line of $W_1$
  • Figure 2.2: The curves $\gamma_1$ (left), $\gamma_2$ (center), and $\gamma_3$ (right)
  • Figure 3.1: Examples of $B=0$ or $C=0$
  • Figure 4.1: A $(1,0)$-ruled surface (left), a $(0,1)$-ruled surface (center), and a $(1,1)$-ruled surface (right)

Theorems & Definitions (18)

  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Example 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • ...and 8 more