Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Counting normals to closed curves in $\mathbb{R}^3$

Gaiane Panina, Dirk Siersma

TL;DR

This work investigates lower bounds on the number of normals emanating from a fixed point to generic closed curves in $\mathbb{R}^3$, distinguishing smooth and piecewise-linear (PL) curves and the role of knottedness. It combines Morse theory applied to the squared distance function $SQD_y$ with the geometry of the focal surface $\mathcal{F}(C)$ and its self-intersections for smooth curves, and develops a PL bifurcation framework via the wedge-based bifurcation set $\mathcal{B}(C)$. The main results are: (i) for every generic smooth closed curve there exists a point $y$ with at least $6$ emanating normals, and at least $8$ if the curve is knotted; (ii) for generic non-planar PL curves there exists a point with at least $8$ emanating normals, and at least $10$ if knotted. These findings extend normal-count phenomena from planar curves to 3D space curves and highlight the richer structure in PL objects, with connections to $ED$-degree considerations in algebraic settings.

Abstract

We prove the following results: (1) For every generic closed smooth curve in $\mathbb{R}^3$ there is a point with at least $6$ emanating normals to the curve. If the curve is knotted, there is a point with at least $8$ emanating normals. (2) For every generic closed piecewise linear curve in $\mathbb{R}^3$ there is a point with at least $8$ emanating normals to the curve. If the curve is knotted, there is a point with at least $10$ emanating normals. The proof is based on the Morse theory for the squared distance function and self intersections of the focal surface.

Counting normals to closed curves in $\mathbb{R}^3$

TL;DR

This work investigates lower bounds on the number of normals emanating from a fixed point to generic closed curves in , distinguishing smooth and piecewise-linear (PL) curves and the role of knottedness. It combines Morse theory applied to the squared distance function with the geometry of the focal surface and its self-intersections for smooth curves, and develops a PL bifurcation framework via the wedge-based bifurcation set . The main results are: (i) for every generic smooth closed curve there exists a point with at least emanating normals, and at least if the curve is knotted; (ii) for generic non-planar PL curves there exists a point with at least emanating normals, and at least if knotted. These findings extend normal-count phenomena from planar curves to 3D space curves and highlight the richer structure in PL objects, with connections to -degree considerations in algebraic settings.

Abstract

We prove the following results: (1) For every generic closed smooth curve in there is a point with at least emanating normals to the curve. If the curve is knotted, there is a point with at least emanating normals. (2) For every generic closed piecewise linear curve in there is a point with at least emanating normals to the curve. If the curve is knotted, there is a point with at least emanating normals. The proof is based on the Morse theory for the squared distance function and self intersections of the focal surface.
Paper Structure (3 sections, 11 theorems, 3 equations, 3 figures)

This paper contains 3 sections, 11 theorems, 3 equations, 3 figures.

Table of Contents

  1. Preliminaries and basic definitions
  2. Smooth curves. Proof of Theorem \ref{['Thm1']}
  3. Piecewise linear curves. Proof of Theorem \ref{['Thm1Knot']}

Key Result

Theorem 1

For every generic smooth closed curve in $\mathbb{R}^3$, there exists a point $y$ with at least $6$ emanating normals. If the curve is knotted, there is a point with at least $8$ normals.

Figures (3)

  • Figure 1: Each vertex of $C$ gives a wedge of $\mathcal{B}(C)$.
  • Figure 2: Lunes on the sphere at infinity: two lunes, and some different positions of three neighbour lunes.
  • Figure 3: The chamber $\sigma$ (left). The proof does not work for quadrilateral $\tau$ (right).

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • proof
  • proof
  • Example 1
  • ...and 15 more