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On comparing the writhe of a smooth curve to the writhe of an inscribed polygon

Jason Cantarella

TL;DR

This work provides rigorous, quantitative control on the error incurred when approximating the writhe of a smooth closed curve by an inscribed polygon. By extending Fuller’s $\Delta\operatorname{Wr}$ formula to polygonal curves via a polygonal tantrix and carefully smoothing corners, the authors establish a concrete bound $|\operatorname{Wr}(C) - \operatorname{Wr}(C_n)| \le \alpha \; n \; x^3 + n O(x^4)$ under an embedded ribbon condition and curvature/torsion bounds, yielding $O(1/n^2)$ convergence for similarly sized edges. They also prove that the writhe of a polygonal curve equals the writhe of any smoothly rounded curve, and provide an explicit spherical-area expression for the writhe difference in the polygonal setting. Numerical experiments on a closed-helix example corroborate quadratic convergence and illustrate the practical utility of the bounds for reliable writhe computation. The work also outlines open problems, notably extending the theory to open curves and to piecewise $C^2$ tantrixes.

Abstract

We find bounds on the difference between the writhing number of a smooth curve, and the writhing number of a polygon inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal curves. The results establish error bounds useful in the computation of writhe.

On comparing the writhe of a smooth curve to the writhe of an inscribed polygon

TL;DR

This work provides rigorous, quantitative control on the error incurred when approximating the writhe of a smooth closed curve by an inscribed polygon. By extending Fuller’s formula to polygonal curves via a polygonal tantrix and carefully smoothing corners, the authors establish a concrete bound under an embedded ribbon condition and curvature/torsion bounds, yielding convergence for similarly sized edges. They also prove that the writhe of a polygonal curve equals the writhe of any smoothly rounded curve, and provide an explicit spherical-area expression for the writhe difference in the polygonal setting. Numerical experiments on a closed-helix example corroborate quadratic convergence and illustrate the practical utility of the bounds for reliable writhe computation. The work also outlines open problems, notably extending the theory to open curves and to piecewise tantrixes.

Abstract

We find bounds on the difference between the writhing number of a smooth curve, and the writhing number of a polygon inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal curves. The results establish error bounds useful in the computation of writhe.

Paper Structure

This paper contains 10 sections, 13 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Fuller's $\Delta\!\operatorname{Wr}$ formula
  4. Justifying Fuller's interpretion of the $\Delta\!\operatorname{Wr}$ formula
  5. Extending Fuller's formula to polygonal curves: I
  6. Extending Fuller's formula to polygonal curves: II
  7. Bounding the $\Delta\!\operatorname{Wr}$ Formula
  8. Example Computations
  9. Further Directions
  10. Acknowledgements

Key Result

Theorem 1

Suppose $C(t)$ is a simple, closed curve of class $\mathcal{C}^4$. We assume $C(t)$ is parametrized so that $|C'(t)| \geq 1$, and that we have upper bounds $B_1, \dots, B_4$ on $|C'(t)|, \dots |C^{(4)}(t)|$. Let $C_n(t)$ be any $n$-edge polygonal curve inscribed in $C$ with maximum edge length $x$ a where $\alpha$ is a numerical constant less than $B_2 (5 B_2^2 + B_3)$.

Theorems & Definitions (23)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Theorem 7
  • proof
  • ...and 13 more