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How regular is the evolute of a plane curve?

Pascal J. Thomas, Nikolai Nikolov

TL;DR

This work investigates how little smoothness is required on a plane curve to ensure a corresponding level of smoothness for its evolute. It develops a bootstrap framework linking evolute regularity to the radius of curvature and its monotonicity, proving that the evolute loses at most one regularity order when the curvature derivative is nonzero. The key results establish necessary and sufficient conditions for C^1 and higher regularity, and provide a detailed local analysis of points where the curvature derivative vanishes, including cusp formation. The paper also discusses global phenomena such as double points in evolutes and the behavior of involutes, enriching the geometric understanding of evolute-involute relationships.

Abstract

We study the relationship between the smoothness of a plane curve and that of its evolute, especially in the cases where the parent curve is no more two or three times continuously differentiable, and exhibit the same kind of apparent improvement in regularity: in the generic local situation, the evolute has one order of regularity less than the parent curve.

How regular is the evolute of a plane curve?

TL;DR

This work investigates how little smoothness is required on a plane curve to ensure a corresponding level of smoothness for its evolute. It develops a bootstrap framework linking evolute regularity to the radius of curvature and its monotonicity, proving that the evolute loses at most one regularity order when the curvature derivative is nonzero. The key results establish necessary and sufficient conditions for C^1 and higher regularity, and provide a detailed local analysis of points where the curvature derivative vanishes, including cusp formation. The paper also discusses global phenomena such as double points in evolutes and the behavior of involutes, enriching the geometric understanding of evolute-involute relationships.

Abstract

We study the relationship between the smoothness of a plane curve and that of its evolute, especially in the cases where the parent curve is no more two or three times continuously differentiable, and exhibit the same kind of apparent improvement in regularity: in the generic local situation, the evolute has one order of regularity less than the parent curve.

Paper Structure

This paper contains 7 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Motivations and outline.
  3. Definitions and notations.
  4. Main results.
  5. Proofs
  6. Local behavior
  7. Multiple points

Key Result

Theorem 1

Let $\gamma: (a;b)\longrightarrow \mathbb R^2$ be a $\mathcal{C}^2$-regular curve such that $\kappa(t)\neq 0$ for all $t\in(a;b)$, and denote its evolute by $\tilde{\gamma}$. Then

Theorems & Definitions (11)

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