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Classifications of cusps appearing on plane curves

Yoshiki Matsushita

Abstract

In this paper, we deal with plane curves with cusps. It is well known that there are various types of cusps. Among them, we investigate criteria for $(n, n+1)$ cusps with respect to several differential conditions and relations between these singularities and evolutes of fronts. We give complete classifications with respect to $(4, 5)$-cusps.

Classifications of cusps appearing on plane curves

Abstract

In this paper, we deal with plane curves with cusps. It is well known that there are various types of cusps. Among them, we investigate criteria for cusps with respect to several differential conditions and relations between these singularities and evolutes of fronts. We give complete classifications with respect to -cusps.
Paper Structure (15 sections, 57 equations, 1 figure)

This paper contains 15 sections, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fronts and singularities
  4. Evolutes of fronts
  5. Classifications of $(n, n+1)$ cusps in the sense of $C^{1}$-equivalence
  6. Invariants of $(n, n+1)$ cusps
  7. $C^{1}$ diffeomorphism forms of $(n, n+1)$ cusps
  8. Proof of Theorem \ref{['thm:C1-diffeo-classification']}
  9. Criterion for $(4, 5)$-cusp
  10. $(4, 5)$-cusps and Whitney's lemma
  11. Eliminations of the $7$-th order and the $11$-th order terms
  12. Proof of Theorem \ref{['thm:criterion for (4, 5)cusp']}
  13. Examples
  14. Classifications for $(4,5)$-cusps
  15. Evolutes of the $(4,5)$-cusp

Figures (1)

  • Figure 1:

Theorems & Definitions (9)

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