Geometric deformations of implicit curves

TL;DR

This work develops an FRS-equivalence framework for deformations of implicit plane curves $f=0$ and provides a codimension-≤2 classification of how vertices and inflections appear under versal deformations. Using the Monge–Taylor map and jet-space stratifications, the authors derive explicit normal forms for the $A_1^-$, $A_1^+$, and $A_2$ singularities and analyze two codimension-2 subcases of $A_1^-$. They show how higher-order inflections and vertices arise or disappear in parametrized families and reveal the corresponding bifurcation sets and tangent structures relative to the original curve, with clear geometric interpretations for plane-section problems. The results yield concrete deformation models, such as $F=f+x^3 t+s$, $F=f+x^4 t+s$, and $F=f+ty+s$, and provide a framework applicable to computer vision and shape analysis where plane sections of surfaces produce singular plane curves. Overall, the paper advances the geometry of deformations of implicit plane curves by linking jet-space stratifications to explicit, verifiable deformation models and their bifurcation behavior.

Abstract

Let f = 0 be an implicit singular plane curve. When deforming f = 0, inflections and vertex emerge from the singularities. In this papper, we classify the deformations of f = 0 with respect to the inflections and the vertices in the cases of codimension less than or equal to 2, that is, in the cases that occur generically in families of implicit curves with 2 parameters.

Figures (15)

• Figure 2: Versal deformation of a $A_1^-$ singularity.
• Figure 3: Exemple of the configuration of $C_f$ and the inflexion curves, when $\eta_1\eta_2>0$.
• Figure 4: Configuration of the vertex and inflexional curves for $\eta_1\eta_2>0$ and $\eta_3\eta_4<0$.
• Figure 5: Possible geometric configuration for the general case (codimension 1). Red cdots are vertices and blue cdots are inflections.
• Figure 6: Geometric bifurcation set of a $A_1^-$-singularity when one of the branches has a inflection at the singular point.

Theorems & Definitions (16)

• Proposition 2.2
• Remark 2.3
• Proposition 2.4
• Proposition 4.1
• Remark 4.2
• Proposition 4.3
• Definition 4.4
• Theorem 4.5
• Proposition 4.6
• Proposition 4.7