Cusp singularities of plane envelopes
Résumé
We consider smooth $1$-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension $2$ singularity of envelopes (two transversal semicubic cusps), and we describe its perestroikas under generic small deformations.
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