2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/105013We 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.13 pp; 3 figuresDifferential Geometry14B05, 14H15, 58K50, 58K60text