  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.
