SINGULARITIES OF THE ENVELOPE OF CURVES TANGENT TO A SEMI-CUBIC CUSP

摘要

We study the envelope of one-parameter families of smooth curves tangent to a curve having a semi-cubic cusp such that the radius of curvature at the tangency point vanishes when this point approaches the cusp. We show that, in general, the closure of the envelope has two semi-cubic cusps at the same point, one of which is the given cusp tangent to the same straight line. We study also nongeneric and dual cases and normal forms of the equation of the envelope.