Sphere-Contact Singularities of Cuspidal Edge–Cusp Pairs

In this study, we examine sphere contact for a cuspidal edge–cusp pair (X,B), where X⊂R3 is a cuspidal edge and B⊂X is a distinguished curve whose germ at the singular point is a (2,3)-type cuspidal space curve. Let f:(R2,0)→(R3,0) be a parametrization of X, and let α:(R,0)→(R2,0) be a source curve whose image under f is the distinguished cusp B. We consider the restricted distance squared family D(X,B)(s,q)=∥β(s)−q∥2,q∈R3, which is determined by the singular space curve β, where β=f∘α. We describe its contact strata with the aim of also reflecting the role of β as a distinguished curve on a cuspidal edge, together with its chosen source curve. We prove that centres with a contact of order at least 2 form a plane, while those with contact of order at least 3 form a line; these are also known as the singular normal plane and the singular binormal line of the pair, respectively. We then show that there exists a unique centre with a contact in the order of at least 4 if and only if the singular torsion does not vanish. In this case, the osculating centre is qosc(X,B)=β(0)+3κsing(β,0)τsing(β,0)bsing(β,0), where bsing(β,0) is the unit singular binormal vector of β. Also, we determine the exact A2- and A3-strata and obtain a criterion for fifth-order contact at the osculating centre, with the lower strata determined by the 3-jet of the distinguished cusp. The fourth-order condition takes the form of a mixed jet expression in adapted normal form, involving both the cuspidal edge and the chosen source curve.