Tools for analyzing the intersection curve between a torus and a quadric through projection and lifting

This article introduces several user-friendly tools for analyzing the intersection curve between a ringed torus and an irreducible quadric surface. Our main focus is the study of the projection of this curve onto the plane , known as the cutcurve, which plays a central role in ensuring correct lifting to the intersection curve. We also provide a detailed characterization of the singularities of both the projection and the intersection curve, as well as conditions for the existence of double tangents. A fundamental tool in our analysis is the theory of resultants and subresultants.