Enumerating the morphology of non-degenerate Darboux cyclides

We provide an enumeration of all possible morphologies of non-degenerate Darboux cyclides. Based on the fact that every Darboux cyclide in R3 is the stereographic projection of the intersection surface of a sphere and a quadric in R4, we transform the enumeration problem of morphologies of Darboux cyclides to the enumeration of the algebraic sequences that characterize the intersection of a sphere and a quadric in R4.