General conical surface ($S(s,t)=A+tD(s)$)

A general conical surface is swept out by straight generators through one apex $A$. An invertible affine map sends it to another such surface, since it sends each generator to a line through the transformed apex. A projective transformation preserves the projective concurrence of generators, but can send the apex to infinity, producing parallel generators in the affine chart. Therefore finite-apex cones and cylinders are not separately projectively invariant Euclidean classes.