A cut through a hyperboloid of two sheets

The image below shows the intersection of a hyperboloid of two sheets with a plane (in magenta color). The intersection is a hyperbola. How many types of cuts are possible when a plane cuts through this hyperboloid of two sheets ?

Note: Ellipses and Circles are considered to be the same type. In addition, do not count the empty set as one of the types although it is possible to pass a plane that does not intersect with this hyperboloid of two sheets at all. Finally, do not count an intersection of one or two points as a type of cut curve.