Defining an odd conical section?

[honi]

I know that a parabola is defined by the intersection of a plane and a right circular cone given that the smallest angle of intersection between the plane and the axis of the cone is equal to the angle of the cone.

I know that a circle is defined by the intersection of a plane and a right circular cone given that the plane is perpendicular to the axis of the cone

Thus, an ellipse is the intersection of a right circular cone and a plane, wherein the smallest angle of intersection between the plane and the axis of the cone is greater than the angle of the cone (this can include circles).

Also, a hyperbola is defined by the intersection of a plane and 2 right circular cones (the apexes and axis of which are the same) in which the plane is parallel to the axis of the cones.

So we have gone from perpendicular to the cone's axis to parallel to the cone's axis... But there's an undefined section here, where the plane intersects both cones but is not parallel to the axis of the cones... So it's not a hyperbola (because it's not parallel to the axis of the cones), but isn't an ellipse (because the smallest angle of intersection between the plane and the axis of the cones is smaller than the angle of the cone), either...

My question is, does that type of intersection have a name? Has it been studied anywhere, or can anyone give me any information about its properties?