If both napped of a right circular cone are intersected by a plane that does not pass through the vertex of the cone what is the resulting curve?

When a plane intersects both nappes of a right circular cone without passing through the vertex, the resulting curve is a hyperbola. This occurs because the plane cuts through both sides of the cone, creating two separate branches of the hyperbola. The precise shape and orientation of the hyperbola depend on the angle at which the plane intersects the cone.