If the directrix of a cylinder a conic section ( a curve of the second order, here a hyperbola), the surface of a quadric (a surface of the second order) The model shows a straight hyperbolic cylinder, which means the surface lines are normal to the the carrier plane of the directrix. On the base the asymptote of the hyperbola can be seen.

If you project a second-grade-curve onto its girder in space, you get a cylinder of second grade as an result.