The way conic sections are often described, if you take a section parallel to the double-cone, you get a parabola, and if you take a perfectly vertical section, you get a hyperbola. But what if you take a section that's in between parallel and vertical?
Given that high school math never mentioned this possibility (nor wikipedia, math sites, etc), I'm guessing it's not some sort of exotic new section. It would still cross the double-cone in 2 different places, so I'm guessing it's just a hyperbola? If so, is there a difference between a hyperbola from a vertical section and a diagonal one?