# What do you get when you take a conic section in between a parabola and vertical?

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?

• you get a hyperbola that is not quite the same as the original – Will Jagy Jun 7 '15 at 19:20
• Meanwhile, the angle between the asymptotic lines of this new hyperbola is the same as the angle formed when a parallel plane passes through the vertex of the cone, in which case the section is a pair of intersecting lines. In contrast, a plane passing through the vertex and parallel to those that create a parabola gives a section that is a single line, as the plane is tangent to the cone along the entire line. – Will Jagy Jun 7 '15 at 19:41