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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?

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    $\begingroup$ you get a hyperbola that is not quite the same as the original $\endgroup$
    – Will Jagy
    Jun 7, 2015 at 19:20
  • $\begingroup$ 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. $\endgroup$
    – Will Jagy
    Jun 7, 2015 at 19:41

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A section with a vertical plane or with a plane between parallel and vertical gives anyway an hyperbola. In the first case the center of the hyperbola is at the same ''height'' of the vertex of the double cone, in the second case it go away with the inclination of the plane.

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