How Is a Hyperbola Formed from Conic Sections?

[barryj]

Conic sections are formed when a plane cuts a double cone, i.e. two cones placed tip to tip along the same axis. A circle is when the plane is perpendicular to the axis, an ellipes when the plane is slightly canted, a parabola when the plane is EXACTLY parallel to the edge of the cones so that the plane cuts only one of the cones. Now the question.

Every diagram I have seen shows the hyperbola being formed when the cutting plane is parallel to the axis of the cones. The plane therefore cuts both the top and bottom cone. What do you get if the cutting plane cuts through both cones but is not exactly parallel to the axis of the cones. Is this still a hyperbola or some other function?

 

[micromass]

It's still a hyperbola.

 

[barryj]

I find this hard to visualize. So the cut on the top cone would be symmetrical to the cut on the bottom cone as a hyperbola?

 

[micromass]

I find this hard to visualize. So the cut on the top cone would be symmetrical to the cut on the bottom cone as a hyperbola?

Yes. Just as when you get an ellipse (tha is not a circle) if the cut is not perpendicular to the axis.

I guess the best way to convince you is to either read a (geometrical) proof of it, or to construct two cones yourself and make the cut.

 

[CellCoree]

I can explain that a hyperbola is formed when a plane cuts through both cones at an angle. This results in two curves that are mirror images of each other and are asymptotic to each other. The shape of a hyperbola is determined by the angle at which the plane cuts through the cones. If the plane is not exactly parallel to the axis of the cones, the resulting shape may not be a perfect hyperbola, but it will still have the defining characteristics of a hyperbola. It may be slightly distorted or elongated, but it will still have two curves that are asymptotic to each other. Therefore, it can still be considered a hyperbola, just with slight variations in its shape.