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Math Help - Angle of Cut Through Cone to Produce Parabola?

  1. #1
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    Angle of Cut Through Cone to Produce Parabola?

    (I am second-guessing myself by asking this question.)

    A parabola is created when a cutting plane passes through one nappe of a right double-cone (but not through the tip.)
    To create a parabola, I believe that this cutting plane must be parallel to the side of the cone. I.e. - only one angle for the cutting plane is possible.
    Is this correct?

    Most of the resources I have found don't specify the exact angle.
    A circle is made if the cutting plane is perpendicular to the cone axis.
    A hyperbola is created if the cutting plane goes through both nappes of the double cone, but doesn't have to be parallel to the cone axis.
    An ellipse is created if the cutting plane goes through one nappe of the double-cone, but is not perpendicular to the cone axis.
    Can a parabola be created if the cutting angle is within a particular angular range with respect to the perpendicular to the cone, or is it restricted to only one angle (parallel to the side of the cone)?

    Can anybody recommend a resource that presents these properties and the specific angles (ranges) required to produce each type of conic?
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Angle of Cut Through Cone to Produce Parabola?

    You are correct - a parabola requires the cutting be parallel to the edge of the cone. If the cutting plane is not equal to the slope of the edge of the cone then it will either be (a) too shallow and thus slice completely through the one lobe forming an elipse (a shape of finite size), or (b) too steep and thus its upper portion is always within the confines of the cone no matter how far you go, and its lower portion slices into tje lower lobe, thereby making two shapess that are each infinite in size. But if sliced at precisley the correct angle (parallel to the edge ) you end up with one shape that is infinite in size. Thus a parabola is the intermediate between an elipse and a hyperbola.

    The wikipedia article is pretty complete: Conic section - Wikipedia, the free encyclopedia
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