Given a plane in 3-space that has a conic section traced on it. Can we find a cone to intersect on the section given a random plane?

2014-07-28T12:48:21Z

In other words, if I were to take a plane with a given gradient, is there a method to find an infinite cone that intersects the plane on a parabola y=x^2, for example, that is traced on that plane?

My problem is that I'd like to be able to construct a parabola on a sheet of paper and a half cone is such a way as to form a representation of the conic section of a parabola on the cone. I'd like to do it mathematically.

Pope2014-08-01T21:16:30Z

Your problem was solved about 2200 years ago. Research the Conics of Apollonius. The work is often called Conica. The definition of a cone to fit a given parabola may be found in Book I, Proposition 52. See also the hyperbola (Proposition 54) and the ellipse, (Proposition 56).

In any of these cases there is no unique cone that will generate the section. In fact, there are infinitely many cones fitting any given conic section.

The constructions quite likely were solved by Euclid about two generations earlier, but Apollonius pressed further with conics, and Euclid's conics works fell into disuse. They are now lost. Most of the Conics of Apollonius still survive.

The link below does not have the construction itself, but it has some background information and suggested reading.

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