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Everybody knows that if I take the intersection of a right circular cone with a plane, I get a conic section. My question is, where does the symmetry axis of the cone intersect the plane? Does this point relative to the conic have a name, or a simple description? For example, for an ellipse I first guessed that it was one focus of the ellipse, but that is false.

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I guess you can see "focus" is wrong by considering the hyperbola made when the plane is parallel to the axis of the cone... – Gerald Edgar Mar 21 '12 at 13:03
Right, for that one specific kind of hyperbola the point goes off to infinity. – Keenan Pepper Mar 21 '12 at 14:07
You don't need a right circular cone to get a conic section, right? I think a skew elliptic cone will work just as well. Then it makes sense to ask if this point you describe is independent of the expression of the curve as a section of a cone. – Jeff Strom Mar 21 '12 at 14:16
@Jeff: The answer is clearly no. If you get an ellipse from a right circular cone, the point is off-center, but if you get it from an elliptic cone that's dead on, the point is the center. – Will Sawin Mar 21 '12 at 16:42
All circular cones whose section is a given ellipse also produce a differents points. The more narrow is the cone, the closer to the center is the point (one gets the center from sections of a cylinder). In general, I'd say the point is between the foci, at certain distances from them, whose ratio is equal to the ratio of the radii of the Dandelin spheres. – Pietro Majer Mar 21 '12 at 19:15
up vote 4 down vote accepted

Following Keenan's suggestion I delete my comment and make it into an answer:

Projectively speaking, there is no distinguished point inside a conic because the group of projective transformations that preserves the conic acts transitively on its interior: if someone gives you a circle and an unmarked ruler, you will never be able to construct the center.

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