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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3$\begingroup$ I guess you can see "focus" is wrong by considering the hyperbola made when the plane is parallel to the axis of the cone... $\endgroup$– Gerald EdgarCommented Mar 21, 2012 at 13:03
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$\begingroup$ Right, for that one specific kind of hyperbola the point goes off to infinity. $\endgroup$– Keenan PepperCommented Mar 21, 2012 at 14:07
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1$\begingroup$ 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. $\endgroup$– Jeff StromCommented Mar 21, 2012 at 14:16
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3$\begingroup$ @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. $\endgroup$– Will SawinCommented Mar 21, 2012 at 16:42
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2$\begingroup$ 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. $\endgroup$– Pietro MajerCommented Mar 21, 2012 at 19:15
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2 Answers
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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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This question and accepted answer are almost ten years old, but in case anybody stumbles upon this question, here's some more information on the topic. (For cone terminology and background, see wikipedia)
First, an edited version of one of the comments:
All [right] circular cones whose section is a given ellipse produce [different] points. The more narrow [...] the cone, the closer to the center is the point [of intersection]. In general, [...] 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
The apex of each cone in this family is on a hyperbola $h$ passing through the foci of the ellipse. The foci of $h$ are the points of intersection of the ellipse and its major axis. This is easily proven in the same context as the proofs for Dandelin spheres, and uses the property that the difference of distances from a point on the hyperbola to its foci is constant for all points. See Salmon, Conic Sections, Art. 367.
For a much more detailed treatment see Armstrong, Where is the Cone?. From the abstract:
Real quadric curves are often referred to as “conic sections,” implying that they can be realized as plane sections of circular cones. However, it seems that the details of this equivalence have been partially forgotten by the mathematical community. The definitive analytic treatment was given by Otto Staude in the 1880s and a non-technical description was given in the first chapter of Hilbert and Cohn-Vossen’s Geometry and the Imagination (1932). [...] Our hope is to revive the lost knowledge of “conic sections” by providing the slickest possible modern treatment, by using standard linear algebra that was not standard in 1932.
