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I have question regarding the possibility of inscribing a cone in a sphere.

Essentially, I am looking for some proper definition of that. When dealing with polyhedra, the matter is simple: each of the polyhedron's vertices must lie on the sphere. However, what conditions need to be met in order for a cone to be inscribed in a sphere? When the base is circular, I understand that the apex of the cone and the edge of the base must lie on the sphere. But can an elliptical cone be inscribed in a sphere, and if yes, would that mean that just two points from the edge of the base are on that sphere?

Intuition seems to say "yes", but as I mentioned, I would be thankful for some definition.

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Perhaps a modified method of Lagrange multipiers would be useful here. The normal to the ellipse will line up with the projections of the normals of the sphere on the plane of the ellipse at the points of tangency. Let's see, if $f=0$ defines the ellipse and $g=0$ is the sphere then we want $\nabla f = \lambda Proj_S(\nabla g)$ where $S$ is the plane of the ellipse. – James S. Cook Sep 4 '12 at 18:05
It really depends on what you mean by cone. It is often understood that a cone is a right circular cone, but, depending on the context, the base might not be circular and the axis might not be perpendicular to the base. A problem might ask for the largest cone inscribed in a sphere, in which case the base will be circular. On the other hand, a problem might ask to inscribe in a sphere the largest cone whose base is elliptical with eccentricity (say) $0.2$. – Théophile Sep 4 '12 at 18:47
Yes, Theophile, a problem was asking for the largest cone inscribed in a sphere. The problem itself is quite easy, and yes, the largest cone is indeed in this case a right circular - for obvious reasons. However, as I was ruling out the other options in the beginning (for example, a cone with the axis not perpendicular to the base), I was wondering if a cone with an elliptical base was even an option. If yes, the explanation is obviously simple (the area of the base can be expanded to a circle). – Johnny Westerling Sep 4 '12 at 19:23

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