Imagine a (half) cone with its bottom pointed vertically down. Now put a ball bearing inside the cone & send it around the inside of the cone. If the ball bearing is sufficiently small, and the friction is sufficiently small, the ball should revolve around the center of the cone in a cyclic manner. I would guess that this cycle would remain in a plane, but this would have to be proven. Q. What shape does this cycle have when the plane of the cycle isn't horizontal? (I don't know the answer.) At 12:05 PM 2/14/2007, R. William Gosper wrote:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval. What about slicing a paraboloid of rotation? More generally how can eliminating a quadratic mumbloid vs a linear (planar) equation yield anything but a quadratic (conic)? --rwg
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