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Starting with a plane parallel to the cone's base (defining a circle with center O), the plane is rotated by φ degrees around one axis that contains the center of the original circle (that is, around a diameter). In the resulting ellipse, how can we calculate this "original center" O, in relation to the ellipse's parameters (center, foci etc) and the angle φ?

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You've seen the Dandelin construction, by any chance? – J. M. Aug 7 '12 at 0:09
I've seen it as I was searching for an answer, although I cannot immediately see how it can be used. If the top sphere is held constant as the other varies, the ellipse will not contain the O point. – derio Aug 7 '12 at 8:10

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