How to calculate the position of the (right circular) cone's vertex projection upon an ellipse?

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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?