Parabola  Red Ellipse  Orange Blue Dots  Foci Green  Path of foci This curve is a strophoid. Thank you to JeanPierre Ehrmann of France for telling me this. 
 Clive Tooth followed Alexander's post with a demonstration using a Dandelin sphere:  Consider a cone. Let the vertex of the cone be V and let g be a generator. Call the axis of the cone x. Let A be a fixed point on g. Let B be the point on x which is closest to A. Let r be the line through A which is tangential to the cone and which is perpendicular to g. (r is not visible in this diagram because it's perpendicular to the screen)  Consider the pencil of planes through r. Let P be one of these planes. P will intersect our cone in some conic section. 
inscribed in the cone and touching P, at a point F. It is well known that F is a focus of our conic section.  Clearly C is on x and the points ABC and F are in the same plane.  
the angle that AF makes with g, on the side of P away from the sphere. Call this angle 2t. Now the angles CBA and CFA are both right angles. So the points ABC & F are concyclic. So the angle ACF is equal to the angle ABF. But the angle CAF is (pi2t)/2 = pi/2  t, so the angle ACF is t, so the angle ABF is t.  Thus, A and B are fixed points such that for any position of the plane P, the angle gAF is twice the angle ABF.  Thus, the point F is the intersection of two rotating lines (BF and AF) whose angular velocities are in the ratio 1:2.  QED 
