The Quetelet-Dandelin Consruction and the Focus-Directrix Description of an Ellipse

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This animation uses the Quetelet-Dandelin construction to show that a plane whose angle from the vertical is larger than the angle made by the generators of the cone will intersect that cone in an ellipse. It uses the focus-directrix definition of ellipse to do so (See notes below.)

The sphere is inscribed in the cone so that it is tangent to the cone itself (along the horizontal red circle) and also tangent to the tilted plane (at the point shown). The horizontal plane is the one which contains the circle where the sphere meets the cone. The two blue segments are tangent to the sphere. The lower one lies entirely in the tilted plane; the upper lies entirely in the cone. Being tangent to the same sphere from the same point (always a point on the yellow curve where the tilted plane intersects the cone), those segments are therefore congruent.

The vertical black segment always extends from a point on the yellow curve to the point on the horizontal plane that lies directly above it. The angle that the upper blue segment makes with the vertical black segment is always the same angle as that made by a generator with the vertical—because the upper blue segment lies entirely in the cone and is therefore a part of a generator. All of the right triangles consisting of two black sides and a blue hypotenuse are therefore similar, and we see that the ratio of the length of either of the blue segments to the length of the vertical black segment is constant.

The red line is the intersection of the two planes, and the green segment is the perpendicular from a point on the yellow curve to the red line. Because the green segment lies entirely in the tilted plane, it makes an angle with the vertical black segment that exceeds the angle that generators of the cone make with the vertical. However, the angle the green segment makes with the vertical black segment remains constant. Thus all of the right triangles consisting of two black sides and a green hypotenuse are similar to each other, and the ratio of the length of the vertical black segment to the length of the green segment remains constant.

It now follows that the ratio of the length of the blue segment that lies in the tilted plane to the length of the green segment remains constant and less than one, so that yellow curve is the ellipse whose directrix is the red line and whose focus is the point where the sphere touches the tilted plane. (11/06/09)