Tracing a Parabola with Quetelet & Dandelin

Tracing an Ellipse with Quetelet & Dandelin

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This animation shows the Quetelet-Dandelin proof that a plane whose angle from the vertical is greater than the vertex angle of a cone meets that cone in a parabola. (See notes below.)

The upper sphere is inscribed in the cone so that it is tangent to the cone along the upper of the two red circles and tangent to the tilted plane at the right-hand red point. The lower sphere is inscribed in the cone so that it is tangent to the cone along the lower of the two red circles and tangent to the tilted plane at the left-hand red point.

The two dark blue segments, both being tangent to the lower sphere from the same point, are always congruent to each other. Similarly, the two light blue segments are always congruent to each other. The two red circles, each being where one of the two spheres is tangent to the cone, are horizontal. Thus, the sum of the lengths of a dark blue segment and a light blue segment remains constant. See the animation Tracing Out An Ellipse. (11/16/07)