Tracing a Parabola with Quetelet & Dandelin

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This animation shows the Quetelet-Dandelin proof that a plane that is parallel to a generator of a cone intersects that cone in a parabola. (See notes below.)

Sometime around 1825, the Belgian geometers Adolphe Quetelet and Germinal Dandelin devised this simple and elegant construction showing that a plane parallel to a generator of a cone intersects that cone in a parabola.

The sphere is inscribed in the cone so that it is just tangent to the tilted plane. The horizontal plane is determined by the circle where the sphere meets the cone. Two of the blue segments are tangent to the sphere; being tangent to the same sphere from the same point, those segments are therefore congruent. The blue line segment that is tangent to the sphere at a point on the red circle is congruent to the blue line segment that lies in the sloping plane because they are the hypotenuses of right triangles which share their vertical sides and have congruent angles at their lowest vertices. Thus, the point where the inscribed sphere is tangent to the plane is the focus of the parabola whose directrix is the horizontal red line where the inclined plane meets the horizontal plane. That parabola is precisely the curve where the inclined plane meets the cone. (12/01/07; revision 10/30/09)