Tracing a Hyperbola with Quetelet & Dandelin

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This animation shows the Quetelet-Dandelin proof that a plane whose angle from the vertical is less than the vertex angle of a cone meets that cone in a hyperbola. (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 upper of the two black points. 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 lower of the two black points.

The two dark blue segments, both being tangent to the lower sphere from the same point, are always congruent to each other. The green segment and the segment obtained by joining the yellow segment to the blue segment it meets are both tangent to the upper sphere from the same point, and so are congruent also. The yellow segment therefore gives the difference between, on the one hand, the distance from the curve to the upper black point and, on the other hand, the distance from the curve to the lower black point. The two red circles, each being where one of the two spheres is tangent to the cone, are horizontal. Thus, the length of the yellow segment remains constant. (10/29/09)