Dandelin Sphere for the Parabola

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The parabola can be defined as the curve formed by the intersection of a cone with a plane parallel to a line on the cone passing through the origin. In this Demonstration, a Dandelin sphere shows the relationship between a parabola and its focus and directrix.


Dandelin's spheres have a special relationship with their associated conic section: their tangency points with the plane cutting the cone correspond to the foci of the conic section, while the planes through their circles of contact with the cone intersect the cutting plane in the conic section's directrices.

To get a very responsive Demonstration, the parabola and the contact circle are represented as nonuniform rational B-splines, avoiding the use of Wolfram Language plotting functions.


Contributed by: Jan Mangaldan (April 2020)
Open content licensed under CC BY-NC-SA




[1] L. Piegl and W. Tiller, The NURBS Book, 2nd ed., New York: Springer, 1997.

[2] C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, New York: Dover, 1963.

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