Dandelin Spheres for the Elliptic Case

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The ellipse can be defined as the curve formed by the intersection of a plane with one nappe of a cone. In this Demonstration, Dandelin's spheres help show the relationship between an ellipse and its foci and directrices.


Dandelin's spheres have a special relationship with their associated conic sections: 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 ellipse and the contact circles are represented as nonuniform rational B-splines.


Contributed by: Jan Mangaldan (April 2019)
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 Publications, 1963.

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