Dandelin Spheres for the Hyperbola

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The hyperbola can be defined as the curve formed by the intersection of a plane with the two nappes of a cone. In this Demonstration, Dandelin's spheres show the relationship between a hyperbola and its foci and directrices.


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 hyperbola and the contact circles 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.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.