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Dandelin Spheres


DandelinSpheres

The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres.

The spheres can be used to show that the intersection of the plane with the cone is an ellipse. Let pi be a plane intersecting a right circular cone with vertex O in the curve E. Call the spheres tangent to the cone and the plane S_1 and S_2, and the circles on which the spheres are tangent to the cone R_1 and R_2. Pick a line along the cone which intersects R_1 at Q, E at P, and R_2 at T. Call the points on the plane where the sphere are tangent F_1 and F_2. Because intersecting tangents have the same length,

 F_1P=QP
(1)
 F_2P=TP.
(2)

Therefore,

 PF_1+PF_2=QP+PT=QT,
(3)

which is a constant independent of P, so E is an ellipse with a=QT/2.


See also

Cone, Sphere

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References

Honsberger, R. "Kepler's Conics." Ch. 9 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., p. 170, 1979.Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 40-44, 1991.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 80-81, 1990.Ogilvy, C. S. Excursions in Mathematics. New York: Dover, pp. 68-69, 1994.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 48, 1991.

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Dandelin Spheres

Cite this as:

Weisstein, Eric W. "Dandelin Spheres." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DandelinSpheres.html

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