Dandelin Spheres

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 Circles are Tangent $F_1$ and $F_2$. Because intersecting tangents have the same length,

$$F_1P=QP$$

$$F_2P=TP.$$

Therefore,

$$PF_1+PF_2=QP+PT=QT,$$

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

See also Cone, Sphere

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.