Dandelin’s Spheres
Classically, conic sections (the ellipse, parabola, and hyperbola) are defined by the intersection of a plane with a cone. (Related article: Conics.) Their more useful definitions, however, are those from plane geometry. This article presents a now classic proof, due to the French/Belgian mathematician Germinal Dandelin (1794–1847), which shows the equivalence of these definitions.
We take the case of an ellipse; the other cases are quite similar. As a conic section, an ellipse is the intersection of a cone and a plane whose angle to the vertical is larger than that of the generator of the cone. That is, it is the curve that results when a plane slices right through one of the nappes of the cone.
In plane geometry, however, an ellipse is the locus of points in the plane, the sum of whose distances from two fixed points, called the foci, remains constant.
At first glance it is not obvious that these are definitions of the same mathematical object. To show that they are, Dandelin peformed an ingenious construction.
In order to see how the construction works, we must first recall an elementary fact from plane geometry. Consider a circle, and a point outside the circle which we’ll call \(V\). Now draw the two tangents to the circle that intersect at \(V\). Then the points of tangency are equidistant from \(V\).
The standard proof of this fact is straightforward, but we'll content ourselves with noticing that it follows more or less immediately from the symmetry of the situation. Indeed, if we rotate the above figure about its axis of symmetry, we get an immediate extension to the case of a sphere inscribed in a cone:
Evidently, as we would expect, the tangent curve between a sphere and a cone is a circle, one whose every point is equidistant from the vertex of the cone.
Now we may tackle Dandelin’s construction. We begin with a cone, and imagine that a plane has passed through the cone, making an ellipse. We then inscribe two spheres in the cone, one above and one below the ellipse, so that they are each tangent to the cone and to the plane of the ellipse simultaneously. (Imagine putting a perfectly spherical balloon in the cone and blowing it up until it just touches the sides of the cone and the surface of the slicing plane, but no more.)
Now take any straight line in the surface of the cone which passes through the vertex \(V\), and note its points of intersection with the first circle, with the ellipse, and with the second circle. (These points are labelled \(A\), \(B\), and \(C\) in the diagram). Since both the segments \(\overline{VA}\) and \(\overline{VC}\) are the same no matter what line through \(V\) we choose, we must have that their difference, segment \(\overline{AC}\), is always the same. Note that \(\overline{AC}\) is the sum of \(\overline{BA}\) and \(\overline{BC}\).
Now notice that, if we consider the points \(D\) and \(E\) to be the points of tangency between the plane of the ellipse and the large and small spheres respectively, then we must have \(\overline{BA} = \overline{BE}\) since each is a tangent from \(B\) to the small sphere.
Likewise, \(\overline{BC} = \overline{BD}\) since each is a tangent from \(B\) to the large sphere.
Therefore, \(\overline{BA} + \overline{BC} = \overline{BD} + \overline{BE}\). But \(\overline{BA} + \overline{BC}\) was shown to be constant, and so \(\overline{BD} + \overline{BE}\) must be constant. Hence the ellipse is the locus of points the sum of whose distances from two fixed points (\(D\) and \(E\)) is constant. QED
The constructions for the parabola and the hyperbola are almost exactly the same, and the deductions analagous. Students should try a quick sketch of these other two cases, to ensure that the logic of their proofs is transparent.
Here, in Dandelin’s elegantly simple constructions, we find the deepest harmony between our spatial intuitions and the formalisms of Euclidean geometry. To see the unification of these with powerful analytic and algebraic methods, be sure to read the article on conics.
