Apollonius and the Dandelin spheres

Defining the ellipse as the locus where the sum of distances is constant is convenient for computation, but it explains neither the name of the curve nor its connection with the cone. To understand that, we must go back to Greek geometry and then jump twenty centuries forward to an elegant nineteenth-century proof.

Apollonius and the conic sections

In the third century BC Apollonius of Perga gathered in his Conics the systematic study of the curves obtained by cutting a cone with a plane. It was he who introduced the names ellipse, parabola and hyperbola, borrowed from the vocabulary of the application-of-areas problems: the word ellipse (from the Greek élleipsis, “falling short”, “deficiency”) signals that a certain area comes out deficient compared with a reference square. Apollonius studied these curves by purely geometric methods, with no coordinates and no reference to foci: the idea that the ellipse was also the locus of constant sum of distances was already known to the ancients, but the link between the two descriptions — the section of the cone and the focal property — remained far from obvious.

The Dandelin spheres

Only in 1822 did the Belgian mathematician Germinal Pierre Dandelin find a proof of that link, as simple as it is surprising. Imagine cutting a cone with a plane that produces an ellipse. In the region between the plane and the apex we insert two spheres, each inscribed in the cone (that is, tangent to it along a circle) and tangent to the cutting plane: one “above” and one “below” the section. These two spheres are today called Dandelin spheres, and they touch the plane exactly at the two foci $F_1$ and $F_2$ of the ellipse.

The argument uses a single elementary fact: two tangent segments drawn from the same point to a sphere have equal length. Let $P$ be any point of the ellipse and draw the generator of the cone through $P$: it touches the two spheres at two points, $A$ and $B$. Then the segment $PF_1$ and the segment $PA$ are both tangent to the upper sphere from $P$, so $PF_1 = PA$; likewise $PF_2 = PB$. Adding, $PF_1 + PF_2 = PA + PB = AB,$ and $AB$ is the distance, measured along the cone, between the two circles of tangency: a quantity that does not depend on the chosen point $P$. Thus, by pure solid geometry, we recover the focal definition $d(P,F_1) + d(P,F_2) = 2a$, with $2a = AB$. This is the bridge joining Apollonius’s ellipse, born from the section of the cone, to the ellipse of the foci that we draw with nails and string (Boyer; Stillwell).

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Topics: Ellipse Concepts: Ellipse · Foci People: Apollonius of Perga · Germinal Pierre Dandelin