Why a sliced cone is an ellipse with two foci

2026-06-09

An ellipse has two famous definitions, and they look nothing alike. The first: take a cone and slice it with a tilted plane - the edge of the cut is an ellipse. The second: pin two points, call them foci, and trace every point whose two distances to them add up to the same fixed total. A loop of string round two thumbtacks draws exactly this.

These are obviously different recipes. One is about cutting a solid; the other never mentions a cone at all. So why do they make the same curve - and if the slice really is a constant-sum ellipse, where are its two foci? Nothing in the cone points at them.

Here is the answer, due to the Belgian mathematician Germinal Dandelin in 1822. Drop two spheres into the cone, one above the cut and one below. Inflate each until it is wedged snug against the cone - touching all the way round a circle - and just kissing the cutting plane. Each sphere touches the flat plane at a single point. Those two kiss-points are the foci. Press play and a point runs round the ellipse; the sum of its distances to the two foci, read out below, never changes.

The one trick: a tangent length, twice

Press why? on the figure. Take any point P on the ellipse. We want to show PF₁ + PF₂ is the same wherever P sits. The whole proof rests on a single fact about spheres:

From a point outside a sphere, every tangent line touches at the same distance. All the tangent segments from that point have equal length.

Now look at P. The segment PF₁ lies in the cutting plane, which is tangent to the lower sphere - so PF₁ is a tangent segment to that sphere. But P also sits on the cone, on one straight line running from the apex down the surface (its generator). That line grazes the lower sphere too, at the point T₁ on its tangency circle. So P→T₁ is also a tangent segment from P to the same sphere. Two tangents from one point:

PF₁ = P→T₁

The identical argument with the upper sphere gives PF₂ = P→T₂. Add them:

PF₁ + PF₂ = P→T₁ + P→T₂ = T₁T₂

But T₁ and T₂ sit on the same generator, one on each sphere's tangency circle - and T₁T₂ is just the slant distance between those two fixed circles measured along the cone. That distance is the same on every generator: the circles live at fixed heights, and the cone has the same slope all the way round. So T₁T₂ is a constant, and therefore so is PF₁ + PF₂. That constant is the ellipse's full width, the 2a in every textbook. The two definitions are one.

What the spheres were hiding

The foci of an ellipse always feel like they were placed by decree - two special interior points, conjured to make the string trick work. The spheres show they were never arbitrary. They are the shadows of where two perfectly fitted balls touch the page. Tilt the cut steeper and the spheres slide apart and the foci spread; tilt it back toward level and they rush together until, at a flat cut, the two spheres meet at the centre and the ellipse rounds into a circle with a single focus.

Keep tilting past the spheres' reach - until the plane runs parallel to the cone's side - and the upper sphere races off to infinity. One focus escapes, the curve fails to close, and the ellipse becomes a parabola. The same two spheres, pushed further, explain every conic section there is.

Two balls in a cone. Each touches the slice once. The string and the knife were drawing the same curve all along - and the foci were just the points where the spheres came to rest.