Why a Slanted Cut Through a Cone Is an Ellipse

Four Cuts

A level plane cuts a cone in a circle. Tilt the plane and the circle stretches into an ellipse. Tilted until it runs parallel to one of the cone’s straight lines, the curve no longer closes: a parabola. Steeper still, a hyperbola.

The model’s glass cone stands on its tip, with a half-angle of 20°. The glass cut turns from 0° to 84°, and the four curves appear in turn. Together they are the conic sections.

A whole hyperbola has two branches; the second is on the other half of the cone, the one that would open upwards from the tip. The model has only one half, so it cuts only one branch.

Two Pins and a String

Another way to draw: put in two pins, tie the ends of a string that does not stretch to them, pull the string tight with a pen and go round. The pen’s distances to the two pins always add up to the string’s length.

The curve drawn this way is an ellipse, and the two pins are its foci. Its distances to the two foci have the same sum everywhere: that is what an ellipse is. Move the pins together and the ellipse rounds out; put them in one place and it is a circle.

The Same Curve?

The cut through the cone and the curve drawn with string look alike, but looking alike is not being the same. To show the cut is an ellipse, you need two points in the plane of the cut such that every point of the curve has the same sum of distances to them.

The cone has no pins and no string. Where those two points are cannot be seen from the cut itself. In 1822 the mathematician Germinal Pierre Dandelin gave a proof that needs almost no calculation.

Two Spheres

Put a sphere in the tip of the cone so that it rests against the cone’s inside and touches the cut from below. Above the cut, put a larger sphere that also rests against the inside and touches the cut from above.

Each sphere touches the cone all the way round, along a circle. Each touches the cut at a single point. Those two points, F₁ and F₂, are the foci of the ellipse.

In the model the small sphere stays in the tip and the glass cut leans on it. When the tilt changes, the large sphere changes size with it, always just touching both the cone and the cut.

Equal Tangents

From a point outside a sphere, every tangent line to the sphere has the same length, and the points of contact form a circle.

Take a point P on the curve. The segment PF₁ lies in the cut, and the cut touches the small sphere only at F₁, so PF₁ is a tangent to the small sphere. The cone’s straight line through P grazes the small sphere too, touching it at Q₁ on its circle of contact, so PQ₁ is a tangent as well. Hence PF₁ = PQ₁. In the same way, PF₂ = PQ₂.

So PF₁ + PF₂ = PQ₁ + PQ₂ = Q₁Q₂. Q₁Q₂ is the piece of the cone’s line between the two circles of contact, and it is the same length on every line of the cone. Wherever P goes, the sum stays the same, and the cut is the ellipse with foci F₁ and F₂.

One Sphere Left

When the cut leans until it runs parallel to a line of the cone, the sphere above it would have to be infinitely large to touch both the cone and the cut. There is no such sphere. Only the small one in the tip is left, with its focus F.

The level plane of the small sphere’s circle of contact meets the cut in a straight line, the directrix. Every point P of the curve is as far from F as it is from the directrix. That is what a parabola is.

The reason is again equal tangents: PF = PQ. PQ runs down the cone’s line from P to the plane of the circle, and PD runs down the cut from P to the same plane. Here the cut and the cone’s line have the same slope, so the two are equally long.

Ellipses and hyperbolas have directrices too, and for all of them PF divided by PD is the eccentricity, e = sin φ / cos α. φ is the cut’s tilt from the level and α the cone’s half-angle, measured from the axis. Below 1 it is an ellipse, at 1 a parabola, above 1 a hyperbola.