One cone, four curves

School introduces these curves one at a time and mostly by their equations, in the way it introduces sine as a ratio — correctly, and with the explanation left out. The circle is $x^2 + y^2 = r^2$. The ellipse is the one with the two denominators. The parabola is $y = x^2$, which turns up in falling objects. The hyperbola is $xy = 1$, or the other one with the minus sign, and is chiefly notable for having two pieces for no clearly stated reason.

Four curves, four formulas, no relationship. Then a cone gets sliced and all four fall out of it.

The reframing is worth stating plainly before the pictures start. On the equation account there are four objects and the task is to remember which is which. On the cone account there is one object and one parameter, and the four names are labels for what happens at different settings of it. Everything in this essay follows from taking the second account seriously.

One dial, four names, and nothing else changing. The four panels below are one experiment photographed at four moments, not four experiments.

Turning one dial #

Start with the plane horizontal, cutting straight across. The section is a circle.

Now tilt the plane slightly. The cut is longer on the low side than the high side, so the section stretches — and it stretches into an ellipse, not into some other egg-shaped thing. The section stays a closed curve for as long as the plane is shallower than the cone’s own side.

Keep tilting. There comes a specific angle — the one where the plane is exactly parallel to the side of the cone — at which the section stops closing. One end of the curve runs away up the cone and never comes back.

Tilt past that angle and something new happens: the plane, extended in both directions, hits the other half of the double cone. The section is a hyperbola, and its two branches are not two curves that happen to look similar. They are one section of one surface, taken on two different sheets of it.

The hyperbola’s second branch, which is genuinely mysterious when the curve is introduced as $xy = 1$, becomes obvious the moment the second cone is in the picture. Mysteries of that kind are usually a sign that a diagram has been cropped. It was never a strange feature of the equation. It was the part of the cone the diagram left out.

The parabola is exactly one angle wide #

Worth dwelling on: of all the tilts available, exactly one produces a parabola.

Anything shallower gives an ellipse. Anything steeper gives a hyperbola. The parabola sits on the boundary, at the single angle where the plane is parallel to a line drawn down the cone’s side. Tilt by a hundredth of a degree either way and it is not a parabola any more.

This is the sense in which parabolas are rare among conics and also the sense in which they are important. Boundary cases usually are: they are where the qualitative behaviour changes, and they inherit properties from both sides. A parabola is the ellipse whose far focus has gone to infinity, and it is the hyperbola whose second branch has done the same. That is why an object escaping a gravitational field at exactly escape velocity follows a parabola — orbits below that speed are ellipses, above it hyperbolas, and the parabola is the knife edge.

The same “everything works except at one exceptional place” pattern turns up in stereographic projection, where a single point of a sphere has nowhere to go. Boundary cases are where the interesting mathematics tends to live — see also the exponential base at which the correction factor vanishes.

The angle is not a matter of taste, and it is worth computing rather than eyeballing. The cone drawn throughout this essay has radius $0.92$ at height $1.3$, so a line down its side rises $1.3$ for every $0.92$ outward and makes an angle of $\arctan(1.3/0.92) = 54.71°$ with the horizontal. That is the critical tilt: shallower cuts close, steeper ones reach the second sheet, and the parabola happens at that number and nowhere else.

The three panels bracket it. The ellipse is cut at $24°$, comfortably under; the hyperbola at $72°$, comfortably over; the parabola at $54.71°$, which is not a round number and was never available to be chosen. Change the cone’s proportions and the critical angle moves with them — a taller, narrower cone puts the parabola nearer to vertical — while the ellipse and the hyperbola remain wherever they are relative to it. The cone owns the angle.

The algebra agrees, and it is worth carrying out on this particular cone rather than deferring it, because it shows that the geometric dial and the algebraic classifier are the same quantity rather than two facts that happen to line up. The cone drawn throughout is $x^2 + y^2 = k^2 z^2$ with $k = 0.92/1.3 = 0.7077$. Cut it with a plane $z = z_0 + x\tan\theta$ and substitute; the section satisfies

$$(1 - k^2\tan^2\theta)\,x^2 + y^2 - 2k^2 z_0 \tan\theta\;x - k^2z_0^2 = 0,$$

a general conic with $A = 1 - k^2\tan^2\theta$, $B = 0$ and $C = 1$. Its discriminant is $B^2 - 4AC = -4\big(1 - k^2\tan^2\theta\big)$, which is negative, zero or positive exactly as $\tan\theta$ falls below, equals, or rises above $1/k$ — and $1/k$ is $1.41304$, whose arctangent is $54.71°$.

Put the essay’s own cuts through it. At $0°$ the discriminant is $-4$, an ellipse at its most negative, which is what a circle should be. At $24°$ it is $-3.60$: an ellipse. At $54.71°$ it is $0.0000$: a parabola. At $72°$ it is $+14.98$: a hyperbola.

So the tilt of the plane and the sign of $B^2 - 4AC$ are one number in two costumes. The classification rule taught with the equations — three cases, memorised, no reason attached — is a report on which side of the cone’s own slope the cutting plane happened to fall. It was never a fact about the equation. It was a fact about the cone the equation came from.

This is a place where a drawing can lie convincingly, so it is worth saying how the figures here are kept honest. A cone drawn to a finite height cuts off every section at its rim, and a sufficiently elongated ellipse, clipped at both ends, is indistinguishable by eye from a parabola: one unbounded-looking branch running off the edge of the picture. The difference between the two is entirely in whether the curve would have closed had the cone continued, which is exactly what the drawing does not show. So the generator computes the type from the cut rather than accepting the label, and refuses to draw a figure whose caption names a curve the geometry does not produce.

Dandelin’s spheres #

The slicing story explains where the curves come from. It does not, on its own, explain why they satisfy the definitions used everywhere else — that an ellipse is the set of points whose distances to two fixed foci sum to a constant, for instance. Those two accounts sat side by side for a long time without an obvious bridge.

The bridge was built in 1822 by Germinal Pierre Dandelin, and it is one of the most elegant arguments in elementary geometry. It is worth noticing what kind of argument it is: nothing is computed, no coordinates appear, and the whole thing turns on one fact about spheres — that the two tangent lengths from an external point to a sphere are equal. Everything else is bookkeeping. Arguments of that shape, where a single well-chosen auxiliary object collapses the problem, are rare enough to be worth collecting; Dandelin’s spheres and the two staircases belong to the same family. Take the cone with its elliptical section. Now inscribe two spheres inside the cone, one above the cutting plane and one below, each fitted so that it touches the cone all the way around a circle and touches the cutting plane at a single point.

Those two touching points are the foci. The proof takes about four lines: for any point $P$ on the section, the distance from $P$ to the upper sphere’s touching point equals the distance from $P$ along the cone’s side to the upper contact circle, because both are tangent lengths from $P$ to the same sphere. Same for the lower. Adding them gives the distance along the cone’s side between the two contact circles — which is a constant that does not depend on $P$ at all.

So the sum of the distances to the two foci is constant, which is the focal definition, derived from the slicing definition in a few sentences and with no algebra. Two descriptions of an ellipse that appear to have nothing in common turn out to be the same statement, and the object that reconciles them is a pair of spheres that were not in either description.

The cylinder gives only two of them #

A good way to see what the cone is contributing is to take it away and use the shape that is nearly a cone: a cylinder, which is a cone whose apex has gone to infinity and whose sides have become parallel.

Tilt a plane through a cylinder and the section is a closed curve at every angle. There is no critical tilt, because there is no side-slope for the plane to match — the sides are vertical, and a plane parallel to them cuts along two straight lines rather than producing anything new. There is no second sheet, so there is no hyperbola. The whole list collapses to circles and ellipses, plus the degenerate pair of lines.

What survives is the more interesting half of the claim, and it is a fact people routinely disbelieve: the tilted section of a cylinder is an exact ellipse. Not an oval, not an approximation, not something ellipse-like that a draughtsman would round off. The intuition against it is strong — a cylinder is wider at the bottom of a tilted cut than at the top, so the section ought to be egg-shaped, narrower at one end.

It is not, and Dandelin’s argument transfers without modification. Drop a sphere of the cylinder’s radius into it from above until it rests on the cutting plane, and push another up from below to do the same. Each touches the cylinder around a circle and the plane at one point. For any point $P$ on the section, the two tangent lengths to each sphere are equal, so the sum of the distances from $P$ to the two touching points equals the distance along the cylinder between the two contact circles — a constant. Constant sum of distances to two fixed points is the definition of an ellipse.

The egg-shaped intuition fails for a locatable reason. It assumes the extra width at the low end comes with extra length, and it does not: the cylinder has the same radius at every height, so the section’s width is the same at both ends of the cut and only the lengthwise extent has been stretched. Uniform stretching in one direction sends a circle to an ellipse and nothing else. The cone earns its four curves precisely because its radius does vary with height, and the cylinder — the same surface with that one variation removed — has only one curve to give.

What the picture cannot show #

The slicing figures are honest about shape and dishonest about extent. A parabola runs to infinity and the drawing stops at the edge of the cone; a hyperbola’s branches do the same. Every conic in these pictures is a finite arc standing in for an infinite curve, and the substitution is invisible.

That is not a flaw that can be designed away — an unbounded curve cannot be drawn — but it does mean the pictures cannot settle questions about asymptotic behaviour. Whether a hyperbola’s branches approach straight lines, and which lines, is not visible here at all; it needs either the algebra or a different picture. The same limit applies to the plane in stereographic projection, where the drawing has an edge and the plane does not.

The cone also quietly hides the degenerate cases: to see them the plane must pass exactly through the apex, and “exactly” is not something a diagram can assert.

Why this keeps happening #

The conics are the first example most people encounter of a phenomenon that recurs constantly: a family of objects that looks like a list turns out to be a single object seen from several angles, and the unified view answers questions the list could not.

The list view has to state separately that ellipses are closed and hyperbolas are not, that hyperbolas have two branches, that parabolas are somehow special. The cone view produces all three as consequences of one dial being turned.

It also settles a question the list cannot even ask cleanly: how many kinds of conic are there? From the equations the answer looks like an inventory. From the cone it is a count of qualitatively different outcomes as one angle sweeps from zero to ninety degrees — three generic ones with a boundary between two of them, plus the degenerate cases where the plane passes through the apex. That is a complete answer with a reason attached, of the kind the corner-angle budget gives for the regular solids.

This is the recurring pleasure of the site’s same thing twice thread, and the conics are its cleanest instance: not two descriptions that happen to agree, but one object that had been catalogued four times. Compare Pascal’s triangle and Sierpiński’s, which turn out to be one recursion in two costumes.

It also predicts things the list does not mention. Push the cutting plane through the apex itself and the section degenerates — into a single point, or a single line, or a crossed pair of lines, depending on the tilt. Those degenerate conics are a nuisance to motivate from the equations, where they look like accidents, and they are unavoidable from the cone, where they are simply what happens when the plane passes through the one point at which the surface is not smooth.

The ladder from here #

The rungs above: Dandelin’s argument in full, with the tangent-length equality drawn rather than described. The focus–directrix definition, and why it produces the same curves a third time. Reflection — why a parabola sends every parallel ray to one point, which is the whole of telescope and headlamp design, and why an ellipse’s two foci hear each other in a whispering gallery. Orbits, and Newton’s demonstration that an inverse-square law forces a conic. Escape velocity as the parabola’s physical meaning. Projective geometry, where the four conics stop being four and become one, because the distinction between bounded and unbounded dissolves once the plane gains its points at infinity. And the general conic $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, whose discriminant $B^2 - 4AC$ decides the type — the tilt angle, arrived at by algebra.

Apollonius of Perga worked all of this out in the third century BC, in eight books, and gave the curves the names they still carry. Ellipse, parabola and hyperbola come from the Greek for falling short, alongside, and exceeding — a description of how the cutting angle compares with the cone’s own slope. The vocabulary has outlived the diagram it described, which is a shame, because the diagram is the part that explains it.