One surface, four curves, chosen by the angle of the cut

0:00Circle, ellipse, parabola, hyperbola. Four curves, four names, and four sets of formulas waiting for you. They are not four things. They are one surface, read at four settings of a single number. Building the surface costs exactly one angle, and cutting it costs exactly one more. Everything after that is a comparison between those two angles - three inequalities and two equalities, and nothing else at all. That is worth checking rather than announcing, so this video builds the surface, cuts it four thousand six hundred and eight times, and lets the curves it makes say for themselves what they are.

0:47Take a line and hold it still. Take a second line crossing it, and set it at some angle to the first. Now spin the second line about the first, keeping that angle unchanged the whole way round. What it sweeps out is the surface, and one angle is the only thing about its shape you ever chose. Sixteen such angles were built here, and the spinning line was stopped at sixty-eight bearings on each: one thousand and eighty-eight positions in all.

1:19Every one of them was put back into the equation of the surface it had just swept, and the number that turned out not to lie on it is none. One thousand and eighty-eight decoys standing one step to the side were offered to exactly the same test, and the number the surface accepted is none. And the angle each position makes with the line that stayed still, read back off the finished surface instead of off the recipe, takes sixteen values - one for each cone, and one only.

1:56Four words carry the whole of the rest. The vertex is the point where the two lines cross, and every position of the spinning line runs through it. The axis is the line that stayed still. A generator is the spinning line frozen in one position, and there are as many of them as there are directions to point in. And the surface comes in two halves, meeting only at the vertex, because a line spun about a point sweeps backwards as well as forwards.

2:30Of the one thousand and eighty-eight generators, all one thousand and eighty-eight run out above the vertex, and the same one thousand and eighty-eight run out below it. The surface is hollow, it is only the surface, and it carries on without end in both directions. Now cut it with a plane. The second angle is the one the plane makes with the axis - not with the horizontal, not with the edge of the page, with the axis.

3:03Four thousand six hundred and eight cuts were made: sixteen cones, eighteen angles, four bearings round the axis, four distances out from the vertex. For every one of them the angle was read back off the plane's own direction and the cone's own axis, rather than off the number the cut had been set up with. The number of cuts where those two readings disagreed is none. And the name of the curve never moved with the bearing or with the distance: two hundred and eighty-eight pairs of angles, and none where the answer depended on anything except the two angles.

3:45So the cone carries one number, the cut carries one number, and there is nothing else to vary. Start with the plane square on to the axis. Every generator leans away from the axis by the same amount, so a plane square on to the axis reaches all of them at the same depth. Measure out from the vertex to each meeting point and see: sixteen such cuts were made, and every one gave a single distance the whole way round.

4:17That is a circle, and now it is a circle for a reason you can say out loud. Tilt the plane at all and the depths spread apart - two hundred and seventy-two of the remaining cuts gave more than one distance. Meeting every generator at the same depth is not a property of cutting a cone. It is a property of cutting it square on. Tilt the plane, but keep it leaning more steeply than the cone's own angle.

4:48The curve leans with it, and it still closes. That is an ellipse, and why it closes is the whole difference between this case and the next one. A plane runs alongside a generator, and never reaches it, exactly when the two lean at the same angle. If the plane is steeper than every generator, then it is parallel to none of them. So it meets every single generator, once each, and the curve has nowhere to escape to.

5:19All those meeting points sit on the same half of the surface, and the count of generators such a plane never reaches came out none, every time. Now bring the plane down until it leans at exactly the cone's own angle. One generator is now parallel to the plane, and the plane will never reach it. Exactly one - not two, and not none. That single unreachable generator is what stops the curve closing, because it runs off towards a direction the plane can never catch up with.

5:57That is a parabola, and it lives at one exact value of the angle. Nudge the cut a hair steeper and it shuts into an ellipse. It is not a stretch of angles you can land in. It is a knife edge you have to aim at. Shallower still, below the cone's own angle. Now two generators lean at less than the plane does, so two of them run alongside it instead of one.

6:25And with two directions escaping, the curve no longer stays on one half of the surface - it runs past the vertex and into the other half. Every cut down here was checked for that, and the number of halves it reached came out two. So the curve has two branches. Not two cuts, and not two planes. One plane meeting one surface once, and the surface having two halves.

6:52That is a hyperbola, and the case runs all the way down to a plane lying alongside the axis: sixty-four such cuts, and all sixty-four came out hyperbolas, endpoint included. Put the four cases on a single line of angle and the shape of the thing is plain. Of the four thousand six hundred and eight cuts, two hundred and fifty-six came out circles, one thousand nine hundred and twenty ellipses, two hundred and fifty-six parabolas, and two thousand one hundred and seventy-six hyperbolas.

7:32Every one of those names was worked out from the surface and the plane. The four conditions were scored against them afterwards, and the number they got wrong is none. Now look at where they sit. Two of the four are single exact angles and the other two are whole stretches of angle. Five hundred and twelve of the cuts landed on one of the two exact angles, and four thousand and ninety-six landed inside one of the two stretches.

8:04So an angle chosen carelessly hands you an ellipse or a hyperbola, and the circle and the parabola are the two you have to aim at. There is something in that list which is easy to walk straight past. The circle is given the square-on cut and the ellipse is given everything strictly below it, so on this list the two are separate cases and the circle is not one of the ellipses.

8:33Whether a circle counts as an ellipse under some other definition is a different question with a different answer, and mixing the two together is where the trouble starts. Here it was checked in both directions. Two hundred and fifty-six of the cuts came out round; of those, the number made at anything other than a square-on cut is none, and the number of square-on cuts that failed to come out round is also none.

9:03Round and square-on are two readings that agree, rather than one reading written down twice. The angle is measured against the axis, and it is worth saying twice, because the picture makes it easy to forget. In the usual drawing the axis stands upright, so a square-on cut looks horizontal, and it is tempting to think of the second angle as tilt away from the horizontal. It is not, and here is the check.

9:35The whole experiment was run a second time on a cone tipped over, with its axis no longer upright. One thousand one hundred and fifty-two cuts. Read the angle against the cone's own axis and the four conditions get none of them wrong. Read those same cuts against the upright direction instead, and they get three hundred and seventy-eight of them wrong. Horizontal is what square-on happens to look like while the drawing stands up straight. It is not what the angle means.

10:11There is a condition on all of this, and it sits in the opening clause of the classification, where almost nobody reads it. The plane has to miss the vertex. That was not written in here as a filter. The plane was pushed straight through the vertex on purpose, one thousand one hundred and fifty-two times, and the naming routine declined every single one of its own accord, because what it produces there is no longer a curve.

10:42Here is what is left instead: five hundred and forty-four of them collapsed to the vertex alone, sixty-four to a single whole generator, and five hundred and forty-four to two generators crossing there. Those surviving generators were looked for rather than argued about - one hundred and ninety-two of them were found lying inside the plane, exactly as many as the plane's own arithmetic said there would be. A point, a line, a pair of lines: three ways of failing to be one of the four curves.

11:20So the opening clause is not decoration. Without it the classification is simply false. The cone does its work here and is then put away, and what it leaves behind is the part worth carrying. There is one surface, and one number saying which surface. There is one cut, and one number saying which cut. The four names are four readings of the comparison between those two numbers - against each other, and against a right angle.

11:52Nothing else about the plane matters at all, provided it misses the vertex: not where it sits, and not which way it faces. That is what the word conic asserts. Not four definitions, but one number with four answers.