Cuts through the vertex, where the curve degenerates

0:00Every classification has a clause in it that nobody reads. The one that sorts the cuts of a double cone carries a small condition: the plane has to meet the surface somewhere other than the point where the two halves join. That point is the vertex, and the clause quietly puts it out of bounds. It is there for a reason. Without it, the four familiar answers stop arriving. So take the clause out.

0:30Leave the angles exactly as they were, and slide each cutting plane down until it passes through the vertex. Nothing else changes - same surface, same tilts, the same comparison of two angles. What comes back is a point, a straight line, and two straight lines crossing. None of them is a curve, and none of them is a mistake. Everything that follows comes from one property of that one point.

1:01The surface is swept by a line. Hold one line still, cross a second one over it at a fixed angle, and spin the second line all the way round. Every position that spinning line takes is called a generator, and a generator lies entirely inside the surface. And every one of them passes through the same place. That is what the vertex is. This was checked rather than assumed.

1:30Two thousand one hundred and seventy-six generators were built and walked out from the vertex in both directions, and the number that missed the vertex or left the surface is none. The same number of decoy lines were built one step off each generator, and the number the surface accepted anyway is none. So a plane through the vertex is a plane through a place every generator already visits. It cannot come away empty, and it cannot come away with a curve that avoids the vertex.

2:07Start with the plane tilted steeply - steeper than the cone's own half-angle, up to and including standing square across the axis. Above the vertex, a plane like that cuts a closed curve. Slide it down onto the vertex and the closed curve shrinks to nothing at all: the vertex, and no other place. The reason is that the plane is leaning too steeply to lie along any generator. It crosses every one of them, and a line and a plane that cross meet at exactly one place.

2:42Each generator meets it at the vertex. So the vertex is the whole of it. To test that, forty directions were laid down inside each such plane - one for every line through the vertex the exact arithmetic can name. Across every steep cut, the number of those directions that turned out to lie on the surface is none. Now the fact this whole topic turns on. Away from the vertex, that steep range is not one case but two.

3:16A plane standing exactly square across the axis cuts a circle. Leaning even slightly less than square, it cuts an ellipse. Nine thousand two hundred and sixteen cuts that miss the vertex were made and named here. Five hundred and twelve of them stand square across the axis, and every single one came out a circle. Three thousand eight hundred and forty lean between the two angles, and every single one came out an ellipse.

3:48That distinction is real, and it is drawn by the square angle. Now pin the plane to the vertex, and watch it disappear. Of the cuts made through the vertex, one hundred and twenty-eight stand square across the axis - and all one hundred and twenty-eight come away with the vertex alone. Nine hundred and sixty lean between - and all nine hundred and sixty come away with the vertex alone.

4:16Two conditions, one answer. The angle that used to divide them has changed hands. It has stopped being a boundary and moved inside the range. Lean the plane back until it makes exactly the cone's own angle. Now it runs parallel to one of the generators. And because it passes through the vertex, which that generator passes through too, parallel becomes contained. The plane holds the whole line. One hundred and twenty-eight cuts came out this way, and each one was made to hand its line over.

4:55The lines were written down and then walked - four places along each - and the number of places that left the surface, or left the plane, is none. So this is not an argument that a line ought to be there. The line is exhibited, and then it is checked. Holding a generator is exactly the condition that turns a curve into a line, and the reason is about sides.

5:23A plane cuts space into two sides. For every cut, the surface was asked which of those two sides it reaches. Where the plane is steep, the surface stays on one side and touches at the vertex only. Where the plane makes the cone's own angle, the surface still stays on one side - but now it touches along a whole line instead of at a single place. One thousand two hundred and sixteen cuts keep the surface on one side.

5:54Of those, one thousand and eighty-eight touch at the vertex alone, and one hundred and twenty-eight touch along a line. The plane is leaning against the cone without ever going through it. That is a very particular position to be in, which is why exactly one angle produces it. Lean the plane back past the cone's own angle, and it loses the ability to lean against the surface at all.

6:24Now it cuts in. It is shallower than every generator, so instead of touching one it passes clean through, and comes away holding two of them, crossing at the vertex. One thousand and eighty-eight cuts came out as a crossed pair. And each of those lines runs through the vertex and straight out the other side, into the second half of the surface. That was checked too. Of every line exhibited, the number that stayed on a single half instead of running through to the other is none.

7:00The cross you draw is not two half-lines meeting at a corner. It is two whole lines passing through each other. There is a distinction hiding inside this case that is easy to draw and easy to miss. The plane may contain the cone's axis, or it may not. Tip it so the axis lies inside the plane, and you get a crossed pair. Tip it off the axis, still shallower than the cone's angle, and you get a crossed pair.

7:31Of the shallow cuts, one hundred and twenty-eight hold the axis and nine hundred and sixty do not, and the number that came out as anything other than a crossed pair is none. The picture changes. The answer does not. Now count what is left. Away from the vertex there are four names: circle, ellipse, parabola, hyperbola. Through the vertex there are three kinds: a point, one line, two lines.

8:03The missing fourth is not missing. It was absorbed. The circle and the ellipse were being kept apart by a condition that no longer decides anything, so two ranges became one range and two answers became one answer. And between them the two lists account for every plane there is. Two thousand five hundred and ninety-two planes were offered to both lists at once. The number that both lists were willing to name is none, and the number that neither list would name is also none.

8:39Either a plane passes through the vertex or it does not. There is no third possibility, and no overlap. Two of the three collapsed answers are usually given a name that points back at the curve they came from. The single line is called a degenerate parabola, because it arrives at exactly the angle that produces a parabola. The crossed pair is called a degenerate hyperbola, for exactly the same reason.

9:11The point is normally left unattached. That is a decision about naming rather than about geometry, and it is worth noticing instead of memorising. The point has two curves standing behind it, not one. There is no single name for it to inherit. A parabola can also be described without any cone at all. Fix a point, fix a line, and take every place that is as far from the point as it is from the line.

9:43Push that description to its own boundary by putting the point onto the line. Four thousand and fifty-six places were offered to a point sitting on its own line, and fifty-nine of them came out equally far from both. Every one of those fifty-nine lies square across the line, through the point - and the number that was equally far from both without lying square is none. So the answer collapses to a single straight line, standing at a right angle to the one you started with.

10:18Lift the point back off the line and the collapse stops immediately: six places came out equally far from both, and they sit at five different positions along the line instead of piling up at one. So a parabola degenerates to a straight line twice over, by two routes that share nothing. One line is a generator of a cone. It lies inside a surface, at the one angle where a plane stops crossing and starts leaning.

10:52The other has no surface anywhere near it. It is fixed by being square to a line, through a point sitting on that line. There is no cone in the second argument at all - no axis, no half-angle, nothing spinning. Same word, same kind of collapse, two independent reasons. That is a far better fact than one line wearing two labels. None of this needed algebra, but it is worth knowing where the algebra would put it.

11:24Write out what the plane cuts, in coordinates laid down inside that plane, and what you get is six numbers. Move the plane onto the vertex, and the vertex lands on the origin of those coordinates, so the last three of the six go to nothing. What survives is a comparison built from the first three, and the sign of that one comparison is the entire classification. Negative gives a point. Nothing gives one line. Positive gives two.

11:58Three answers out of one sign - which is why there are three of them and not four. The curves did not break here. One condition stopped being able to separate two of them, and everything else went on doing exactly what it always did.