Replacing the fixed total by a fixed difference

0:00Here is the sentence that defines an ellipse. Two fixed points. Take any place, add its two distances to them, and that sum is the same everywhere on the curve. Now change one word. Sum becomes difference. Two fixed points. Take any place, SUBTRACT one distance from the other, and that difference is the same everywhere. It looks like a small edit. It is not. Adding two numbers is a settled operation. It does not matter which one you write first.

0:35Subtracting is not. Nine take away one is eight; one take away nine is minus eight. So the new sentence is incomplete as it stands. It says 'the difference', and a difference between two lengths has a direction that nobody has chosen yet. Before this definition describes anything at all, that choice has to be made. Here is the choice. Whichever of the two fixed points is FARTHER away supplies the length, and the nearer one is taken off it.

1:10Farther minus closer. That is a convention. Nothing forces it. It is picked, and picking it is what makes the sentence say something. Watch what happens without it. Take the first point's distance and subtract the second's, every time, in that order. On one side of the picture that comes out positive. On the other side it comes out negative. So the same curve would need two conditions to describe it, one for each side, and the constant would be plus eight in one place and minus eight in another.

1:48The convention replaces two conditions with one. And it does something else, which is the whole of the next part. Because farther minus closer never says WHICH point is the farther one, either of them can play that role. And that alone is enough to break the curve into two pieces. Measure it rather than say it. Score a window of four thousand six hundred and forty-one places against the definition, with the two fixed points at minus five and plus five and the constant eight.

2:25Places where the difference is exactly eight: two. Now score the one-way condition on the same window - first distance minus second, in that order. It holds at one place. The other way round, also one. Places where both hold at once: zero. And places where the convention and the two one-way conditions taken together disagree: zero. One condition. Two pieces. Neither piece is a separate curve; they are what a single sentence returns.

3:01You can watch the reversal happen. Take three places of the curve and ask, at each one, which fixed point is the farther. At the first: the left one. At the second: the left one again. At the third: the RIGHT one. The third place sits on the other piece, so the subtraction runs the other way about. Over forty-five exact places of the curve - fifteen on one piece and thirty on the other - the left point is the farther at fifteen of them and the right point at thirty.

3:36Places where neither is farther: zero. And every one of the forty-five satisfies the definition. Not one is rejected. So the reversal is not an exception to the rule. It is the rule, doing exactly what it was chosen to do. Between the two pieces there is a gap, and nothing crosses it. The two places where the curve meets the line through the fixed points are at minus four and plus four.

4:08Of the four thousand six hundred and forty-one places in the window, one thousand nine hundred and eighty-nine lie strictly between them. Places of the curve among those: zero. Not few. None. So there is no animation to be made of a point sliding from one piece to the other, because there is no path. The strip is eight units wide, and the curve is absent from all of it.

4:36A closed curve you can walk around. This one you cannot. The vocabulary comes over with the names changed. The two fixed points are still the foci. The place midway between them is the centre. The line through them is the transverse axis, and the two places where the curve meets it are the vertices. The line through the centre at right angles to that one is the conjugate axis.

5:07Three lengths get names. Twice c between the foci. Twice a between the vertices. And twice b along the conjugate axis. Two of those three you can point at on the curve. The third one you cannot, and that is not a slip of the drawing. It is what the second half of this is about. The definition never said what the constant is. It only said there is one.

5:35So find out. Walk along the line through the two fixed points and ask the definition, step by step, where it is satisfied. With the points at minus five and plus five and the constant eight, it answers at exactly two places: minus four and plus four. The distance between them is eight. Which is the constant. That was not put in by hand. It was walked to, and it could have come out anything at all.

6:07So the constant IS the distance between the vertices, and calling it two a is now earned rather than assumed. Without that, writing two a into the algebra that comes next would be a substitution nobody had paid for. The argument that gets you there has a middle step people lose, so here it is slowly. At the right vertex, the farther focus is the left one, and that distance is nine. The nearer focus is one away. Nine take away one is eight.

6:42At the left vertex everything mirrors. The farther focus is the right one, at nine; the nearer is one away; nine take away one is eight. The two OUTER distances - each vertex to the focus on the far side - came out equal. Nine and nine. So did the two inner ones. One and one. And the outer less the inner is eight, which is the distance between the vertices.

7:10Every length in that argument lies along a single line, which is why they can be added by looking instead of by formula. That, and not anything special about the shape there, is what makes the vertices the convenient place to stand. The constant is not free either. Take any place off the line, and it makes a genuine triangle with the two fixed points - so the difference of two of its sides is smaller than the third.

7:43The third side here is the distance between the two points: ten. Count it. Places in the window where the two distances differ by more than ten: zero. Not a few. Not near misses. Zero. At exactly ten, forty-two places - and every single one of them lies on the line through the two points, none off it. Past ten, for each larger constant offered: nothing at all. So the constant has to be smaller than the separation, which is to say a has to be smaller than c.

8:19And c squared minus a squared is positive exactly when that holds, which matters in a moment. Now the sharp point of the whole topic. For the ellipse, b was FOUND. Walk out from the centre along the axis across the middle, ask the definition where it is satisfied, and it answers: minus three and plus three. Run the identical search on the difference set. It answers nowhere. The same routine, the same rule, the same steps - two places on one definition and none at all on the other.

8:58Because along that axis the two distances are always equal, so their difference is nothing, and nothing is not the constant. So b cannot be measured here. It is written down: b is the square root of c squared minus a squared. For one curve a relation was proved. For the other a formula is defined. That is not a small difference, and it is worth saying out loud rather than letting the symbol carry it.

9:30Which leaves a fair question. If b is not measured on the curve, what is it? It is a constructed length that gets a geometric job afterwards: it marks off the conjugate axis. But the ends of that axis are not places of the curve. Of the two places a formula would put there, at nothing-and-three and nothing-and-minus-three, the difference set accepts zero. Offer the same two places to the sum set, and both are accepted.

10:03So on the ellipse those really are points of the curve, and here they are not. They mark a length. They do not sit on the thing. Put the two side by side, because they can be built from the same three numbers. A difference set with points at plus and minus five and the constant eight has a equal to four, b equal to three and c equal to five.

10:29A sum set with points at plus and minus four and the total ten has a equal to five, b equal to three and c equal to four. Three, four and five both times, with two of the roles swapped. Over six frames of each: the offset c is the largest of the three every time for the difference sets. Six out of six. And a is the largest every time for the sum sets. Six out of six.

10:59So the biggest quantity is the one that sits alone on its own side of the relation, and that is the whole of the sign flip people misremember. One more contrast. b beating a is impossible for a sum set - zero of the six - and it happens for three of the six difference sets. So what has this actually delivered? A definition, with the direction of the subtraction settled.

11:28Two pieces, which followed from that choice rather than being assumed alongside it. The names. The constant, shown to be the distance between the vertices. A formula for b, and the condition that makes it mean anything. What it has NOT delivered is an equation. Not one line of algebra has been written, and the curve has not been put on axes. Nor is there yet a number measuring how open or closed the shape is.

12:00Both of those are the next step, and they follow the same road the ellipse's did: choose the frame, write the definition down, square twice. For now the whole thing rests on one changed word and one chosen direction.