Two fixed points and a fixed total distance

0:00A circle is one fixed point and one fixed distance. Every place at exactly that distance from the point, and nothing else. Change two things about that and you get the next curve in the family. Instead of one fixed point, take two. And instead of asking for one distance, ask that the two distances add up to a fixed total. That is the entire definition. Two points, one number, and a question you can ask of any place in the plane.

0:32Do my two distances add to that number? If they do, the place belongs to the set. If they do not, it does not. Notice what the definition does not say. It does not say the answer will be a nice closed curve. It does not even say there will be any places at all. Both of those turn out to hang on a single comparison, and that comparison is what this is really about.

1:02Take two points eight apart, and a total of ten. Here is a place the definition accepts. Measure to each point. Five and five. Move to another one. Nine and one. A third. Seven point four and two point six. Every time, ten. That is not a theorem about ellipses. It is the definition doing exactly what it says. The total stays the same because we only let in the places where it was ten to begin with.

1:33To see what shape that carves out, every place on a grid of twenty five hundred and one was put to the question. Twelve of them satisfy it exactly. Eleven hundred and sixty one fall short. Thirteen hundred and twenty eight overshoot. And those three add back to twenty five hundred and one, so nothing was counted twice and nothing was missed. One more thing about how that was checked. None of it used decimals.

2:04Adding two square roots and comparing the answer to a number is done here by an exact test on whole numbers, so equal really means equal. Offer four hundred and one different fractions to a sum of two square roots that is not itself a fraction, and the number of them it calls exactly equal is zero. Now the condition, and it is the whole point of this topic. The total is not free. It has to be bigger than the distance between the two fixed points.

2:40Bigger. Not equal to. Not at least as big as. Strictly bigger. It is easy to read past that as small print and go on to the picture. It is not small print. It is the sentence that decides whether the definition defines anything at all. With the two points eight apart, a total of ten is allowed and a total of eight is not. Nothing about that is arbitrary, and you already know why.

3:10Take any place off the line through the two points, and join it to both of them. You have a triangle. And two sides of a triangle always come to more than the third. The two distances are two of the sides. The gap between the fixed points is the third. So the two distances always come to more than that gap. Always. Which means a total smaller than the gap is asking for something the plane cannot supply.

3:40That is the condition, and it is not a fact about this curve. It is the triangle inequality wearing a different hat. Counted over the whole grid, against a gap of eight. Two thousand four hundred and sixty places have the two distances adding to more than eight. Forty one have them adding to exactly eight. And the number adding to less than eight is zero. Not one place in twenty five hundred and one. That zero is the triangle inequality, counted rather than quoted.

4:18So what happens at the boundary? Set the total equal to the gap. Eight and eight. The definition still runs. It just stops being a curve. Forty one places pass, and every one of them lies on the line between the two fixed points. The number lying off it is zero. There is no inside either. The number of places held inside is zero. And the two ends of what survives are the two fixed points themselves.

4:48The curve has flattened onto the segment joining them. This is exactly the flattened triangle from a moment ago. A place between the two points does not make a triangle at all. Its two distances add to the gap, because it is sitting on the gap. Push the total below the gap and there is nothing to flatten. Six, with the points still eight apart. Places of the grid that pass: zero.

5:18Places held inside: zero. And along the whole line through both points, where the last survivors would have to be if there were any: zero again. The set is empty. Not small, not degenerate. Empty. So the condition is doing real work. Above the gap you get a curve. At the gap you get a segment. Below it you get nothing. One inequality, three completely different answers. There is one more edge worth visiting, in the other direction.

5:54Slide the two fixed points towards each other until they are the same point. The gap is now zero. Any positive total beats zero, so the condition still holds. And the definition still runs. Both distances are now the same distance, so asking them to add to ten is asking for distance five. Fourteen places of the grid pass, and the number of those at anything other than five from that point is zero.

6:25It is a circle. So a circle is not a squashed anything, and this curve is not a squashed circle. They are the same definition, with the two points either apart or on top of each other. Back above the boundary, where there is a curve, and time to name the parts. The place halfway between the two fixed points is the centre. The line through both fixed points cuts the curve at two places. Those two are the vertices, and the piece between them is one axis.

7:01The line across the centre, square to that one, cuts it at two more. That piece is the other axis. Every one of those was found by searching, not by a formula. Forty different sets were built, at two different centres, along four different directions, in five different shapes. The number where the search along the line of the fixed points did not return exactly the two ends the total names is zero.

7:31The number where the search across it did not return exactly two is zero. The number where a fixed point turned out to lie on the curve is zero. They are inside it, never on it. And the number whose halfway place was not held inside is zero. All of that was then repeated with the whole picture turned by an exact angle and slid. Seven thousand five hundred and three scorings, zero disagreements.

8:03One of those two axes gets called the major one. And it is worth being precise about why. It is the axis through the two fixed points. That is the naming rule, and it is a decision, not a discovery. It is tempting to say it is called major because it is the longer one. In all forty sets built here it did come out longer. But that is forty measurements, not a proof, and nothing so far has shown it has to be.

8:34It does have to be, and showing it is the next step. Until then, major means through the fixed points, and the fixed points are on it by definition rather than by argument. Three lengths, and the notation is where people lose marks. The whole axis through the fixed points is written two a. So a is half of it. The whole axis across is two b, and b is half of that.

9:03The gap between the fixed points is two c, and c is half of that. Take the set we started with. Searching along the first axis returns ends at minus five and plus five. So that axis is ten long. Which is the total itself, not half of it. Searching across returns minus three and plus three, so that axis is six. And the two fixed points are eight apart.

9:31So a is five, b is three, and c is four. The constant everybody stores as a is two a. The letter b runs to the end of the short axis. The letter c runs to a fixed point. Those two get swapped constantly. And now the honest state of play. Three lengths have names. There is no stated relation between them at all. Nothing so far has needed one. The definition never mentions b. The condition only compares the total with the gap.

10:07Every measurement in this video came from searching a set of places, never from a formula linking a, b and c. That was on purpose. Look at the numbers though. Five, three and four. Across the forty sets there were only five different pairs of axis lengths, and in every single one the axis through the fixed points was the longer. Five, three and four is not a coincidence, and the reason is the next thing to build.

10:40Finally, what all of this is for. A runner on a course, whose distances to two flag posts always add to ten metres. The posts are eight metres apart. What path does the runner trace? The first move is not algebra. It is the comparison. Ten is more than eight, so the condition holds and a path exists. Then read it off. The total is two a, so two a is ten and a is five. The posts are two c apart, so two c is eight and c is four.

11:15Now the same two posts, and a runner whose distances always total six. Six is less than eight. Places anywhere on the grid: zero. Places on the line between the posts: zero. That runner does not exist. No amount of algebra was going to produce a path, because there is not one. So check the comparison first. It costs one line. It is the difference between a question with an answer and a question that only looks like one.