The gap between two parallel lines as one distance measured once

0:00Two lines run alongside each other and never meet. How far apart are they? Stand at a point of the first line and measure across to the second, and you get a length. Walk a little way along the first line and measure again. Nothing so far says the two lengths have to agree. The phrase everybody uses - the distance between two parallel lines - is quietly claiming that they do.

0:28Before that phrase means anything at all, somebody has to check it. In a window of one hundred and sixty-nine points there are one hundred and forty-eight thousand and fifty pairs of distinct lines running in the same direction. For every one of those pairs, the shortest way across was measured from every single point of the first line, and then from every point of the second. Pairs where two points of a line gave different answers: none.

1:02Pairs where the two directions of measuring disagreed: none. And to be sure the test can say no, the very same machinery was run on thirteen thousand seven hundred and seventeen pairs of lines that do cross. Forty-seven of those came out constant as well - every one of them a line the window sees at exactly twice, whose two points sit either side of the crossing line. Of the four thousand and thirty-two crossing pairs whose first line has three points or more, the number that came out constant is none.

1:41Write the two lines with the same slope and different heights, and draw them. The first one crosses the flat axis somewhere, and that crossing is a point of the first line that costs nothing to find. Its second coordinate is nothing, and its first coordinate is minus the first height divided by the slope. That is the point the argument measures from. Drop the perpendicular from there onto the second line and its length is the answer.

2:12It is the point-to-line formula, run once, at a point chosen because it was easy to write down. Put that point into the second line's equation and watch what happens. The second line, tidied, is the slope times x, minus y, plus the second height. At our point, y is nothing, so the middle term goes. And x is minus the first height over the slope, so the slope multiplying it cancels the slope underneath, leaving minus the first height.

2:48What is left upstairs is the second height minus the first, and the size of that. Underneath is the root of one plus the slope squared. So the answer is the difference of the two heights, divided by that root. There is a hole in that, and it is exactly where the argument was most convenient. The point we measured from was a crossing with the flat axis, found by intersecting.

3:17Ask the intersection routine for that point on every line the window sees, and it refuses thirteen times. Twelve of the refusals say the two never meet. One says something different: that line does not miss the flat axis, it IS the flat axis, so every point of it is a crossing and no single point is the crossing. Two distinct reasons, and the routine produced both without being told to look for them.

3:47Those thirteen lines are exactly the ones every point of which stands at one height - none refused that is not level, and none level that was not refused. Which makes seventy-eight of the parallel pairs impossible to run this argument on at all. But now look at what the formula itself says about those seventy-eight. If the two lines are level, the slope is nothing, so the root of one plus the slope squared is just one.

4:20The answer collapses to the difference of the two heights, full stop - which is the vertical gap you could measure with a ruler. It is plainly right, and the argument that produced it could not be run. Scored properly, one pair in every seven was put through the whole derivation: twenty-one thousand one hundred and thirty-eight of them went through, with no disagreement anywhere against the smallest way across.

4:50Twelve refused to start. So the exclusion belongs to the choice of point, and not to the result. Write both lines in the general shape, with the same two leading coefficients and different constants. Now take any point of the first line at all - do not say which one, do not write its coordinates down. Because that point is on the first line, the first coefficient times its x plus the second times its y is minus the first constant.

5:24So when you feed it into the second line's left-hand side, those two terms collapse and what is left is the second constant minus the first. The numerator is the difference of the two constants, and no coordinate of the point has survived. Divide by the root of the two leading coefficients squared and added, and that is the whole thing. Over all six hundred and eighty-three thousand eight hundred and forty-four points stood at, the number where that left-hand side was not the difference of the two constants is none.

6:02No point was chosen, and there is no case to leave out. We asked whether the answer depends on where you stand, and here is the reason it cannot. The derivation began at an arbitrary point of the first line and ended with an expression the point is not in. That is not the same as measuring at one convenient place and hoping. It is why the phrase is allowed to say THE distance rather than A distance.

6:34What is the same, every time, is the answer within one pair. The two equations must carry the same pair of leading coefficients. Two parallel lines can perfectly well be written with coefficients that are proportional but not equal - multiply one equation through by three and it is the same line, still. Forty-five thousand five hundred and fifty-six such rescalings were tried, and the number that stopped naming the line they came from is none.

7:09The equation is fine. It is the subtraction that is not. Subtract the constants anyway and the answer is wrong forty-four thousand nine hundred and twenty-one times. It is right six hundred and thirty-five times, which is the dangerous part, and those six hundred and thirty-five are not random. Three hundred and sixteen of them are pairs whose second line passes through the origin, so its constant is nothing and multiplying it changed nothing.

7:44The other three hundred and nineteen are pairs where the rescaling happened to land on the mirror image of the right constant. Accidents of any other kind: none. So scale first, every time. A related habit is to measure between the two heights, straight up the page. That is not the distance, and the formula tells you exactly by how much it misses. Of one hundred and forty-seven thousand nine hundred and seventy-two parallel pairs that both reach the upright axis, the vertical gap between them equals the true distance in seventy-eight cases.

8:27Those seventy-eight are the level pairs - the only ones where up the page and across are the same direction. Everywhere else the vertical gap is too big, and the number it is too big by is the divisor in the formula. Pairs where that relation failed to hold: none. So the root underneath is not decoration. It is the correction between the gap you can see and the gap you want.

8:57Once the two constants are matched, the line lying midway between the two is almost free. Keep the leading pair, double it, and take the sum of the two constants. Four thousand seven hundred and seventy-six of those were built and then checked by minimising onto them. Ones whose distance varied along either line: none. Ones not equally far from the two: none. And the squared distance is a quarter of the pair's, every time, which is what halfway means.

9:31Here is the version that catches people: nine x plus six y minus seven, and three x plus two y plus six. Those are parallel, but the leading pairs do not match, so scale the second by three - nine x plus six y plus eighteen - and only then average. The midway line is eighteen x plus twelve y plus eleven, and its squared distance from each of the two is six hundred and twenty-five over four hundred and sixty-eight.

10:05Average the constants without scaling and the line you get is not equally far from the two at all - it sits much nearer the first. Three x minus four y plus seven, and three x minus four y plus five. The leading pairs already agree, the constants differ by two, the root of nine plus sixteen is five, so the distance is two fifths. Fifteen x plus eight y minus thirty-four, against fifteen x plus eight y plus thirty-one: the constants differ by sixty-five, the root of two hundred and twenty-five plus sixty-four is seventeen, and the answer is sixty-five over seventeen.

10:54Some letter times x plus y, plus one number; and the same letter times x plus y, minus another. Eighty whole-number cases were tried and every one gave the sum of the two numbers over that letter times the root of two. But sixteen values were refused outright, because when the letter is nothing neither equation names a line at all, and there is nothing to be a distance between.

11:23And to draw a parallel through a given point, keep the leading pair and solve for the constant. Parallel to three x minus four y plus two, through minus two and three, is three x minus four y plus eighteen. Keeping the leading pair and changing only the constant walks you along the parallel family; three hundred and twenty-four of those were built and the given point sits on every single one.

11:54And if the two constants are equal, the two equations are one line and the distance is nothing, which the formula gets right without being asked. So: there is no new formula in any of this. There is the point-to-line formula, used once, and one observation - that the answer does not depend on where you used it. That observation is not obvious, it is not usually proved, and proving it is exactly what makes the phrase mean something.