Recovering the angle between two lines from their two slopes

0:00Two lines cross. How big is the angle between them? That sounds like a complete question. It is not. A crossing does not make one angle. It makes four. Counted over three thousand three hundred and sixty crossings, the number offering more than two different sizes is zero. And two hundred and twenty-four of those crossings offer only one size - every one of them a pair meeting at a right angle.

0:30So everywhere else there are two answers on the table, and the question as asked does not say which one it wants. Here is the picture: two lines, each crossing the horizontal somewhere of its own, meeting above it. At each of those there is an inclination: sweep anticlockwise from the positive horizontal direction round to the line. Call the shallower one alpha and the steeper one beta. Now the meeting point itself. Two angles are marked there, one directly above the other. Call them theta and phi.

1:10The two facing each other across the crossing are equal, and the two side by side fill a straight angle between them. Both were checked at every one of those crossings, and the failures number zero either way. Now the move it all rests on, and it is a fact about the picture, not algebra. The two lines and the horizontal cut off a triangle. Theta sits outside it at the meeting point, and an exterior angle is the two interior angles it does not touch, added together.

1:47Work that through and theta is exactly the amount by which the steeper line's inclination exceeds the shallower one's. Theta is beta minus alpha. Not a formula yet. A difference of two angles. That claim was tested by building the sectors at each crossing from the four rays alone, with no inclination allowed anywhere near them. Crossings where those sectors were not the two differences of the two inclinations: zero.

2:20So the angle you want is a difference of two angles, and the tangent of a difference has a standard expansion. The tangent of beta minus alpha is the tangent of beta less the tangent of alpha, all over one plus the two tangents multiplied together. And each of those tangents has a name already. The tangent of an inclination is that line's slope. Put the two names in. Above the line: the second slope less the first. Below it: one plus the two slopes multiplied together.

2:58The tangent of the angle at the crossing, out of nothing but the two slopes. One thing had to be true before any of it started: both tangents had to exist, so neither line may stand upright. The expansion arrives carrying a condition, worth more attention than it usually gets. One plus the two slopes multiplied together must not be nothing. Written like that it looks like housekeeping about division. It is a geometric condition wearing algebraic clothes.

3:35One plus the two slopes multiplied together is nothing exactly when the two slopes multiplied together are minus one. Two slopes multiplying to minus one is what it means for two lines to meet at a right angle. So the formula's single refusal falls on perpendicular lines, and on nothing else. That was checked without letting either slope decide anything: a right angle was settled by a dot product, over seven thousand six hundred and fifty-six ordered pairs of directions.

4:13Eighty-eight of those pairs are at a right angle. Pairs where the quotient has nothing underneath but the lines are not perpendicular: zero. Pairs at a right angle for which the quotient still hands back a number: zero as well. So the exclusion is free. A formula refusing a case nobody could otherwise handle would be a defect; this one refuses the case you already know the answer to. Time to run the quotient against the angles it claims to compute.

4:50Seven thousand three hundred and ninety-six pairs of directions the formula can speak about: neither line upright, and the two not at a right angle. The angle each pair makes was built the long way, from the rays at the crossing, and its tangent from an upright leg over a signed flat leg. Pairs where the quotient is not that number: zero. And the other angle at the same crossing? Its tangent is the same number with its sign turned over, on every one of the seven thousand three hundred and ninety-six.

5:29Which is the picture again: the two angles fill a straight angle, so their tangents can only differ in sign. That turns the sign of the quotient into a decision procedure. Exactly one of the two angles at a crossing is sharp, unless the lines are perpendicular and both are right. Pairs where neither of the two is sharp: zero. Pairs where both are: zero as well. So the quotient comes out above nothing exactly when the angle it named is the sharp one.

6:05Three thousand six hundred and ninety-eight pairs answered with a sharp angle, three thousand six hundred and ninety-eight with a blunt one, and disagreements with the dot product: zero. Note what decided sharpness there. Not the quotient. A dot product, which knows nothing about slopes. Checking a sign every time is tiresome. There is a way to stop. Put the whole quotient inside a pair of bars and throw the sign away.

6:38If the quotient was already above nothing, nothing changes. If it was below, the bars turn it over, which is the tangent of the other angle at the same crossing. Either way what comes out is the tangent of the sharp one: the case distinction has been collapsed rather than checked. Measured: pairs where the bars do not return the tangent of the sharp angle: zero. The blunt one is then a subtraction: whatever is left of a straight angle once the sharp one is taken away.

7:15It is easy to drop the bars and treat whatever comes out as the angle between the lines. Here is what that costs. Of the seven thousand three hundred and ninety-six pairs, the bare quotient names the blunt angle on three thousand six hundred and ninety-eight of them. Half. Not an edge case: half of all pairs. But the negative sign is not an error. A value below nothing is a piece of information: it is telling you that the angle you happened to name is the blunt one, and its partner is the sharp one you wanted.

7:53The bars do not fix a mistake. They decline a choice you would otherwise have to make. One more habit worth dismantling: subtracting the smaller slope from the bigger, so the answer comes out positive. Swap the two slopes and the top of the quotient turns over, while the bottom, being a product, does not move. So the whole quotient turns over, and turning it over is exactly what swaps which of the two angles you have named.

8:26Measured: pairs where swapping changes the bare quotient: all seven thousand three hundred and ninety-six of them. Pairs where swapping changes the quotient inside the bars: zero. So there is nothing to order. The formula never needed to know which line you called the first. There is a second silence in this formula, and unlike the first one it is not free. Both lines had to have slopes before any of this began, so an upright line is outside it entirely.

9:02That is not a small class: a hundred and seventy-four of the ordered pairs here have an upright line in them. Every one of those pairs has a perfectly good angle at its crossing. The number with no angle to speak of is zero. The number the quotient can say anything at all about is also zero. You cannot rescue it by giving the upright line a very large slope. It has none, and a formula built on tangents has nothing to put there.

9:36Now run it backwards. The angle between two lines is half a right angle. One of them has slope a half. What is the other slope? Take the bars off and there are two equations, one for each sign, and each of them has an answer. Rather than solve either, the check searched eighty-eight directions and asked which of them sit at half a right angle from the given line.

10:03It found exactly two. Their slopes are three, and minus a third. Multiply those two together and you get minus one, which says something the question never asked: the two answers are perpendicular to each other. Two answers is not an accident of that question. The picture says why. You have a line and an angle. Turn the line through that angle one way, and you have an answer. Turn it the same amount the other way, and you have another.

10:37Both were built that way, over two hundred and forty combinations of a line and an angle, and the cases where either construction failed to make the wanted angle number zero. The two answers fall together on thirty of those cases, and on exactly the ones you would expect: where the angle asked for is nothing, or a right angle. And here is the tie back to the beginning. Ask instead for the supplement of that angle, and the search turns up the very same two lines.

11:12Cases where it does not: zero. The ambiguity at the crossing and the two answers to the question are the same ambiguity, seen twice. A harder version of the same trap. One line's slope is double the other's, and the angle between them has tangent a third. Find the slopes. Searching a hundred and eighty-two rational slopes, the number that work is four, not one and not two. Search again insisting the quotient come out above nothing, as it would without the bars, and only two survive - two answers thrown away by an operation people think is cosmetic.

11:56One more of this shape, because its ending is unusual: two given lines, and you want a third leaning equally against both. Taking the bars off gives two branches. One of them asks for a slope whose square is minus one: under the root sits minus one hundred, and there is no line to be had. The other has two hundred under the root, and hands back two lines. The branch that failed is not a mistake in the working; it is the answer to a question with no picture behind it.

12:33Finally, the case where the sharp choice is only a convention. Two lines whose slopes are not rational numbers at all: one is minus the root of three, the other minus one over that same root. The arithmetic was done exactly, with nothing turned into a decimal and no root ever taken. Above the line the difference is two thirds of the root of three; below it, exactly two. The quotient is one over the root of three, and it is above nothing, so the angle it names is the sharp one.

13:10How big is that angle, without a protractor? Add it to itself and add it again, and the flat part of the result vanishes while the upright part does not. Three of them make a right angle. So the sharp angle here is a third of a right angle, and the other one is what is left of a straight angle after it. The question that asks for the angles, in the plural, wants both. The bars would have handed over one.

13:42So what was bought. Not a formula for the angle between two lines. A formula for one of the two, and a rule for deciding which. The chain is short and every link is a picture. The angle at the crossing is a difference of two inclinations; the tangent of a difference expands; and the tangents of inclinations are slopes. The condition that arrives with the expansion is not about division. It says the two lines are not perpendicular, and it declines the one configuration that never needed a formula.

14:21The sign of the quotient is a complete answer to which angle you named, because exactly one of the two is sharp. The bars are how you stop asking. And the price of forgetting them is not small: half of all pairs named the blunt angle, and a question with four answers left showing only two. Two lines make two angles. Everything here is that one sentence taken seriously.