A third photograph closes the hole the second leaves, if it brings a building

A second photograph reads the lean the first cannot took a row of eight posts in front of a building and the camera that graded them for lean, and asked where a second picture should stand to read the part of each lean that runs along the first camera’s sight — the part a picture cannot see. The answer was in the angle at which the two cameras’ sight lines cross at each post: two readings of a lean are two independent equations where the lines cross widely and one equation where they run parallel. Ninety degrees round the row was best. Sixty or seventy-five was nearly as good and had the advantage, without a building behind the posts from the second camera’s side, of not looking straight down the row.

At any angle other than ninety degrees, the second picture left one post read badly — the post whose sight lines to the two cameras ran nearest parallel. That essay closed by proposing a third picture taken from the other side of the first, whose own worst post would be somewhere else, and asked three things of it. How much better the worst post is read. Whether three pictures sixty degrees either side, which see the row spread across their frames and need no building at right angles, catch as many of the hidden posts as two at a right angle with a building of their own. And whether, with three equations for each post’s two unknowns, the posts themselves can fix the further pictures’ verticals.

Where each further picture leaves its hole #

The street is the earlier essays’: posts 2.5 m tall in a row 7 m long, the first camera 7.5 m in front of the row’s middle, a building’s two plumb corners behind the posts, every post’s edge read across itself to 0.4 pixels. A further picture stands 10 m from the row’s middle, walked round it by a stated angle, with plumb corners of its own unless the arrangement says otherwise.

A camera sixty degrees round to the right looks across the row from that side, and its sight line to the post at the far end — the leftmost — runs closest to the first camera’s line through the same post. There the two lines cross at 43°; at the near end, at 71°. A camera sixty degrees round to the left is the mirror of it, and its nearly-parallel post is the rightmost. With both, every post has at least one pair of sight lines crossing widely: 71° at the ends of the row, where one further camera’s line is nearly perpendicular to the first’s, and 62° in the middle, where both further cameras look in at an angle.

The hole moves because it is a property of geometry, not of the post. A post is read badly by a pair of pictures when the line from the first camera through it, carried on, passes near the second camera. For a camera to one side of the first, that line leaves the row from the end farther from it. Put a camera on each side and the two lines that would have been bad lead away from different cameras.

The worst post, and the rest #

Least squares on each post’s three equations — one angle read in each picture, each a known mixture of the post’s two lean components — gives the lean, and rows of posts at random leans show how well.

Sixty degrees round, one further picture reads the median post’s lean along the first sight to 0.13° and its worst post to 0.23°. A picture on each side reads the median post to 0.086° and the worst to 0.097°: the worst post is now read as well as the median one, and about as well as one picture read every post’s lean across its sight. At forty-five degrees the worst post improves from 0.31° to 0.125°; at thirty, from 0.47 to 0.17; at fifteen — a few paces along the pavement either side — from 1.3° to 0.35, still poor, because three sight lines nearly parallel are three versions of one equation.

At ninety degrees round, where one further picture already left no post badly read, the third improves the worst post from 0.16° to 0.087° and the median from 0.104 to 0.077 — two good readings averaged, not a hole filled. Past ninety the single further picture starts to look back along the first camera’s lines again, and at 150° its worst post is not read at all, 2.9°; the third picture, from the mirror bearing, brings it to 0.16. There is nothing special about sixty degrees except that it is the bearing the earlier essay recommended when no building stood at a right angle, and there a third picture turns a survey with a hole in it into one without.

Post by post the picture is plain. Sixty degrees to the right: the leftmost post 0.23°, falling steadily to 0.09 at the right-hand end. Sixty degrees to the left: the same, reversed, from 0.20 at the right-hand end to 0.10 at the left. Both: every post between 0.080 and 0.096°. Each further picture alone has a slope across the row, and the slopes run opposite ways.

Three pictures without buildings are not two with one #

The second question was the one with a practical stake. A picture at a right angle to the first looks straight down the row, the posts stand one behind another in its frame, and a building behind them from that side is what reads its vertical; without one, the earlier essay found the hidden posts caught falling from 65 in 100 to 21. Two pictures sixty degrees either side see the row spread across their frames and can read their verticals from the posts themselves, taking the row as plumb on average. Do they catch as many?

They do not. Half the posts in each row lean a quarter of a degree in a random direction and half within a twentieth; each post’s read lean is weighed by its own scatter, learned on separate rows of plumb posts, and called leaning above a cut set to clear nine straight posts in ten. Three pictures sixty degrees either side, each with a building of its own, catch 82 in 100 of the leaning posts and 77 of those leaning along the first sight — better than two at a right angle with a building, 75 and 65. Without buildings behind the further pictures, the same three catch 62 and 49. Seventy-five degrees either side without buildings, 67 and 55. Two at a right angle with a building of its own remain the better of the two practical arrangements the question set against each other.

The reason is in what reading a vertical from the posts gives up. A picture whose vertical is taken from its posts assumes the row stands plumb on average, and so cannot see any lean the whole row shares — or, as the next section finds, any lean that grows steadily along it. Rows in which half the posts lean a quarter of a degree in random directions have such a shared lean, of a few hundredths of a degree, every time, and two pictures that read their verticals from the posts both lose it. The extra crossings the third picture supplies buy back less than that costs.

With buildings, the third picture is worth a great deal: 77 hidden posts in 100 against 58 for a pair at sixty degrees, and three pictures ninety degrees either side catch 88. A photographer with a building behind the posts from both sides of the first camera should take both pictures. One with a building from neither side should take the single picture at a right angle and find a building for it.

The spare equation cannot read the verticals #

Each post gives three equations in its two lean components, one to spare. Eight posts give eight spare equations, and the two further pictures’ verticals are four unknowns. On a count, the posts should be able to read the further pictures’ verticals with four equations left over.

The count is right and the answer is no. Solved with every post’s lean, the further pictures’ verticals leave two things unread: the lean the whole row shares along the first camera’s sight, and a lean that grows steadily from one end of the row to the other. Both come back as numbers thousands of degrees wide — not read at all. What is left of each post’s lean after those two is read to 0.19°, against 0.062° when the further pictures have buildings and 0.063° when their verticals are taken from the posts as plumb.

The arithmetic is the geometry of a vanishing point. A picture’s vertical, moved a little, turns the plumb line through every post’s foot by an amount that varies smoothly along the row: very nearly a constant plus a slope, since the posts stand in one line across the picture. A lean the row shares turns every post’s image by a constant amount in a given picture, and a lean that grows along the row by a slope. In the further pictures the two are indistinguishable. The first picture could separate them — it has its building — except that the lean in question runs along its own sight, the one component it cannot see. Eight spare equations cannot read two unknowns that every equation sees in the same combination.

Reading the verticals from the posts as plumb on average is the cruder fix and the better one: it decides the row’s shared lean is zero rather than leaving it free, and the error it makes is the row’s actual shared lean and growing lean, 0.076° and 0.14° at the row’s end over these rows. That is a decision about the posts, not a measurement of them — a leaning post is worth its lean made the same decision for the first picture’s horizon and found its price in the same place.

Where to stand, with what behind the posts #

Put as an instruction to someone with a camera on a real street, the three findings make a short decision. A photograph grades its posts across the sight is the first picture, taken square to the row with the building behind. What the photographer does next depends on what stands behind the posts as seen from elsewhere.

If a building, a wall or any long plumb edge stands behind the posts from a point at right angles to the row — down a side street, or from the end of the row looking along it — take one picture there. It reads every post’s lean along the first sight to about a tenth of a degree and catches 65 hidden leaning posts in 100. If such edges stand behind the posts from both sides of the first camera, take two, sixty degrees round either side: 77 in 100, and no post left with only parallel lines. If they stand behind from neither side, the pictures can still be taken, but their verticals then come from the posts, and the survey can no longer tell a row leaning as a whole from a row standing plumb.

That last case is the common one, and it is worth saying what it costs plainly. A row of posts in front of a building is usually seen against open sky or a car park from every direction but the building’s. Two further pictures without buildings still read each post’s lean against its neighbours — 0.063° for each post’s own part, as well as with buildings — and a post that leans when its neighbours do not still stands out against them. What is lost is the row taken together: a fence of posts all leaning the same way, or leaning more towards one end, is read as plumb. A survey looking for one bad post can take the pictures it has; a survey of whether a row of posts has been pushed over as a whole cannot.

A turn of the head is not a step sideways made the same kind of distinction for a stereo pair: what a second view is worth depends on what it changes, not on how far away it stands. Here what a further view changes is the direction of sight through each post, and what it needs besides is a reference for up.

Why three is not the same as more angles #

Two rays that do not meet found a point read from two pictures well across both rays and poorly along them where they nearly coincide, and a third ray is worth what its picture is worth found what a further view adds depends on where it looks from. A lean is a direction rather than a point, but its two components are read through sight lines in the same way, and the third picture behaves as a third ray does: it is worth most where the first two are weakest, and that is a place that can be predicted from a plan before anybody lifts a camera.

What the third picture cannot change is the floor under every reading of a lean. Each picture’s vertical comes from its corners, a facade’s corners see the horizon measured how well, and every post that picture reads shares that error. The floor a better camera cannot reach is the general statement: a measurement closed by an assumption has a floor the assumption sets. Three pictures with three buildings have three such floors and average them, which is part of why three at ninety degrees either side read the median post to 0.077° where two at ninety read it to 0.104. Three pictures with one building have one floor and two assumptions about the posts, and the assumptions are what cost.

A post read both ways sets its lean aside is the opposite case, worth setting beside this one. There a post read from opposite directions had its own lean cancel and a shutter’s lean add, because the thing to be removed reversed between the views. Here every further picture sees some of the lean to be read and some of the vertical to be removed, and neither reverses — which is why more pictures help the reading only when each brings its own building.

What the three pictures take on trust #

Every picture’s building is plumb. Three pictures with three buildings have three chances for a corner taken as plumb to lean, and a corner’s lean appears in that picture as a lean the row shares, in the component that picture reads best.

Every post is seen whole in every picture. Sixty degrees round, the posts at the far end of the row are seen at a slant behind the near ones, and here each was read as if nothing stood in front of it.

The posts are straight and the readings independent. As in the earlier essays; a post bent at a bracket is fitted a different line from each side.

The cameras’ positions are known. Each picture’s gains — how much a post’s image turns for each component of its lean — depend on where the camera stood, and here they were taken from the true positions. A further picture whose position is guessed from the street turns its gains with the guess.

Still open: whether the posts can read the further pictures’ verticals if one post is known plumb #

The spare equations failed because two kinds of lean — one shared by the row and one growing steadily along it — look like a vertical misread. Both are kinds of lean the row could have. A survey that knew any one post to be plumb, or measured one post’s lean with a spirit level, would remove the first; a second post known plumb, at the other end of the row, would remove the second. Then the spare equations would have only the verticals left to read.

The measurement that settles what that is worth fixes one post’s lean — or two — as known, solves the further pictures’ verticals with every other post’s lean from the three pictures, and asks how well the other posts are then read along the first sight against the arrangement with buildings; whether one known post near the middle of the row does more than one at an end, since a lean growing along the row pivots about somewhere; and whether a post known only to a tenth of a degree, as a level held against it would read it, still gives the verticals back.