A second photograph reads the lean the first cannot, if it is taken round the corner

A photograph grades its posts across the sight, and not along it took a street with a row of posts in front of a building and read each post for its own lean. Two plumb corners fix the picture’s vertical vanishing point, and each post’s line is compared with the plumb line through its foot: the angle between them is the post’s lean, seen through the camera. Posts 2.5 metres tall at 7.5 metres were read to about a twelfth of a degree. But only the part of a lean that lies across the line of sight turns a post’s image. A lean towards the camera keeps the post on its own plumb line near the centre of the frame and turns it only a little at the edges, so a post leaning straight at the camera was caught five times in a hundred near the centre.

That essay ended by proposing the obvious remedy, a second photograph, and two places to take it: across the street, or a few paces along the same pavement. It asked how far apart the two viewpoints must be before the lean along the first sight is read as well as the lean across it, and whether the second picture needs plumb corners of its own.

The measurement says both proposed places are poor, and why: the second picture reads the same kind of thing the first does, and what matters is the angle between what the two read.

Two pictures, two equations for every post #

A post’s lean has two components on the ground: how far its top stands to one side of its foot, and how far forward or back. Call them, for the first camera, across and along its sight. In any picture, the angle between the post’s line and its plumb line is a sum: a gain times the across component plus another gain times the along component, where the gains depend on where the post stands in that picture. The first essay measured them. Across the sight the gain is about one — a post leaning a degree sideways turns its image by about a degree — and along the sight it is the post’s distance from the centre column over its distance from the camera, nearly zero in the middle of the frame.

A second camera standing elsewhere has gains of its own. Its across direction is the across direction of its own sight line, which is some mixture of the first camera’s across and along. So each post gives two equations — one angle read in each picture — in its two lean components, and the pair can be solved post by post. The solution is exact when the readings are; with reading error it is good when the two equations are independent and poor when they are nearly the same equation. They are the same equation when the two sight lines through the post run parallel, in either direction: two cameras on one line through the post see the same across component, one of them reversed.

So the question is not how far apart the two cameras stand but at what angle their sight lines cross at each post. The street here is the first essay’s — eight posts 2.5 metres tall in a row 7 metres long, the first camera 7.5 metres in front of the row’s middle, a building’s two plumb corners behind them — and the second camera stands 10 metres from the row’s middle, walked round it by a stated angle from the first camera’s bearing, with a pair of plumb corners of its own behind the posts as it sees them, and every post’s edge read across itself to 0.4 pixels in both pictures. At ninety degrees round, the two sight lines cross at the posts at 65 degrees on the left of the row to 115 on the right. At 180 degrees — directly across the street, looking back — they cross at 136 degrees at the ends of the row and 173 in the middle, nearly back along each other. A few paces along the pavement, ten degrees round, they cross at a few degrees everywhere.

Read best at right angles, worst near either bearing #

The reading follows the crossing angle.

At ninety degrees round, the pair reads each post’s lean along the first sight to 0.10 of a degree and across it to 0.08 — the across reading as good as the first picture alone gave, 0.084, and the along reading nearly as good. At sixty degrees and at 120 the along reading is 0.13 and 0.14. At forty-five, 0.16. Short of that it falls away quickly: 0.24 at thirty degrees, 0.35 at twenty, 0.72 at ten, and a degree and a half at five. Past ninety it falls away too, more slowly: 0.15 at 135 degrees, 0.21 at 150, and 0.30 from across the street.

A few paces along the pavement is the near end of that curve. Ten degrees round from 10 metres is a step of about a metre and three-quarters to the side, and from there the pair reads the along component to 0.72 of a degree — three times the quarter-degree lean the first essay set out to catch. The second picture has hardly changed the angle of anybody’s sight line through any post, so it has hardly added an independent equation, and solved as a pair the near-duplicate equation even drags the across reading down with it, to 0.16 of a degree from the first picture’s 0.08; a reader would keep the first picture’s across reading there.

The far end — across the street — is the end the question proposed, and it is poor for the opposite reason. The second camera looks back along very nearly the first camera’s sight lines and reads, of each post’s lean, very nearly the same across component, reversed. What it adds is a second reading of the across component — which is why the across reading is best there, 0.069 of a degree, two readings averaged — and very little of the along one.

The post it reads worst moves along the row #

The median hides that the posts are not alike. The crossing angle differs from post to post, and so does the reading.

At ninety degrees round no post’s two sight lines run parallel, and every post is read to between 0.07 and 0.16 of a degree, the worst being the leftmost, where the lines cross at 65 degrees. At 135 degrees the rightmost post sees the two cameras almost opposite, and it is read to four degrees — not read at all — while the leftmost six are read to between 0.12 and 0.16. At 165 degrees the blind post has walked in to the sixth from the left; at 180 it is the two in the middle, read to a degree each, while the posts at the ends, whose sight lines still cross at 136 degrees, are read to 0.2.

That is worth knowing before choosing where to stand, because a second picture that reads most posts well and one post not at all is a survey with a hole in it at a place the photographer can predict. The hole is wherever a line from the first camera through a post, carried on through the post, reaches the second camera. A camera standing at right angles to the first, seen from the row, puts that line nowhere among the posts.

Caught at the same false alarm #

The point of the first essay’s grading was a decision about each post: leaning, or straight. The pair can make it for the posts the first picture could 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 how well that post is read — its scatter learned beforehand from rows of plumb posts — and called leaning above a cut set so that nine straight posts in ten are cleared in every arrangement, so the arrangements are compared on what they catch at the same false alarm. One picture catches 57 in 100 of the leaning posts and 26 of those leaning within thirty degrees of its own sight, which are the posts it cannot see. A second picture ninety degrees round catches 75 of all leaning posts and 65 of those. At sixty degrees round, 73 and 58; at 120, 67 and 49. From a few paces along the pavement, 62 and 32; from across the street, 65 and 29 — barely more of the hidden posts than one picture alone.

The overall share is held down by what was already limiting the first essay: a quarter of a degree is only two and a half times the reading’s scatter, and at a fixed false alarm some leaning posts always slip under the cut. The along-sight share is the one that measures what the second picture adds, and it more than doubles at right angles and hardly moves anywhere else.

A second picture needs a vertical of its own #

The question also asked whether the second picture needs plumb corners of its own, or whether the posts the two pictures share could fix its vertical.

They cannot quite. Each picture’s vertical vanishing point has two unknowns, every post adds two, and two pictures give two equations a post; the posts alone leave the second picture’s vertical two equations short. What a second picture without a building can do is read its vertical from the posts themselves, taking them as plumb on average — the reading the first essay found loses whatever lean the whole row shares. At sixty degrees round that costs the along reading 0.04 of a degree, 0.17 against 0.13. At ninety degrees it costs nearly everything, 0.64 against 0.10, and the hidden posts caught fall from 65 in 100 to 21. At ninety degrees the second picture looks straight down the row, the posts stand one behind another on one column of its frame, and lines that nearly coincide barely fix a vanishing point at all. Turned fifteen degrees short of the right angle, the row spreads across the picture again — 75 degrees round, the along reading is 0.15 without corners against 0.11 with them.

So the instruction has a second half. A second picture taken at right angles needs a building behind the posts from its side; without one, take it at sixty or seventy-five degrees round instead, where the posts themselves can stand in for a building at a small cost.

Where the rule comes from #

None of this is particular to posts. A picture measures directions across its own line of sight and not along it, and the same is true of every measurement made with rays. Two rays that do not meet found a point’s position read from two pictures to be good across both rays and poor along them where they nearly coincide, and a third ray is worth what its picture is worth found that what a further view adds depends on the direction it comes from as well as how well it is read. A lean is a direction rather than a position, but its two components are read through the same two sight lines, and the conditioning is the same: the sine of the angle at which the lines cross. That is why ninety degrees is best, why a few paces is a few degrees and worth little, and why across the street — which sounds like the most different view a photographer can take — is the same sight line seen from its other end.

There is a kindred result in another field. A post read both ways sets its lean aside separated a post’s own tilt from a rolling shutter’s lean by driving past it twice, in opposite directions, so that the shutter’s contribution changed sign and the tilt’s did not. Opposite views help there because the thing to be removed reverses and the thing to be kept does not. Here the thing to be read reverses too, so opposite views read it no better than one.

A step is not an angle #

It is tempting to measure a second viewpoint by how far it is from the first, and the street shows why that is the wrong ruler. From a few paces along the pavement the second camera stands a metre and three-quarters from the first and reads the along component to 0.72 of a degree; from across the street it stands seventeen metres away and reads it to 0.30; from round the corner, ninety degrees round, it stands twelve and a half metres away and reads it to 0.10. Distance between the cameras orders those three wrongly. The angle between their sight lines at the post orders them rightly, because the angle is what decides whether the two readings are two equations or one. A turn of the head is not a step sideways made the corresponding point for a stereo pair’s baseline: what fails a short baseline is not the algebra but the ratio of what the step changes to what the reading can resolve, and here what the step changes is a direction of sight.

On a real street that turns into a simple instruction. The second picture belongs at the end of the row, or down the side street at a corner, looking along the row rather than across it — wherever a line from the first camera through the middle of the row, turned through a right angle at the row, leads. Sixty to seventy-five degrees round is nearly as good and keeps the posts from hiding each other.

What the second picture cannot change is the floor the first essay found under every reading of a lean. Each picture’s vertical comes from its corners, and a facade’s corners see the horizon is where that vanishing point’s precision was first measured: a building’s corners fix it far better than its widths, but not exactly, and the error is shared by every post that picture reads. It sets a floor no number of posts or pictures goes under, which is the shape the floor a better camera cannot reach found for every measurement closed by an assumption. Two pictures with their own corners have two such floors, in the two components each reads best, and together they are the floor of the pair.

What the pair takes on trust #

Each picture’s vertical is read correctly. Both pictures rest on a pair of plumb corners, as the first essay’s did, and a leaning post is worth its lean is the reminder that a corner taken as plumb when it is not moves every reading in that picture together. Two pictures with two buildings have two such chances to be wrong, and a corner’s error in one picture appears as a lean shared by the whole row in the components that picture reads best.

The posts are straight. A post that bends, or whose top is a lamp bracket overhanging to one side, has no single line; each picture fits its own line to whatever part it sees, and two pictures from different sides can fit different lines to the same post.

Every post is seen whole in both pictures. At ninety degrees round the posts stand one behind another in the second picture. Here each was read as if the one in front were transparent; in a real picture the nearer posts hide parts of the farther ones, and which parts depends on the row’s spacing and the posts’ width. That is one more reason to stand at sixty or seventy-five degrees rather than ninety.

The ground is level. A lean is measured against the vertical, not the ground, so a sloping pavement changes nothing about the reading; but a photographer who expected the posts to stand square to the pavement on a slope would call every one of them leaning, by the slope, in the direction it runs.

Still open: whether a third picture fills the hole the second leaves #

At any angle other than ninety degrees a second picture leaves one post, or a pair, read badly: the post whose sight lines to the two cameras run nearest parallel. A third picture, taken from the other side of the first — sixty degrees round to the left where the second was sixty to the right, say — has its blind post somewhere else, and between the three no post need be read badly.

The measurement that settles what that is worth reads every post from three pictures with three equations in two unknowns, solves each post by least squares, and asks how much the worst post’s reading improves over the best pair; whether three pictures at sixty degrees either side of the first, 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 its own; and whether, with three pictures, the posts can after all fix the second and third pictures’ verticals between them, since three equations a post leave one to spare.