A turn at the landing makes the mistake a scale, and a scale is weak evidence

A staircase winds the mistake round the lattice put a third camera on a straight stair along a facade of identical windows, which a level stereo pair had matched one to four columns out. Every rebuilt window — nearer than the wall, because every such mistake pulls the facade forward — slides through the climbing camera’s picture along the stair’s own direction, at 98 px for each metre and each column of the mistake, and a matcher sees the slide wrapped at the columns across and the storeys up: a line wound round a torus. The stair names the mistake from almost any single frame, and is blind only at the few places where that line passes back through a real window.

Every flight in that essay ran along the wall, so the camera never changed its distance from it. Most outside staircases turn. They climb one wall to a landing and then rise towards the building, or away from it, or round a corner. The essay ended by asking what such a turn does: on the second flight a camera sees near and far parts of the facade at different depths, so the slide is no longer the same at every window. Is that variation a nuisance the reading must model, or evidence — enough that one oblique picture from the landing does what the whole straight flight did?

The answer splits the question into two things that a turn does and that the question ran together. Turning the camera does nothing. Moving towards the wall does something exact and weak.

Towards the wall, the slide becomes a scale #

The facade is the one of the straight-stair essay: five columns of windows 2.4 m apart and three storeys of 3 m, twenty metres off, faced squarely by a pair 1.1 m apart with a focal length of 900 pixels. A rebuilt window lies on the left eye’s ray through its real window, at the depth ratio $\rho = B/(B - kp)$ for a $k$-column mistake — 0.314 for one column out, 0.186 for two, 0.133 for three, 0.103 for four, so the rebuilt facades stand 6.3, 3.7, 2.7 and 2.1 m from the pair.

A camera moved by $(T_x, T_y, T_z)$ from the left eye — along the wall, up it, and towards it — and still facing the wall squarely draws a real window at $x$ and its rebuilt copy at $x'$, and the two are related by one line of algebra:

$$x' - e = s\,(x - e), \qquad s = \frac{\rho\,(Z - T_z)}{\rho Z - T_z}, \qquad e = c + \frac{f}{T_z}\,(T_x,\,-T_y).$$

The rebuilt facade’s picture is the real facade’s picture scaled by $s$ about $e$, and $e$ is the focus of expansion: the point in the picture the camera is moving towards. A dolly zoom is a step and a zoom found the same structure for a camera stepping towards a subject — every mark moves along the line from the point stepped towards, outward if it is nearer than the subject — and the rebuilt facade is a nearer subject. When the camera does not move towards the wall, $T_z = 0$, the focus runs off to infinity and the scale about it becomes the uniform slide of the straight stair.

On the camera drawn here — a 30° stair along the wall for 0.8 m to a landing, then a level metre straight towards the wall — the one-column reading’s rebuilt windows are the real windows’ picture scaled by 1.130, the two-column reading’s by 1.298, the four-column reading’s by 1.850. The focus is 624 px to the right of the picture’s centre and 360 px above it, far off the canvas, because the camera has moved further along and up the wall than towards it; seen from the facade, each rebuilt window is pushed away from a point out beyond the top right corner by an amount that grows with its distance from that point. The windows nearer the focus move less, and the gradient across the picture is the variation the question was asking about.

The scale grows as the walk over the rebuilt facade’s distance #

How large is the variation? The closed form says, and the projection agrees with it to the last digit.

The scale is $1 + (1 - \rho)\,T_z/(\rho Z - T_z)$, so while the walk is short against the rebuilt facade’s distance it grows as $T_z/\rho Z$ — the walk measured against 6.3 m for the one-column reading. A metre and a half towards the wall scales the four readings by 1.21, 1.55, 2.13 and 3.42; the first ten centimetres change the one-column reading’s picture by 1.11 per cent. On a facade drawn a few hundred pixels across, that is a few pixels. The nearer mistakes grow faster because their rebuilt facades stand nearer, and every reading grows at a different rate — which is the whole of the evidence a turn can supply.

Two things follow at once. The first is that no picture on a flight towards the wall is blind. On a straight stair, two readings agree at a frame when their slides differ by a whole number of columns and storeys, and then every window agrees at once, because every window carries the same slide. Under two different scales the windows carry different offsets, growing with their distance from the focus, and they cannot all differ by a whole lattice step at once unless the scales differ by a whole number — a doubling, a walk of metres. The second is that the evidence comes slowly. A slide grows as the walk over the wall’s distance multiplied by the baseline ratio; a scale grows as the walk over the rebuilt facade’s distance, and its effect on a window’s offset is that times the window’s distance from the focus.

Turning the camera adds nothing #

Before the walk towards the wall, there is the turn itself. A photographer on a landing turns to face the next flight, and a camera that turns sees the facade obliquely: near columns larger, far ones smaller, the window lattice drawn as a converging grid. That is the variation across the picture the question described, and it is present before a single step is taken.

It carries nothing. A rebuilt window and its real window lie on one ray from the left eye, and a camera at the left eye’s position draws them at the same point however it is turned; a camera displaced from it draws them apart by the displacement and nothing else. Turning a camera about its own centre re-draws the whole picture through one homography — the same warp applied to real windows and rebuilt ones — and a matcher that undoes the warp has the unturned picture back, pixel for pixel. The turn changes how the evidence is drawn, not how much of it there is.

The measurement confirms it exactly. The straight-stair essay’s diagonal stair, rising at 51.3° — a storey over a column — carries the one-column reading back onto the real lattice 1.76 m along it: there the interior of a picture reads that mistake exactly as it reads the honest facade. Turned at that place by 0°, 2° or 4° — turns small enough to keep the whole facade in a picture whose half-width spans 21° — the camera still reads the two identically, to three millionths of a millionth of a pixel. A landing whose turn is made by the photographer rather than by the stair is a landing on the straight stair.

A step towards the wall breaks the blindness, slowly #

So the question becomes what a step does, and the blind place is where it shows most clearly: there the slide has put the one-column reading exactly on the honest one, and whatever separates them afterwards is what the step bought.

Ten centimetres on along the stair, or straight up, separate the two readings by 9.8 px; the slide in the wall’s plane runs at 98 px a metre for this reading, and any direction in that plane gives the same. Ten centimetres level towards the wall separate them by 1.8 px, and away from it by 1.7. Forty centimetres towards the wall reach 7.6 px, which along the stair took eight centimetres. The two readings sit at a fifth of the separation, at the cost of the same walk.

Between the two directions, a step does some of each — slides the rebuilt facade by its component along the wall and scales it by its component towards it — and the separation falls smoothly as the step turns: 9.8 px along the wall, 9.1 and 7.7 at 30° and 45° towards it, 5.9 at 60°, 1.8 straight at it. The slide is worth about five and a half times the scale for the same distance. A flight that turns towards the facade at an angle short of a right angle keeps most of what the straight flight had, by keeping most of its run along the wall.

Enough of a weak thing is still evidence #

A fifth of the separation is not none, and the reading error is half a pixel. The question with a number in it is how far from the blind place one picture has to be taken before it names the right reading.

Set the facade’s edge aside — an edge names a mistake even at a blind place, as the climbing essays found, and the point here is what the interior says. One picture a centimetre on along the stair names the right reading 95 times in 100, and two centimetres on every time. Towards the wall, two centimetres name it 45 times — a coin — three centimetres 60, five centimetres 95, seven and a half centimetres every time. Away from the wall the figures are almost the same. A turn towards the wall does what the question hoped: one picture on that flight, anywhere beyond a few centimetres from the landing, names the mistake from the variation across it. It needs two to three times the distance a step along the wall needs to do it.

So the answer to “does one oblique picture from the landing do what the whole straight flight did” is that a single picture already did that on the straight flight. The climbing essay found one frame enough almost anywhere, and the straight stair kept that. What the straight flight could not do was escape its blind places by any means but another frame elsewhere. The turned flight escapes them within its own picture, slowly.

The same weakness is a tolerance #

The straight-stair essay found the real cost of reading on a climb was what the camera knows of its own position: a camera that believes it is 70 cm up when it is 75 predicts every reading’s offset 5 cm of slide from where it is. On a flight towards the wall, the same 5 cm of error is 5 cm of walk towards the wall, and that moves every prediction by a fifth as much.

A camera that knows where the landing is and misjudges how far up the second flight it has come by 20 cm names the reading from one picture 93 times in 100 on a level flight towards the wall, 99 on a level flight away, 89 on a flight climbing towards the wall at 30°, and 81 carrying on along it at 30°. At 50 cm of error the figures are 84, 94, 73 and 66; at a metre, 77, 90, 61 and 58. The level flights are the robust ones because their whole motion is towards or away from the wall, where error is cheap. The climbing flight towards the wall rises as it goes, and its rise is in the wall’s plane, where error is expensive, so it sits between.

That is a weakness and a tolerance stated twice. A move towards the wall changes the picture little, so it is poor evidence and an error in it costs little; a move along the wall changes the picture a great deal, so it is strong evidence and an error in it costs a great deal. A photographer who knows which step of a flight they are on to a tread should want the flight along the wall; one who has lost count should want it running at the wall, or better, away from it.

Away from the wall keeps the facade #

Away is better than towards in that figure for a reason the scale makes plain. Walking towards the wall enlarges the rebuilt facade’s picture about the focus, and its outer windows are pushed off the real facade one after another. Walking away shrinks it towards the focus, and the rebuilt windows crowd in over the facade rather than off it.

Of the thirty rebuilt windows the four nearer readings have between them, 24 land on real windows at the landing. A metre and a half up a flight turned towards the wall, 11 still do; carrying on along it, 7; turned away, 16. Three metres up, the flights towards the wall and along it have 4 left each, and the flight away has 11. A facade matched a column out comes forward and the climbing essays found the lattice’s edge the binding limit on how long a slide can be read, and a straight stair can spend its slide only against the facade’s width and height. A flight away from the wall spends almost none of it: the scale it adds pulls the rebuilt facade in rather than out. It is the one thing a turn does that no straight flight can.

The geometry the reading leans on #

The camera faces the wall squarely on both flights. The scale about the focus of expansion is exact for a camera that only translates. A camera that also turns redraws everything through one warp; that adds nothing and loses nothing, except that a facade seen obliquely is drawn with fewer pixels across its far columns, and resolution lost there is reading error gained.

The landing is known. Every position error here is along the second flight. A camera that does not know where the landing is carries an error in the wall’s plane as well, and the tolerance of a flight towards the wall then applies only to the part of the error that is towards the wall.

The facade is one plane. A projecting bay or a recessed porch puts some real windows nearer than others, and the scale of each depends on its own depth: depth is a reciprocal is why, since the scale is set by $T_z$ over each window’s distance. A facade of several depths would show several scales even for the honest reading, and the reading would have to model them.

The mistake is a clean shift of whole columns. A wrong match is not a small error is the standing reason: a coherent mistake has one $\rho$ and so one scale, and a scatter of errors has neither.

The reading error is half a pixel, independent from window to window. A third ray is worth what its picture is worth states the general rule. A focal length wrong by a per cent scales every window’s picture together, about the centre rather than about the focus of expansion — a different centre, which is the only thing that keeps it from being read as a walk towards the wall.

What the turn adds to the straight stair #

A mismatch on its own line needs a third eye found that a repeat along one baseline is broken by a baseline not parallel to it. A third eye that lands on the next post found that such a baseline is blind only at the single placement where the repeat recurs. The straight stair extended that to a lattice, blind at the slopes the lattice repeats along. The turn adds the third dimension. A move towards the wall is a baseline the lattice cannot repeat along at all, because it does not slide the rebuilt facade across the lattice but scales it, and no lattice repeats under a scale. That makes it the one direction with no blind places — and, because the facade is far, the direction that buys least for each step.

Still open: whether the scale names the mistake for a camera that does not know where along the wall it stands #

Every reading here knew where the camera was, up to an error along one flight. The scale and the slide depend on the camera’s position differently: the slide on its move along and up the wall, the scale only on its move towards it, and the focus of expansion on the ratio of the two. A camera that knew its distance from the wall — from a rangefinder, or from the facade’s own known spacing — and nothing about how far along or up it stood would have no use for the slide, whose every offset is a guess at its own position, but might still have the scale.

The measurement that settles what that is worth reads one picture taken from a camera whose distance towards the wall is known and whose position along and up the wall is unknown: for each candidate mistake it fits the focus of expansion freely, as two extra unknowns, and asks whether the scale left over names the mistake. The questions with numbers in them are how far towards the wall the camera must be before the scales of the five readings are told apart from one picture with the focus free; whether fitting the focus costs the reading more than the slide’s position error did; and whether a picture taken walking away from the wall, which keeps the rebuilt windows on the facade, reads the scale as well as one taken walking towards it.