A staircase winds the mistake round the lattice

A climbing eye names the column mistake, if it knows how high it is took a level stereo pair that had matched a facade of identical windows one to four columns out, and put a third camera on a vertical climb above the left eye. The rebuilt windows — nearer than the wall, because every mistake on this facade pulls it forward — slide straight down the climbing camera’s picture at 98 pixels a metre for each column of the mistake, and a matcher reads that slide against the nearest real window, wrapped at each storey. A metre of climb from level names every one of the four mistakes. The cost was knowledge: the climb had to begin before the rebuilt facade slid off the bottom of the lattice, and the camera had to know its own height to about a quarter storey of slide.

Few cameras climb straight up. A photographer on an outside staircase rises along the wall as well as up it, and a drone climbs on a slant. A third camera displaced up and across from the left eye has a slanted baseline with it, and the slide that was vertical now has an across component too. Across, the facade repeats at its columns; up, at its storeys. So a frame reads the offset on a two-dimensional lattice rather than round a single storey, and the question the last essay ended on was whether that is worth anything — and whether some pitches of stair line the slide up with the lattice and are blind.

Both happen, and the answer to “worth anything” is not the one the question expected.

The slide on a stair is the climb’s slide, pointed along the stair #

The facade is the one of the climbing 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 at standing height with a focal length of 900 pixels. A column is 108 pixels across in the picture and a storey 135 pixels up. The third camera starts at the left eye and climbs a straight stair along the wall at a stated pitch, facing the wall squarely the whole way.

The geometry is the climb’s, generalised by one line. 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. A camera moved by $(u, h)$ from the left eye — $u$ along the wall, $h$ up — sees that point displaced from the real window by

$$\frac{f}{Z}\Big(1 - \frac{1}{\rho}\Big)\,(u,\;h) \;=\; \frac{f}{Z}\,\frac{kp}{B}\,(u,\;h),$$

the vertical climb’s slide with the stair’s run added across. Per metre of stair it is the same 98 pixels for each column of the mistake, and it points along the stair. A matcher sees it against the nearest real window, so it wraps across at a column and up at a storey.

A lattice cell whose opposite edges are the same place is a torus, and a straight slide wrapped on it is a line wound round the torus. That is the picture to hold. Two things follow from it at once.

The first is that every reading travels the same line. The $k$-column mistake at a given distance up the stair is exactly where the one-column mistake is at $k$ times that distance, so the four mistakes are not four lines but four travellers on one, moving at one, two, three and four times the same speed. The honest reading sits still at the real window. A frame tells the readings apart by where on the line it finds each of them.

The second is that a line on a torus closes only at certain slopes. If the stair’s rise over its run is a ratio of whole storeys to whole columns, the slide comes back exactly onto a real window — so many columns across and so many storeys up — and a frame there reads that mistake as no mistake. At any other slope the line winds round forever without passing through the window again. A vertical climb is the extreme closing case: its line is a single vertical circle round the cell, and it comes back onto the real window at every blind raise a raised eye reads the storeys counted.

Blind only at pitches the lattice keeps #

The closing slopes can be listed and checked against the measurement. For each pitch from 1° to 90°, the figure below takes every frame on the first 1.6 m of stair and finds the nearest that any two of the five readings come — the honest one and the four mistakes — wrapped on the lattice: the closest the climb ever comes to a frame that cannot tell two of them apart.

At most pitches the readings never come within 8 pixels of one another. The dips sit at 17.5°, 22.5°, 32°, 40°, 51.5°, 68° and 90°, and each is a ratio the lattice keeps: a storey over four columns has a slope of 0.31 and an angle of 17.4°; a storey over three, 22.6°; a storey over two, 32.0°; two storeys over three columns, 39.8°; a storey over a column, the diagonal, 51.3°; two storeys over a column, 68.2°; and the vertical, a storey over no columns at all. The diagonal the question singled out is one of these and not the only deep one: the vertical climb, the stair at a storey over two columns and the diagonal all bring two readings to within a fraction of a pixel inside 1.6 m — exactly together, at the right frame, and the figure’s 0.16 px on the diagonal is only how near its sampled frames fell to that one.

How far off the diagonal must a stair be? Near it the nearest approach grows by 3.0 pixels for every degree of pitch: 0.77 px a quarter of a degree off, 1.52 half a degree off, 3.02 a degree off and 6.04 two degrees off. With the half-pixel reading error this essay assumes throughout, a stair a degree off the diagonal never puts two readings closer than six times the error. Ordinary stairs are pitched between about 30° and 42°, which lands most of them between the dips at a storey over two columns and two storeys over three; neither dip is wide, and a stair is a degree off either with no effort at all.

There is a limit to how much the dips matter, and it is the climbing essay’s own finding. A blind frame is one frame. At the diagonal, the four-column mistake lands back on a real window — one column across and one storey up — 0.44 m along the stair, the three-column mistake at 0.59 m, the two-column mistake at 0.88 m and the one-column mistake at 1.76 m; at each of those places one frame reads one mistake as the honest facade. Any second frame elsewhere on the stair tells them apart, exactly as a second frame anywhere but at a blind raise does on the vertical climb. A blind pitch is a hazard for a camera that takes one picture at one unlucky place, not for a climb.

The readings sit further apart in a frame #

The question asked whether two wraps together name a mistake from fewer frames than the vertical climb needed. They do not, because the vertical climb needed only one: a single frame anywhere on the first metre names the mistake made 99 times in 100 on the vertical climb and 99 to 100 on a stair, at half a pixel of reading error. There is no room below one frame.

What the second wrap changes is how far apart the readings sit in the frame that does the naming, and that is what decides how much else can go wrong before the naming fails.

In a typical frame on the first 0.85 m — low enough that no reading has slid off the lattice — the two nearest readings sit 14 px apart on a vertical climb and 18 to 30 px apart on a stair: 25.5 px at 30°, 28.0 at 60°, 29.9 at 70°. The reason is the dimension of the space the readings are scattered over. On a vertical climb the five residuals are five points on a circle 135 px round, and five points on a circle cannot be much further apart than a fifth of it; most of the time two of them crowd. On a stair they are five points on a 108 by 135 px torus, and five points spread over an area keep their distance far more easily than five spread round a line. The diagonal gives less of the gain than other pitches, 17.7 px, because its line closes after one turn and holds the readings near one diagonal of the cell — it is a little like a circle again.

What the camera must know of where it is #

That spacing is spent where the climbing essay found the climb’s real cost: in what the camera knows of its own position. A camera that believes it is 70 cm up the stair when it is truly 75 predicts every reading’s residual 5 cm of slide away from where it is, along the stair. A reading is lost when that error carries it nearer another reading’s residual than its own, and readings that start further apart can be carried further.

With its position known to 5 cm every arrangement names the reading at least 99 times in 100. At 10 cm the vertical climb names it 82 times; a stair at 60° 94, at 30° 95, the diagonal 90. At 20 cm the vertical climb is at 55 and the 30° stair at 72. By half a metre of error every arrangement has fallen to between 35 and 43, close to what is left when only the slowest mistake can still be named. The order is the spacing’s order, diagonal included: the stair whose readings sit furthest apart survives the largest error.

That is the stair’s first real gain, and it matters because the climbing essay’s knowledge cost was the binding one. A photographer on a staircase knows roughly which step they are on and how high the steps are, which is a few centimetres of uncertainty at best and a tread’s depth at worst. The vertical climb needed its height to about a quarter storey of the fastest mistake’s slide, 9 cm for the four-column mistake; the stair asks for less.

A stair keeps the slide on the facade #

The climbing essay’s other cost was the lattice’s edge — the same edge that a facade matched a column out comes forward and a railing matched one along stands underground found convicting a mistake, here limiting how long one can be named. A frame reads a mistake only while some rebuilt window still lands on a real one, and on a vertical climb the slide runs straight down three storeys: the four-column mistake has left them by 0.86 m, the one-column mistake by 3.44 m, and a climb begun above those heights cannot name them. On a stair only the slide’s rise is spent against the storeys. Its run is spent across the columns, and the facade has five.

Four frames on the metre beginning a metre along the climb name the reading 78 times in 100 on the vertical climb and every time on a 30° stair. On the metre beginning 1.5 m along, the vertical climb names it 53 times, the diagonal 60, a 60° stair 59 and a 30° stair 95. By three metres along every arrangement is down near 40, which is the share left when only the honest reading and the slowest mistake can still be told — the other three have left the lattice in every direction.

The 15° stair, dashed, is the turn in the curve: 83 at 1.5 m, below the 30° stair. Too shallow a stair spends nearly all its slide across the columns, and the five columns of a facade run out as surely as three storeys do — sooner, for the faster mistakes, whose rebuilt facades have only one or two columns of windows to slide. The best pitch for keeping a slide on a facade depends on the facade’s shape: a stair should spend its slide across the lattice’s longer side. On this one, five columns of 2.4 m by three storeys of 3 m, the best of the pitches tried is 30°, which is also where ordinary stairs are built.

What the stair adds to the vertical climb #

Set side by side, the vertical climb and the stair agree on the essential and differ on the costs. Both refuse the column mistake from almost any frame and name it from almost any single frame low on the climb. Both are blind only at countable places — the vertical climb at its blind raises, a stair at the places where its line on the torus passes back through the real window, which happens only at pitches whose slope is a ratio of whole storeys to whole columns. A degree off such a pitch removes the blindness, and a second frame removes it at any pitch.

The stair is better on both of the climbing essay’s costs. Its readings sit 18 to 30 px apart in a frame where the vertical climb’s sit 14, so it survives about twice the error in what the camera knows of its position; and its slide crosses the facade as well as descending it, so a stair near 30° goes on naming a metre and a half along where a vertical climb has lost half its readings. Neither gain is free of the facade. A tall narrow tower keeps a vertical climb’s slide longest; a long low terrace keeps a shallow stair’s.

The result is an instance of the oldest finding in this sequence. A mismatch on its own line needs a third eye observed 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 the one placement such a baseline must avoid is the one at which the repeat recurs — a single blind placement, not a blind direction. The stair has the same shape one dimension up: a lattice repeats in two directions, so a climb is blind not at a direction but at the slopes the lattice itself repeats along, and the blindness is still a matter of single frames.

The geometry the reading leans on #

The camera faces the wall squarely and the stair runs along it. A stair that turns towards the facade changes the camera’s distance as it climbs, and the slide rate, which depends on the depth, then changes with it; a camera that pans to keep the facade framed turns the slide by an amount that varies across the picture. Both would have to be modelled frame by frame. Depth is a reciprocal is why the first changes the rate: every disparity here is a difference of reciprocal depths.

The mistake is a clean shift of whole columns, as the pair’s matcher makes it. A scatter of window-by-window errors has no single slide and no single line on the torus; every frame refuses it, and no candidate names it. A wrong match is not a small error is the standing reminder that the mistakes worth naming are the coherent ones.

The position error is along the stair. A camera that knows its height but not how far it has walked along the wall — or the reverse — has an error that is not along the slide, and the readings it confuses are then the ones lying across the line on the torus rather than along it. That case was not measured.

The reading error is half a pixel, independent in each direction and from frame to frame. A third ray is worth what its picture is worth states the general rule; a correlated error, such as a focal length wrong by a per cent, scales every slide together and is indistinguishable from a different mistake.

Still open: a stair that turns at a landing #

Every stair here is straight. Most outside staircases are not: they rise along one wall, turn at a landing and rise along the next, or wind round a corner. A camera on such a stair faces the facade squarely on one flight and obliquely on the next, and its distance to the wall changes at the turn.

The measurement that settles what a turn is worth takes a stair of two flights — the first along the facade at a stated pitch, the second turned towards or away from it — and carries the pair’s mistakes into frames on both. On the second flight the slide is no longer the same at every window, because a camera that has turned sees the near and far columns at different depths, and the rate changes across the picture; the mistake then draws not a line on the torus but a curve, different for each window. The question with a number in it is whether that variation is a nuisance the reading must model, or evidence — whether the way the slide changes across one oblique frame names the mistake by itself, as the edge of the lattice did at a blind raise, so that a single picture from the landing does what the whole straight flight did.