A steady creep draws the rule's bow

The rule’s bow is read by its shape found a better way to convict the constant-ratio rule than measuring how far a pavement’s diagonal strays from straight. The rule does not merely bend the diagonal; it bends it one way, in one smooth arc, zero at both ends and largest near the middle, with a shape fixed by the ratio. A painter’s random scatter has no preferred shape. Fitting the rule’s own arc to a painted diagonal with one amplitude — one for the rule, nought for a correct pavement — ranked the rule above a construction 96 times in a hundred on five braccia at two pixels of scatter, where the chord’s size had managed 83.

That essay’s first assumption was a painter whose errors were independent from one transversal to the next. It ended on the way that assumption fails: a painter who places each transversal from the one before, or who tires as the pavement recedes, makes errors that are correlated along the pavement, and a correlated error can have a shape. If some ordinary habit of a painter’s hand has the rule’s shape, the arc reading would convict the hand and call it the rule.

One does, almost exactly, and at a size a real hand can reach. What survives is a reading of the rule’s definition rather than its bow.

A creep with the rule’s shape #

The pavements below are correct ones — the distance point’s transversals — painted by a hand that creeps: each gap between one transversal and the next is drawn a stated fraction larger than the correct gap, compounding from the ground line, as a tiring hand draws a different habit had a tiring painter do on divided strips. The painter places each transversal from the one before and does not know where the last one should fall, so the pavement’s far end moves with the creep.

On eight braccia, a creep of 4.49 per cent a braccio bows the correct pavement’s diagonal by exactly as much as the rule does, in the arc reading’s measure — that is how the creep was chosen — and it bows it in the same shape. The two sets of departures from the chord differ by 0.31 pixels at worst, over a bow that reaches eight. Taken as two vectors of seven numbers each, their cosine is 0.99939. A second shape fitted beside the rule’s, to separate the two, would be fitting two copies of nearly the same curve. Drag the pavement’s length and the match holds at every one: the creep that counterfeits the rule falls from 7.1 per cent a braccio on four braccia to 2.1 on sixteen, and the cosine between the two bows never drops below 0.9994.

The reason is not a coincidence of these numbers. The rule is exact for a floor that lengthens found that the constant-ratio rule draws a correct picture of a room whose tiles grow in depth as they recede. A creep does the same thing to a correct pavement: each tile drawn a little deeper than the last is a tile that lengthens as it recedes. The arc reading was built to recognise a pavement whose tiles lengthen; it recognises this one.

Why every lengthening floor bows alike #

It is worth seeing why the match is so close, because a cosine of 0.9994 between two shapes made by different rules looks like a coincidence and is not one. A pavement’s diagonal runs through one tile corner per transversal. Relative to the chord from its first corner to its last, the corners depart by an amount that depends on how the transversal gaps are spaced relative to a correct pavement’s. Any spacing that is close to correct and differs from it smoothly — gaps a little longer as they recede, by any smooth law — shifts the interior corners off the chord in a smooth curve that is pinned at both ends, largest somewhere in the middle, and one-signed.

On seven interior corners, a smooth one-signed curve pinned at both ends has very little room to differ from another. Its peak can sit a little nearer one end or the other, and that is almost all. The rule’s bow and a creep’s bow are two such curves, and they differ only in where the lengthening is concentrated: a compounding creep lengthens the far tiles relative to the near ones slightly more than the rule does, so its bow leans a little towards the far end — the creep’s corners sit 0.2 to 0.3 pixels further off the chord over the far three, and a little less over the near three, which is the whole of the 0.31 pixels by which the figure’s two curves differ. The rule that draws another room found the rule to be a correct perspective to within a fifth of a pixel of a room it did not draw; a creep is the same kind of near-miss from the other side, a correct drawing of the right room made slightly wrong in a smooth way.

The lesson is general. A reading that fits one shape to a handful of points pinned at both ends can distinguish that shape from noise, as the earlier essay found, but not from another smooth shape of the same sign. Any smooth, one-way distortion of the transversals is a candidate counterfeit, and the creep is only the one a painter’s hand most naturally makes.

How much creep it takes #

A creep of four and a half per cent a braccio is a large one on eight braccia. The question is how the counterfeiting creep changes with the pavement’s length, since the rule’s bow grows with the pavement and so does a creep’s.

On four braccia the counterfeiting creep is seven per cent a braccio; on eight, four and a half; on twelve, under three; on sixteen, 2.1; on twenty-two, 1.5. The shapes agree to a cosine of 0.9994 or better at every length. The creep compounds, so a longer pavement needs less of it to accumulate the same lengthening, while the rule’s bow grows with the pavement’s length at a rate set by its ratio.

Against the two per cent a step that this collection’s tiring painter was given on divided strips, the counterfeit is out of reach on short pavements and within it on the longest: from about twenty-two braccia a hand tiring at that rate bows the diagonal as the rule does. The long pavements were the ones on which the rule was easiest to convict by size, in a straightedge convicts the rule on five braccia and after; they are also the ones on which a modest creep is enough to counterfeit it.

The arc reading measures a lengthening #

Between no creep and the counterfeit, the arc reading’s amplitude is not a verdict but a measurement.

A two per cent creep scores 0.41 of the rule’s bow on eight braccia, four per cent scores 0.88, six per cent 1.41. A hand that crowds its far gaps — a negative creep, each gap a little smaller than the correct one — bows the diagonal the other way, to −0.36 at two per cent and −0.95 at six, which the rule never does. The amplitude is nearly proportional to the creep, so the arc reading is a gauge: it reads how much the pavement’s tiles lengthen as they recede, whatever did the lengthening.

That changes what the earlier essay’s result means without changing its numbers. Against correctly constructed pavements painted with independent scatter, the arc reading convicts the rule, as it found. Against pavements whose painter crept, it convicts the creep in proportion to its size, and at the counterfeiting size it cannot tell the creep from the rule at all. A reader who finds an amplitude near one has found a pavement whose tiles lengthen by the rule’s amount; the rule is one explanation, a tiring hand another.

The rule’s definition, read directly #

The creep and the rule bow the diagonal alike because they lengthen the tiles alike overall. They do not lay the lengthening out alike, and the rule’s own definition says how its layout differs.

The constant-ratio rule makes each transversal gap a fixed fraction of the one before, so the logarithm of each gap over the last is one number repeated: −0.0815 at every gap on eight braccia, its trend along the pavement zero to the last digit. A correct pavement’s gaps shrink by a ratio that changes along the pavement — quickly at first, then more slowly — so its log-ratios run from −0.207 to −0.128, rising by 0.0131 a braccio. A steady creep multiplies every gap by the same factor more than the last, which adds one number, $\log(1 + c)$, to every log-ratio: the crept pavement’s run from −0.163 to −0.084, and their trend is the correct pavement’s 0.0131 exactly.

So the lengthening that the diagonal sees as one bow is laid out differently by the two. The rule lengthens every tile by the same ratio; a creep lengthens a correct pavement whose own ratios change. No steady creep of a correct pavement can produce the rule’s constant ratio, because the creep cannot remove the correct pavement’s trend. The rule’s definition is a test the counterfeit fails by construction.

Read under a painter’s hand #

A trend in seven log-ratios is a subtle thing to read, and a painter’s scatter falls hardest on the far gaps, which are short.

Against its counterfeit, the arc reading is a coin at every length and every scatter: 0.46 to 0.55. Both pavements bow with amplitude one, so it has nothing to rank by.

The gaps can rank them if they are read well. The statistic that works best is the curvature of the gaps’ logarithms down the pavement — straight for the rule, bent for a correct pavement however crept — fitted with each gap weighted by its length squared, since a hand’s scatter of $\sigma$ pixels is an error of about $\sigma\sqrt{2}/g$ in the logarithm of a gap $g$ pixels long, and the short far gaps would otherwise decide it. Read that way, a quarter-pixel hand’s pavements are told apart every time from eight braccia; a half-pixel hand’s 94 times in a hundred; a one-pixel hand’s two times in three; a two-pixel hand’s not at all.

That is a narrower result than the earlier essay’s, and an honest one. The rule’s definition names it against its counterfeit only where the painter placed transversals to about half a pixel on a picture of this size — a careful hand, or a large panel, since the reading error is what matters against the gaps’ lengths in the picture. On the rough, small pavements where the arc reading did its best work against independent scatter, nothing in one diagonal separates the rule from a hand that crept.

A lens is another counterfeit #

The creep is a painter’s counterfeit. A photographed pavement has one of its own. The lens a pavement can hide found that a photograph of a pavement taken through a lens with barrel distortion reads as a correct drawing up to a radial coefficient of about four tenths, because a pavement lies near the picture’s middle where a radial term is small. Where it is not small, a radial term can bend the transversals smoothly and one way too, and its bow on the diagonal is then a third candidate for the arc reading to mistake for the rule.

The gaps’ ratios deal with it no better and no worse than with the creep. A radial term moves each transversal by an amount that grows with its distance from the principal point, which should change the gaps’ ratios along the pavement in a pattern of its own — neither the rule’s constant nor a correct pavement’s trend shifted by a constant — and a well-measured pavement should say so; that is a prediction this essay did not measure. The practical point is the same: a single bowed diagonal is evidence that something lengthened the tiles, and the list of things that can is longer than the rule.

Two sequences about one hand #

The painter’s hand has now been met from two directions. On divided strips, a stepped hand passes for a tiring one on a short strip found that a hand that steps its divisions from one to the next leaves an error that the strip’s own fitted habit and creep absorb, and that only long strips or many of them tell stepping from tiring. Here, on a pavement, a hand that tires leaves a creep that the rule’s bow absorbs, and only a steady hand’s gaps tell the creep from the rule.

The two results are the same fact seen twice. Whatever a reading fits first — a strip’s habit and creep, a pavement’s arc — takes the smooth part of any error that happens to share its shape, and the reading that follows sees only the remainder. The rows count hands, not cameras was the first essay to find that divided rows record a painter’s procedure rather than a projection; the recurring discovery since is that the procedure is recorded in the fine structure — scatter, ratios, curvature — and not in the overall shape, which many procedures share.

What one diagonal can say #

The sequence this essay closes began by showing that the rule draws a correct picture of a different room, and has been sharpening the question of what a picture can say about how it was made. The arc reading answered it for one pair of alternatives: correct perspective under a random hand, or the rule. With a creeping hand as a third alternative, the diagonal alone says less. It says how much the pavement’s tiles lengthen as they recede, in one number, and nothing about why.

The transversals say more, because they carry the lengthening’s layout. A reader with a well-painted pavement can read its gaps and ask whether their ratios are constant; if they are, the painter used the rule, and if they drift, the painter did not, whatever the diagonal’s bow. A reader with a roughly painted one is left with the bow and two stories: a workshop that taught the rule, or a painter who tired. A tiring hand draws a different habit found on divided strips that a creep counterfeits a habit on one strip and not across a picture; here too the way out is more than one pavement. A painter’s creep varies with the painter and the day; the rule’s ratio does not.

What was assumed #

The creep is steady and compounding, from the ground line. A creep that begins partway, or grows linearly, or tires unevenly, has a different shape from the rule’s, and the arc reading would separate it partly. The steady compounding creep is the hardest case because it is the closest to the rule’s own multiplicative law.

The pavement’s far end is not fixed. A painter who rules the last transversal first and divides the space back to the ground line cannot let the far end drift; a creep then has to be spread over a fixed height, which bends the gaps differently and would not match the rule’s bow so closely.

The scatter is independent on every line and foot. Scatter correlated along the pavement — the painter’s stepping error — adds a second correlated shape on top of the creep, and makes every reading here harder.

Still open: whether a panel’s several pavements convict a creep that one cannot #

A painted interior rarely has one pavement. Floors, a loggia’s ceiling coffers, a tiled step, a table’s inlaid top: each is a receding grid drawn, if by the rule, with the same ratio for the same viewing distance, and, if by a tiring hand, with a creep that depends on how the painter’s hand moved on each and need not be the same.

The measurement that settles it paints a panel with several pavements of different lengths at different heights, either all by the rule or all correctly with a creep that varies from pavement to pavement, and asks whether the pavements’ arc amplitudes and gap curvatures, read together, separate the rule from the creep where no single pavement could: the rule’s amplitudes should follow the pavements’ lengths by one law, and a creep’s should scatter about it.