The rule's bow is read by its shape

A straightedge convicts the rule on five braccia took the one mark the taught constant-ratio rule leaves that a correct construction does not — its pavement’s tile corners bow off the diagonal — and asked whether a painter’s hand would bury it. At a pixel of scatter it did not, on any pavement of five braccia or more. At two pixels the straightedge levelled off near nine in ten, and on short pavements painted roughly it barely separated the rule from a construction at all. It gave out once the hand scattered its lines by about a third of the bow it was looking for.

That test read the bow’s size: the worst corner’s distance off the chord from the first corner to the last. But the rule’s corners do not merely fall off the chord. They fall off it in one direction, with the largest departure near the middle and none at the ends — a smooth arc whose shape the ratio fixes. A hand’s corners fall off it in both directions at random. The two differ in shape, not only in size, and the earlier essay closed on the lesson a different test had already taught this sequence: the roughness test tells two hands apart by the shape of what they leave, and its separation did not depend on the hand’s scatter at all.

So the question was whether a test that reads the arc’s shape keeps the rule separable where the straightedge failed — or whether, at three or four braccia, the arc has too few corners to have a shape at all. It keeps it, at every length, by a wide margin. Three corners have a shape; faintly, but enough to beat their own worst distance.

One arc, and one number to fit it #

The figure takes an eight-braccio pavement drawn to one page width, as the earlier essay drew them, and plots each interior tile corner’s signed distance off the chord.

Unpainted, the rule’s seven interior corners trace one arc, zero at both ends where the rule’s pavement was calibrated to agree with a correct one, largest near the middle at 8.15 pixels, all on one side of the chord. Painted with a pixel of scatter on every line and every orthogonal’s foot, the rule’s corners still sit near the arc, jostled; a correctly constructed pavement’s corners scatter about the chord on both sides.

The arc reading takes the rule’s own unpainted departures on this pavement as a template and fits it to the observed departures with one free number, its amplitude, by least squares. The rule’s own bow has amplitude one. A correct pavement’s has amplitude nought. On the painted pair in the figure, the rule comes out at 0.84 and the construction at −0.17 — the construction’s scatter happens to lean slightly the wrong way, which the signed amplitude counts against it.

That is the whole of the method. It is a matched filter, in the language of signal detection: the best linear test for a signal of known shape in noise that has no shape. It uses three things the straightedge ignores. It uses every corner, not only the worst. It uses the side each corner falls on, not only its distance. And it uses the weighting the shape itself supplies — corners near the middle, where the bow is largest, count for more than corners near the ends, where the rule and a construction nearly agree.

Ranked right at every length #

The test the earlier essay used to compare readings is the one used here: over two hundred painted pairs, one pavement by the rule and one by a construction at the same length and scatter, how often does the reading rank the rule’s the more guilty?

The arc reading is the better at every length and every scatter in the sweep. At a pixel it is already near certainty from four braccia, where the chord reading was at 0.96. At two pixels it ranks the pair correctly 76 times in a hundred on three braccia, 89 on four and 96 on five, against the chord’s 63, 74 and 83. At four pixels, where the chord never passes 0.79, the arc reaches 0.92 on eight braccia and holds there.

Past eight braccia neither reading improves much, and for the reason the earlier essay found. A pavement drawn to one page width bows no more at twenty braccia than at eight: the page is full, and more braccia only make the tiles smaller. The arc reading has more corners to average over on a longer pavement, but each is displaced by the same scatter and the bow they share has stopped growing. The ranking plateaus where the bow does.

Why more corners help the shape and hurt the size #

The plateau is worth seeing from the earlier essay’s side, because it shows why the two readings respond so differently to a pavement’s length.

The rule’s bow stops growing at about eight braccia, once the page is full. The hand’s wander off the chord — the worst of a correct pavement’s corners — keeps rising, because a longer pavement has more corners and the worst of more corners is further out. That rising wander is the chord reading’s noise floor, and it is why the chord reading’s ranking levels off and even slips on the longest pavements: the signal has stopped and the noise has not.

The arc reading’s noise does the opposite. It is the amplitude a correct pavement’s scatter lends the arc, and a projection onto a fixed shape averages the scatter over every corner: more corners, less lent. So on a long pavement the arc reading’s noise falls while its signal holds, and its ranking stays at the top of the chart. The two readings take the same corners and draw opposite conclusions from their number, because one asks for the worst of them and the other for what they share.

This is a general property of the two kinds of statistic, and a painter’s panel is a clean place to see it. A worst-case statistic is right when the signal is concentrated in one place and the noise is not; an averaging statistic is right when the signal is spread out in a known pattern. The rule’s bow is spread over every corner of the diagonal in a pattern the ratio fixes. The straightedge was the wrong statistic for it from the start, and the right one only needed the pattern written down.

The threshold a reader would use #

A reader of one panel does not rank pairs; they decide whether one pavement is the rule’s. The fitted amplitude supplies the natural threshold: halfway between nought and one. A pavement whose corners fit the rule’s arc at an amplitude above a half is called the rule’s; below, a construction’s.

On four braccia at two pixels, over four hundred painted pavements of each kind, that threshold convicts the rule’s pavement 79 times in a hundred and wrongly convicts a construction 19 times — one in five, which is what three corners and two pixels of scatter allow. On eight braccia at the same scatter it convicts the rule 97 times in a hundred and a construction fewer than two. A reader who wants fewer false convictions on a short pavement can raise the threshold and accept more acquittals of the guilty; the amplitude’s two distributions say exactly what each choice costs. The threshold does not depend on the page, the pavement’s length or the painter’s precision, since the amplitude is measured in units of the rule’s own bow; a reader needs only the number of braccia, to draw the template, and a ruler.

The shape holds to half the bow #

The earlier essay’s clearest number was where the straightedge gave out: at a scatter of about a third of the bow. The same number for the arc reading says what reading the shape buys.

The chord reading falls below nine in ten at 0.81 pixels of scatter on three braccia, 1.68 on five and 2.79 on eight — about a third of the rule’s unpainted bow on each, 2.0, 4.6 and 8.1 pixels. The arc reading holds to 1.15, 2.65 and 4.33: about half the bow, and one and a half times the chord’s reach at every length. A hand that scatters its lines by two pixels is beyond the straightedge’s reach on any pavement under six braccia; the arc reading reaches it at four and a half.

The ratio is not an accident of these pavements. The chord reading’s statistic is the worst of several noisy departures, and the worst of several is dominated by the noise once the noise is a fair fraction of the signal; the arc reading’s statistic averages the departures with the weights the shape supplies, so the noise averages down as the square root of the number of corners the bow is spread over. On a short pavement there are few corners and the advantage is modest. It never becomes a disadvantage, because the arc reading contains the chord reading’s information and more.

What three corners can say #

The earlier essay’s worry about short pavements was that a bow of three or four braccia has too few corners to have a shape. The distribution of the fitted amplitude on four braccia answers it.

Four braccia leave three interior corners. With a hand at two pixels, the rule’s fitted amplitude scatters about one, with its middle half between 0.60 and 1.35, and a construction’s about nought, between −0.40 and 0.40. The two distributions overlap, but their centres are a full unit apart and their spreads are under half a unit each, so a pavement drawn from one is ranked above one drawn from the other 87 times in a hundred. Three corners have a shape. They have it faintly, but the shape is in the side of the chord they fall on and in which of them falls furthest, and the straightedge’s worst-distance reading can use neither.

The limit is two braccia, which leave one interior corner. One corner has a distance and a side and no shape; the arc reading then reduces to its signed distance, which still beats the unsigned worst distance, but only because it knows which way the rule bows.

Why the earlier readings were a special case #

The earlier essay compared three ways of laying the straightedge and found that the chord — pinned at the two ends, where the rule agrees with a correct pavement — beat a straightedge fitted through all the corners, which splits the bow in two and loses part of it, and that averaging every corner’s distance beat both on long pavements.

The arc reading is where that sequence of improvements was heading. Pinning at the ends is right because the rule’s bow is zero there. Averaging every corner is right because the hand’s scatter averages down. Signing each departure is right because the bow has a side. And weighting each corner by the bow’s own shape is right because the bow is largest in the middle. Each of the earlier readings used some of those facts; the arc uses all four, and a matched filter is, by construction, the linear reading that uses them best.

The roughness test belongs in the same family, applied to a different question. Stepped, or measured from the zero and one hand step each found that a measured hand and a stepped hand leave residuals of different shape — one rough, one smooth — and that reading the shape made the test independent of the hand’s size. Here the shape belongs to the rule rather than to the hand, and reading it does not make the test independent of the hand’s size, since the hand’s scatter is the noise the rule’s bow must be seen through. What it does is get the most out of every corner the pavement has.

What a reader of a real panel should do #

The procedure is short and needs no computer. On a painted pavement suspected of the constant-ratio rule, rule the chord from the first tile corner along the diagonal to the last, measure each interior corner’s distance off it with its sign, and compare the pattern with the arc the rule would bend on a pavement of that many braccia — which is the same arc on any page, scaled. If the corners follow the arc, to one side and largest in the middle, the rule is convicted at half the scatter a straightedge needs; if they fall about the chord on both sides, the pavement was constructed.

Three procedures, one panel set the rule beside the constructions a painter could have used instead, and the rule that draws another room established that the rule’s pavement is a correct picture of a different floor, and the rule is exact for a floor that lengthens that no reading of the transversals alone can convict it. The diagonal is the rule’s only confession, and it is a confession made in a shape. Reading it for its size alone was listening to half of it. The other half was always on the panel — in which side of the chord each corner falls, and where along the diagonal the largest departure sits — and a reader with a ruler and the template for that many braccia can hear it without any statistics beyond a sum of products.

What was assumed #

The hand’s scatter is independent from line to line. Each transversal and each orthogonal’s foot is placed with its own error. A painter who drifts — whose lines creep steadily one way as the pavement recedes — produces departures with a shape of their own, and a smooth drift can mimic part of the rule’s arc. A tiring hand draws a different habit is the reminder that a creep and a rule can counterfeit each other on one strip; here the arc’s fixed shape, zero at both ends, is what a steady creep does not share.

The ratio is known. The template is the rule’s own bow on a pavement of that length. A rule applied with a different ratio bends an arc of nearly the same shape and a different size; the amplitude absorbs the size, and the shape’s small change with the ratio costs the reading little.

The ends are placed well. The chord is ruled through the first and last corners, and a painter whose first or last corner slipped moves the whole chord. The arc reading inherits that error as the straightedge did.

Still open: whether a steady creep can counterfeit the arc #

The first assumption is the one a real panel is most likely to break, and it has a specific form. A painter who places each transversal from the one before, rather than from a construction, accumulates error along the pavement; a painter who tires creeps one way. Either leaves the corners’ departures from the chord correlated from corner to corner, and a correlated departure can have a shape.

The measurement that settles it paints correct pavements with a stated steady creep in the transversals’ spacing — the creep a tiring hand draws a different habit introduced for divergent strips — and asks two things. First, how large a creep produces a fitted amplitude as large as the rule’s own bow, on pavements of four to twelve braccia: if it takes a creep far beyond anything a painter shows, the arc reading is safe. Second, whether the creep’s departures differ from the rule’s in a way a two-parameter fit could separate — a creep’s departures are not zero at both ends in the same way, since the chord through the first and last corners absorbs a linear trend but not a curved one — and if they do, whether a reader can then name both the procedure and the painter’s fatigue from one diagonal.