What a panel says about its maker

The row began with a question that sounds like connoisseurship — which procedure drew this floor — and ends with a table. The table is mostly refusals, and the refusals are the part worth having.

The reading, assembled #

Everything the row has built comes down to three instruments applied in order, each cheaper than the last is decisive.

The panel’s horizon against the pavement’s. Find where the orthogonals meet with a straightedge; fit the horizon the transversals imply; compare. If they disagree by more than a slipping hand could account for, the pavement was not drawn to this panel.

The projective test. Four consecutive transversals, their cross-ratio, against the four thirds four evenly spaced points must give. It needs no horizon and no assumption, and it says whether the marks could have been projected at all.

The residual’s shape. Fit the best correct perspective with the horizon free, subtract, and measure the roughness of what is left — the size of its second difference against its own size. Independent errors give the square root of six; a walk gives less; a rule gives much less.

Three instruments, three questions, and none of them is the question the row started with.

The table #

Sixty drawings by each procedure, at a hand precision of a pixel and a fifth on an eight-braccio panel, classified.

The distance point’s row is entirely “correct”, and that is right: a slip in placing it produces another exactly correct drawing, so there is nothing in the picture to find. The constant-ratio rule’s row is also entirely “correct”, and that is the finding the wrong field’s rung turns on — its pavement is internally consistent to under a fifth of a pixel.

The two hand rows are separated on about half the drawings each, which is the honest number and is discussed below.

Admitting the horizon convicts the rule and convicts a strong lens, and leaves both hand rows exactly where they were. That is the reading getting sharper rather than merely stricter — a test that also moved the hands would be a test that convicts anything.

The four claims that survive #

Set out in decreasing order of confidence, which is the order a reader should trust them in.

A drawing can be shown not to be a projection. The cross-ratio, four transversals, decisive, no assumptions. This is the only claim in the row with no failure rate attached.

A drawing can be shown to disagree with its own panel. The implied horizon against the drawn one, at any pavement size, with one straightedge. It convicts the taught spacing rule at a glance and it convicts nothing that a slipping hand does.

A drawing’s class of error can be read. Independent, cumulative or systematic, from the roughness — reliably for the systematic case and about half the time for the split between the two hands.

A drawing’s procedure cannot be named. Not the distance point against Alberti’s section; not a well-executed anything against a well-executed anything else; and not a photograph against a spacing rule without a second family of marks somewhere else on the page.

Why the two hands blur #

The row’s most-asked question and the one with the least satisfying answer.

The roughness of a scatter and the roughness of a walk are two distributions, not two numbers. At six braccia their medians are 3.12 and 2.80 and their spreads are wider than the gap; at thirty-two they are 2.51 and 1.14 and the gap is comfortable. In between the answer is a coin weighted by the pavement’s length.

Where the two numbers come from #

Both medians can be derived, and deriving them says why the gap opens at the rate it does.

The roughness is the size of the residual’s second difference against the size of the residual itself. For independent marks with error $\varepsilon_k$, the second difference is $\varepsilon_{k+1} - 2\varepsilon_k + \varepsilon_{k-1}$, whose variance is $(1+4+1)\sigma^{2} = 6\sigma^{2}$ against a residual of $\sigma$. So the ratio is $\sqrt6 = 2.449$, exactly, and at every pavement length — which is the constant drawn on the figure and the reason it is drawn as a constant.

For a walk, the errors accumulate: $\varepsilon_k = \sum_{j\le k}\delta_j$. The second difference is then $\delta_{k+1} - \delta_k$, of variance $2\sigma_\delta^{2}$ — unchanged by the length — while the residual itself grows, reaching about $\sigma_\delta\sqrt{n/3}$ once the fitted perspective has removed its linear part. So the walk’s roughness falls as

$$\sqrt{\frac{6}{n}},$$

and the gap between the two opens as $\sqrt n$. Against the sweep: from six braccia to thirty-two is a factor of 2.31 in $\sqrt n$, and the walk’s median falls from 2.80 to 1.14, a factor of 2.46. The scatter’s median over the same range goes 3.12 to 2.51, settling onto $\sqrt6$ from above as the sample gets long enough for the median to reach its asymptote.

Two consequences, and the second is the one that decides what the row can be used for.

The scatter’s number is a theorem and the walk’s is a measurement. $\sqrt6$ has no free parameter in it — not the hand’s precision, not the panel, not the pavement’s length — so a reading that returns 2.449 is not reporting a calibration. The walk’s number carries $n$, which is why the two curves are drawn against length at all.

And the separation needs a pavement nobody painted. A factor of two between the medians needs $n \approx 24$; a factor of three needs about fifty. Real painted floors show three or four rows before the figures cut them off, which is the regime where the two medians are 3.1 and 2.8 with overlapping spreads. So the distinction between a hand measuring from a zero and a hand walking with dividers is available in principle, opens as the square root of the floor, and is not available on any panel anybody actually made.

So the separation is real and it is a property of the ensemble. A reading of one panel that reports “stepped” is reporting a draw from an overlapping distribution, and stating that is not a hedge — it is the difference between a measurement and an assertion. Stepped, or measured from the zero fits the exponents that say why the gap opens and how fast.

What the matrix does not contain #

Three absences, each deliberate, each worth naming because a matrix looks complete and this one is not.

An ordinary lens. The photograph row in the table above is taken through a strong wide-angle, deliberately: an ordinary lens leaves a departure smaller than the reader’s own ruler, and its photographs read as correct drawings. Including an undetectable lens in the matrix would report it as a confusion when it is a measurement.

A mixed procedure. Real panels are laid out one way near the reader and another way in the distance, and the matrix runs each procedure alone. The reading handles mixtures better than might be expected, because the residual is read as a shape along the pavement, but the failure rates quoted here are for pure cases.

And a drawing adjusted afterwards. A draughtsman who lays out a pavement correctly and then nudges two transversals to look right has produced a residual with no characteristic shape, and the reading will report whatever the nudges happened to resemble. There is no defence against that and none is claimed.

The verdicts, and why there are five #

The table has five columns and it is worth saying why each exists, because a classifier’s vocabulary is a claim about what distinctions the data supports.

Correct. The drawing is a projection to within the reader’s own ruler. Nothing further can be said, and something further is what every attribution in the literature claims.

No shape. The drawing departs from every correct perspective and does not say how, because a five-transversal pavement carries two statements and two numbers have no shape. This is a real verdict rather than a failure: it distinguishes “there is evidence and it is uninformative” from “there is no evidence”.

From a zero. The marks were made from a common origin and each error is its own. Read from the roughness sitting near the square root of six.

Stepped. The marks were walked from the last, so each carries the ones before it. Read from a roughness below that, and unreliably.

A rule. The spacing follows a law the projection does not have. A recipe or a lens, and the pavement does not separate those two.

Five verdicts for five procedures, and they do not line up one to one. Two procedures share the last column, two share the middle pair, and one of them is never reported at all — which is what a classifier looks like when it is measuring what the data contains rather than what the question asked for.

What a sixth column would have needed #

There is an obvious missing verdict — “drawn with the distance point” — and it is worth being explicit about what would have to change for it to exist.

The distance point’s slip maps the set of correct drawings onto itself. So a column for it would require an instrument that is not invariant under a change of focal length, and every reading of a pavement’s spacing is invariant under exactly that, because a change of focal length is a projective map of the receding line and the readings are projective.

The instrument would therefore have to come from outside the pavement. A vertical of known height does not supply it; a second pavement does not; a wall does, because a wall gives a third direction and three directions determine the focal length. So the sixth column exists on a panel containing a box or a room and does not exist on a panel containing a floor.

That is a clean and slightly unusual result: the missing verdict is missing for a stated reason, and the reason names exactly what a panel would have to contain. A refusal that says what would lift it is worth more than one that does not.

A note on running the reading on a real panel #

Everything above is measured on synthetic drawings, where the truth is known and the hand’s precision is set. Two things change when the drawing is a photograph of a painting.

The marks are found rather than given. A transversal on a painted floor is a joint, an edge or a change of colour, and those are three lines a few millimetres apart in the world. Reading near edges in the foreground and far edges in the distance mixes two families and puts a systematic error into the residual that looks exactly like a rule. Pick one feature of the tile and read it every time.

And the panel is not flat. A photograph of a painting taken off-axis is a homography of the painting, and a homography preserves the cross-ratio — so the projective test survives it exactly. What does not survive is the residual measured in pixels, because a homography changes distances differently in different parts of the frame. Rectify first, or quote the residual in the ratios rather than in pixels and accept the dilution.

Flattening a façade out of the photograph is the rectification, and it needs four points whose world shape is known — which on a painting is the panel’s own rectangular border.

The one number that is a choice #

Everything in the row is stated against a tolerance, and it is the only quantity anybody picked.

The residual is reported as a displacement in pixels: how far the worst transversal is from where a correct perspective would have put it. A fifth of a pixel is a mark read under a glass on a good reproduction; a whole pixel is a mark read across a room. Every other threshold in the row — the square root of six that separates independent errors from cumulative ones, the three unknowns that set the count of testable statements — comes out of the arithmetic.

That single choice moves the answers a long way, which is why it is printed on every figure that uses it rather than folded into a constant. Four marks before anything is said sweeps it against the pavement’s length and the result is blunt: outside a narrow band of care, the reader’s eyes decide the answer and the drawing does not.

What a reader should actually do #

For somebody standing in front of a panel with a reproduction and a straightedge, in the order that costs least.

Count the transversals. Below four, stop. A three-tile pavement is a correct perspective of something whatever its spacing, so no claim about its construction has support.

Find where the orthogonals meet. One straightedge, no arithmetic, and it is the step nearly every published reading skips. It converts a pavement into a panel and it supplies the one statement that separates a rule from a hand.

Then read four marks. The cross-ratio against four thirds, at whatever precision the reproduction allows — and quote the precision, because it decides more of the answer than anything about the painter.

And expect a null result. Most panels will report a correct perspective and mean it. That is not a failure of the reading; it is what a correct perspective looks like from the inside, and it is what a distance-point construction, an Alberti section, a measuring line and a modern photograph all look like alike.

What the row changed about the collection’s own claims #

Two things, and the second is a correction rather than an addition.

The free parameter is not merely unstated, it is unrecordable. The viewing field has argued from its first rung that a construction chooses where the reader stands and does not say so. This row establishes something stronger: the choice leaves no trace in the finished drawing at all, so a reader cannot recover it even in principle from the family it was used to construct. Recovering it needs a second thing in the picture — three vanishing points, a circle, a stated proportion — and the slip that leaves no trace is where that list is made.

And “not a projection” is a weaker verdict than it sounded. The wrong field’s method is to run a taught construction and measure how far it falls short. This row found a construction that is not a projection and whose output is a projection to within a reader’s precision — so “falls short by nine and a half pixels” and “reads as correct” are both true of the same drawing, and which one a reader gets depends entirely on what they compare it against.

The reading against the collection’s other readings #

It is worth placing this one, because the collection contains several readings of a single picture and they differ in exactly the way that matters.

Recovering the camera from the picture it drew returns five numbers from three vanishing points and has internal redundancy — three independent estimates of the focal length that must agree, which is a check the reading performs on itself. A height, out of one photograph returns one number from a vertical, a horizon and a known length, with no redundancy at all and therefore no internal check.

This reading sits between them. It has redundancy — a pavement of n transversals makes n − 3 statements that must all agree — and it returns no number at all, only a verdict about a class. That is an unusual shape for a measurement and it is the right one here: the quantity being asked about, the procedure, is not a number, and a reading that returned one would be inventing a scale for something that does not have one.

The short version #

A perspective pavement will tell a reader whether it is a projection, whether it agrees with the panel it sits on, and what kind of error is in it. It will not tell them which recipe produced it, and on one recipe — the distance point — it will report a correct drawing however badly the one free mark was placed.

The instrument that does the most work is the cheapest and the least used: the horizon the orthogonals meet at, compared with the horizon the transversals imply. One straightedge, no arithmetic, and it separates a construction that is not a projection from a hand that slipped.