One hand step each

A perspective construction is a recipe, and a recipe is followed. Most of the following is mechanical — draw a line between two given points, mark where two lines cross — and the mechanical parts introduce no error worth the name, because a ruler laid between two drawn dots lands between them.

Exactly one step in each recipe is not mechanical. Somewhere a person decides where a mark goes, and that is the only place a hand can slip. Which step it is differs from recipe to recipe, and everything a finished drawing can say about the recipe that made it comes from that difference.

The four steps #

Alberti’s lateral section puts the hand on the braccia marks along the section’s ground. Each mark is set from the panel with a scale, so each one is its own measurement and its own error. A slip on the fourth mark moves the fourth transversal and leaves the others exactly where they were.

The distance-point construction puts the hand on a single mark on the horizon. It is the only step that is chosen rather than measured, and a slip in it changes the panel’s focal length and nothing else — which, as the slip that leaves no trace measures, produces another exactly correct drawing.

A measuring line walked with dividers puts the hand on each length in turn, set from the previous mark rather than from a zero. A slip on the fourth step moves the fourth transversal and every transversal after it, by the same amount.

A photograph puts the hand nowhere. Its departure is a lens, which is a smooth function of where on the page a mark is rather than of which mark it is.

Four steps, four kinds of trace, and the traces differ in kind rather than in size. That is the useful part: the size of a hand error is a fact about the draughtsman and the kind of it is a fact about the recipe.

What a residual is here #

To compare the traces they have to be in one currency, and the currency is a residual.

Take the finished transversals. Fit the best correct perspective to them — the horizon, the scale and the origin that come closest — and subtract. What is left, mark by mark, is what the drawing has that no correct perspective has.

The subtraction has to be done carefully in one respect. The best correct perspective is fitted with the horizon free, because a hand slip moves the implied horizon as well as the transversals, and a fit that pinned the horizon would attribute part of the slip to the pinning. So the residual measured here is what survives after every correct drawing has had its chance to explain the marks.

The residual is then converted back into pixels on the page. That conversion matters: a ratio error on a far transversal is a tiny displacement, because a far transversal is squashed, and reporting the residual in ratios would make a long pavement look worse-drawn than a short one by the same hand.

The three shapes #

Run the same hand at the same precision through the four recipes and the residuals look nothing alike.

Independent. Marks measured from a common zero give errors with no relation to each other. The residual scatters, crossing the axis about as often as a coin comes up heads.

Cumulative. Marks stepped from the last give a random walk. Neighbouring residuals are close because they share every step before them, so the residual wanders across the page instead of scattering.

Systematic. A rule applied throughout, or a lens, gives a smooth curve. It crosses the axis twice on a pavement of any length, because a smooth departure from a projective law is a low-order function and low-order functions have few zeros.

The statistic, which is arithmetic rather than a threshold #

The three shapes need one number to separate them, and the number is not a threshold somebody chose.

Take the residual’s second difference — each value minus twice its neighbour plus the one beyond — and compare its size with the residual’s own. Call the ratio the roughness.

For independent errors the arithmetic is exact. The second difference of three independent quantities has one plus four plus one times their variance, so its root-mean-square is the square root of six times theirs, and the roughness of an independent residual is √6 — about 2.449, whatever the hand’s precision, whatever the panel, whatever the scale.

For a walk it is lower, because a walk’s neighbours are correlated, and the amount lower depends on how long the walk is. For a smooth curve it is much lower still, and it tends to zero as the pavement is drawn finer, because a smooth function sampled more finely has second differences that shrink faster than its amplitude does.

Measured on a thirty-two-braccio pavement, marks from a zero give 2.51 and stepped marks give 1.14. The first is √6 within the sampling error of a hundred and twenty drawings; the second is nowhere near it.

Three shapes, three powers of the length #

The three cases are usually described as “high, lower and much lower”, and they are better described as three different rates, because the rates are what a longer pavement exposes.

For independent marks the roughness is $\sqrt6$ at every length — no $n$ in it at all, as the arithmetic above shows.

For a walk the second difference is the difference of two steps and does not grow with the length, while the residual itself grows as $\sqrt n$ once the fitted perspective has removed its trend. So the roughness falls as $n^{-1/2}$.

For a smooth departure — a rule, or a lens — the second difference over a sample spacing $L/n$ is about $g''(L/n)^{2}$ while the residual’s own size is fixed by $g$ and $L$. So the roughness falls as $n^{-2}$.

$$n^{0}, \qquad n^{-1/2}, \qquad n^{-2}.$$

Three powers, and they are the whole classification. A short pavement compresses all three into a narrow band and a long one spreads them by construction, which is why the row’s readings improve with length and why they improve fastest on the systematic case: a lens is easy to catch on a long floor and a walked hand is only slowly easier.

It also says what a mixed drawing would look like. A hand that measures the first few marks from a zero and steps the rest produces a residual whose roughness sits between two of the three rates rather than at one of them, and the crossover point along the pavement is where the procedure changed.

What this cannot do #

The separation is real in the ensemble and it is not a verdict about a single panel, and saying so is most of what the reading is worth.

The two roughness distributions overlap. A walk that happens not to wander looks like a scatter; a scatter that happens to drift looks like a walk. Measured over sixty drawings at a hand precision of a pixel and a fifth, a walked measuring line is read as walked on about half of them.

What is real is the difference between the distributions, and four marks before anything is said is the count that says how few statements a short pavement makes to begin with. The difference widens with the pavement’s length: the gap between the two medians goes from a third of a unit at six braccia to nearly one and a half at thirty-two. Stepped, or measured from the zero is where that is measured and where the exponents behind it are fitted.

So the honest statement to a reader holding one panel is: a hand can be told from a rule, and which hand cannot. The first is worth having. The second would be worth more, and claiming it would be reporting a distinction that is not in the drawing.

Why the photograph has no hand at all #

The fourth row is different in kind and it is here because leaving it out would make the reading look better than it is.

A photograph’s departure from a correct perspective is a lens, and a lens is radial about the principal point — a smooth function of a mark’s distance from the centre of the frame, and not of which mark it is. Straight lines that are not is where this collection measures it and a lens destroys the invariant is where it establishes that the cross-ratio does not survive it.

That makes a photograph’s residual smooth, so the roughness reads it as a rule. It is a rule, in the sense that matters here: a systematic departure that has nothing to do with anybody’s hand. What separates it from the taught spacing rule is a second family of marks somewhere else on the page, and the lens a pavement can hide is that separation.

The hand precision that is in play #

Every number above is quoted at a stated hand precision and it is worth knowing what the numbers mean.

A draughtsman placing a ruler on a half-metre panel at arm’s length is working to something like a third of a millimetre. Scaled onto the six-hundred-and-ninety-pixel canvas these figures use, that is about half a pixel. Careless work is two or three times that; work under a magnifier is half of it.

At half a pixel the reading sees a slip on about two panels in five. At one pixel it sees nearly all of them. At a quarter of a pixel it sees almost none. So the whole apparatus lives inside a narrow band of care, and outside that band the answer is decided by the draughtsman’s steadiness rather than by anything about the recipe.

That is not a limitation of this reading in particular. It is the ordinary situation for any measurement whose signal and whose noise are the same size, and the useful response is to state the band rather than to quote an answer from inside it.

Why the mechanical steps really are mechanical #

The claim that only one step in each recipe carries error deserves a defence, because a recipe has a dozen steps and calling eleven of them error-free sounds like an idealisation.

It is not, and the reason is projective rather than practical. Once two points are on the paper, the line through them is determined; a ruler laid across two dots may wobble, but the wobble moves the drawn line by a fraction of the dots’ own placement error, and the intersection of two such lines inherits that fraction. The errors of the mechanical steps are therefore second order in the errors of the marks — they are the errors of the errors — and on a panel where the marks are placed to half a pixel they are hundredths of a pixel.

The one exception is an intersection taken at a very shallow angle, where a small displacement of either line moves the crossing a long way. That is the leverage this collection measures elsewhere: the lamp, out of the picture has the same structure, with drawn lines meeting far outside the frame, and there the leverage between the clicked points and the answer runs from five to fifty. In a pavement the diagonal crosses the orthogonals at a healthy angle and the leverage is close to one, so the intersections are honest.

That is a thing worth checking rather than assuming, and it is the reason the constructions here are run with drawn lines and drawn intersections rather than with the closed forms they satisfy. A recipe evaluated in closed form has no intersections to be shallow.

The order the marks are made in #

There is a second difference between the recipes that is easy to miss and turns out to matter as much as which mark is chosen: the order.

Alberti’s marks can be made in any order, because each is set from the panel. The dividers’ marks cannot: the fourth depends on the third. That is what makes one set independent and the other a walk, and it means the distinction is not really about the method at all — it is about where the zero is.

A draughtsman using a measuring line and a scale rather than dividers is measuring from the origin every time, and the marks are independent. A draughtsman using Alberti’s section and stepping a pair of dividers along the ground line is accumulating, and the marks are a walk. So the recipe names one thing and the drawing records another, and the second is the one that leaves a trace.

That is a correction to the framing rather than a detail, and it is why the next rung is called what it is. The reading tells stepped marks from measured ones. It does not tell Alberti’s section from a measuring line, because either can be executed either way.

What the reading is not sensitive to #

Three things a reader might expect to matter and which do not, each worth a sentence because assuming otherwise leads to a wasted measurement.

The panel’s scale. Everything here is a ratio of a residual to the marks’ own spacing, so photographing a panel at twice the resolution changes nothing but the units. A reading that improved with magnification would be measuring the reproduction.

The viewing distance the panel was built at. The three shapes are the same at a focal length of three hundred and of twelve hundred, because the residual is compared against the best correct perspective and every correct perspective is available to the fit. A short-focus panel has more sharply converging transversals and the same roughness.

How many braccia across the front. The pavement’s width is set by where the ground line was divided and enters nothing above. A wide shallow pavement and a narrow deep one with the same number of transversals carry the same statements.

What the reading is sensitive to is the number of transversals and the hand’s precision, which are the two axes of the surface above — and, more than either, the reader’s own ability to place a mark on the reproduction.

What changes if the recipe is mixed #

Real draughtsmen do not follow one recipe. A panel may have its near transversals stepped with dividers and its far ones interpolated by eye; a workshop may lay out one bay by measurement and repeat it by construction.

The reading handles that better than might be expected, and the reason is that the residual is read as a shape along the pavement rather than as a single number. A panel stepped for its first six braccia and measured for the rest has a residual that wanders and then scatters — the same reading the residual has a shape makes of a lamp recovery, where what the leftovers look like says more than how large they are, and the roughness computed over the whole is a mixture of the two.

What it handles badly is a panel whose transversals were adjusted afterwards to look right. That is a hand step applied to the finished drawing rather than to the construction, its errors are neither independent nor cumulative, and it leaves a residual with no characteristic shape at all. Dividing depth by eye measures what a good eye actually produces, and the answer is a residual that is smooth in the middle of the pavement and rough at its ends — which is a fifth shape, and one this reading would report as a rule.

The short version #

Each classical construction has one step a hand performs, and the four constructions perform four different steps: a mark per braccio from the panel, a single mark on the horizon, a mark per braccio from the last, and none at all.

Those four leave three kinds of trace — independent, cumulative and systematic — separated by one statistic whose threshold is arithmetic rather than chosen. A hand can be told from a rule on nearly every drawing. Which hand it was cannot be told from one drawing, and the gap that would tell it only opens on pavements longer than anybody painted.