Copying square by square

The oldest working method for getting a drawing off one surface and onto another is to rule both into squares and copy one square at a time. It is in the drawing manuals, it is in the workshop practice they record, and it is the arrangement Alberti’s gridded veil makes physical — a pane ruled into cells, a matching grid on the paper, and a mark transferred cell by cell. The method’s whole claim is that a hard problem has been broken into small easy ones.

The claim is half true, and the half that fails is not the half anybody checks. Setting out the cell corners on a receding pavement is exact: the corners come from a vanishing point and a measuring point, both of which are constructions this collection has already measured to a fraction of a pixel. What happens inside a cell is not a construction at all. A copyist looks at where a line crosses the cell’s boundary, judges the proportion, and puts a mark in proportion between the four corners — which is a linear map. The map the camera used is projective.

A projective map and its linear part agree to first order and differ at second order, so the discrepancy is zero along the cell’s edges, zero at its corners, largest in the middle, and invisible to every check the method performs on itself. This essay measures it on one pavement, finds its law, finds where on the floor it is worst, and then draws the neighbouring case in which it is exactly zero — which is the case almost everyone learns the method on.

The corners are exact, and that is what makes the trap #

It is worth being precise about which part of the workflow is being defended, because a reader who takes this as an attack on the grid method has taken the wrong half.

The corners of the cells are laid out by construction. The transversals — the lines running across the pavement — are placed by a measuring point, and the essay that took the measuring point apart shows the construction agreeing with a pinhole camera to the limit of the arithmetic. The orthogonals converge on a single vanishing point. Alberti’s own recipe does the same job by a different route and lands in the same place. Nothing in this essay finds a fault in any of that.

What the corners give is a set of exact samples of the map from plan to picture. Sixty-four cells across eight rows means eighty-one exact points. Between them there is nothing but the copyist, and the copyist is doing linear interpolation with a pencil.

That is the same shape of problem as dividing depth by eye, and the difference is instructive. Dividing depth by eye has no exact points in it at all — the whole estimate is a judgement — so the error is large and everybody knows it is there. The grid method surrounds every judgement with exact points, which shrinks the error and, much more importantly, removes the feeling that a judgement is being made. A copyist filling a cell is not conscious of estimating anything. The cell is small, the marks are close to the corners, and the work feels like transcription.

The error is therefore not a large one hiding in plain sight. It is a small one that has no way of announcing itself, on a method whose reassurance comes from a part of itself that is genuinely exact.

Inside the cell, magnified #

The departure has a definite shape, and seeing the shape is what makes its law obvious afterwards.

The pattern is a bulge. Along each edge of the cell the two lattices agree, because along an edge the fill is interpolating between two exact corner values of a map that is itself very nearly linear over that short run. Away from the edges the disagreement grows, peaks near the centre, and is signed — the fill puts every interior point slightly nearer the far edge of the cell than the projection does, because the projective map compresses depth and its linear part does not.

That the disagreement is zero on the boundary is the reason the method survives inspection. A copyist checking work does it by looking at whether the lines meet the cell edges in the right places, and they do, exactly. There is no check available at the interior of a cell that does not amount to redoing the projection.

The scale is worth stating plainly. Three point three pixels on a picture 690 across is under half a per cent of the picture’s width, and read back onto the pavement it is nineteen millimetres of ground. Neither number is dramatic. What makes the measurement worth having is not its size but that it is a systematic displacement with a known law, present in every cell of every drawing made this way, and reported by nothing.

The law is the square of the cell #

A second-order error has a rate, and the rate is what says whether refining the grid is a fix or a palliative.

An exponent approaching $-2$ is exactly what the algebra predicts. Write the projective map as a ratio of two affine functions of the plan coordinate. Over a cell of width $h$, expand about the cell’s centre: the constant and linear terms are reproduced by the fill, the quadratic term is not, and it carries a factor $h^{2}$. So halving the number of cells across the floor multiplies the worst error by about four.

The measured exponent starts shallower than $-2$ and steepens, which is the ordinary behaviour of an asymptotic law measured before its asymptote. At four cells across the floor a cell spans enough depth that the third-order term is not negligible either, and the error reads 10.79 px rather than the four times 3.30 the pure law would give. By sixteen cells the higher terms have died away and the local slope is close to $-1.93$.

There is a second consequence of the exponent that is easy to miss and matters more to a workshop than the first. Because the error is quadratic in the cell and the cell count is quadratic in the linear division, the total error summed over a drawing falls only linearly with the number of cells across. A finer grid puts a smaller error in each of a larger number of places, and the two effects nearly cancel in aggregate; what improves sharply is the worst reading anywhere on the sheet. So the right thing to promise a copyist is not that a finer grid makes the drawing more accurate overall but that it removes the outliers, which is a different and more honest claim.

The practical reading is straightforward and slightly deflating. Refining the grid does work — it is not a case where the error is stubborn — and it works at a rate that makes accuracy expensive in labour. Halving the cell quarters the error and quadruples the number of cells, so the cost of a factor of ten in accuracy is a factor of ten in cells, which is a hundredfold in cell count. That is the trade a workshop is actually making when it rules a finer grid, and it is a better trade than most approximations offer.

The worst cells are the nearest ones #

Where on the floor the error lives is the part of this that contradicts what the method’s users would guess.

The expectation is that the far cells are the dangerous ones, because they are crowded and small and hard to see. The measurement says the opposite twice over. On the paper the nearest row is worse than the deepest by a factor of about nine — 3.30 px against 0.36 — and on the floor, where a pixel near the horizon buys a great deal of ground, it is still worse, by about a factor of two.

The reason is that the fill’s error is a curvature, not a slope. The image of a receding coordinate is a reciprocal, and a reciprocal is steepest far away in plan and most sharply curved near to. A cell in the deepest row occupies a large stretch of pavement, but over that stretch the map is close to its own tangent, so a linear fill is close to right. A cell in the nearest row occupies a small stretch, over which the map bends hard.

It is worth checking that against the other reading of the same numbers, because the two disagree about how alarming the result is. On the paper the near-far ratio grows with the grid: at six rows it is about eight and at sixteen rows about eleven, because a finer grid pushes the deepest row further into the flat tail of the reciprocal, where the map is closest to its own tangent and a linear fill is closest to right. Read back onto the pavement the ratio barely moves, sitting near two whichever grid is used. Both readings agree on the sign, which is the finding, and they disagree on how much of the story the paper tells — which is the ordinary situation whenever a quantity is measured in one space and consumed in another.

That inverts where a copyist’s attention goes. The near foreground of a pavement is the legible part — large cells, clear lines, nothing crowded — and it is where nobody slows down. It is also where the method is least accurate on the paper, in absolute pixels, which is the currency a viewer of the finished drawing is spending.

The control, and it is the arithmetic floor #

The claim so far is that filling a cell in proportion is an approximation. The test that could refute it is the neighbouring case where the same fill is not an approximation at all, and the case exists.

A residual of that size is not a small number in the sense that three pixels is a small number. It is what a double-precision arithmetic returns for a quantity that is mathematically zero, and it stays there whether the wall is ruled into twenty-four cells or two hundred and sixteen. There is no rate to measure, because there is no error to have a rate.

This is the whole explanation of the method’s reputation. Squaring up is taught and practised on flat work — copying a cartoon onto a wall, enlarging a design, transferring a portrait — and on flat work it is exact. Everyone who has used it has used it in the case where it cannot fail, and has then carried the confidence across to a receding pavement, where the identical procedure is a second-order approximation. Nothing in the method’s description distinguishes the two situations, because the description is about the grid and the difference is in the plane.

It is the same distinction the essay on what survives being copied draws between operations that a copy preserves and operations that it does not. A grid transfer between two parallel planes preserves everything, because the map is an affinity and the fill is that affinity. A grid transfer that crosses a projection preserves the corners and nothing between them.

A machine makes the same mistake, and knows it #

The error has an exact analogue in rendering hardware, and the analogue is worth having because it was met there, diagnosed there, and fixed there.

A renderer that interpolates a texture coordinate linearly across a triangle on the screen is doing what the copyist does inside a cell, on a cell the size of a polygon. The result is the affine-texture warp that early real-time graphics is remembered for, and the essay that priced it gives the closed form in the depth ratio.

Two things transfer. The first is the fix — interpolate a quantity that is linear on the page, which for a renderer means dividing through by depth, and which for a copyist would mean placing interior marks by construction rather than by proportion. The second is the mitigation everybody actually used before the fix was affordable, which is to subdivide: smaller triangles, smaller cells, error falling as the square. That is precisely the falloff measured above, arrived at independently by people who had never ruled a pavement.

The difference is that a renderer’s error is between two exactly known maps and can be removed, while a copyist has only a pencil and a judgement. What a copyist can take from it is the knowledge of where subdivision is worth spending — near, not far — which is the opposite of the intuition.

The veil, and the eye it assumes #

The instrument the method comes from carries a second unstated premise, and it belongs here because it is not the same premise and it fails differently.

Alberti’s veil is a pane ruled into squares, held between the draughtsman and the subject, with the drawing surface ruled to match. What the veil records is where the line from the eye to a point of the subject crosses the pane, so the veil’s centre of projection is the draughtsman’s eye — which means the method needs the eye returned to one sighting point between every pair of marks. The manuals say so, and they treat it as a matter of care.

It is not a matter of care, and it is not the same fault as the fill. The fill’s error is a property of the geometry and is there however steady the hand. The sighting point’s error is a property of the operator, is zero if the eye is exactly returned, and grows with how far it wandered. Two independent sources of error in one instrument, one of which discipline can remove and one of which it cannot.

They also enter at different places. A cell’s fill is wrong inside the cell and right at its boundary. A wandering eye moves every mark on the pane, including the ones on cell boundaries, so the corners of the copied figure move too — and a copyist who checks the work by looking at where lines cross the ruled squares will see nothing wrong, because the ruling moved with everything else. The essay on the string frame and the sighting point measures what that wander costs; here the point is only that the grid does not protect against it.

What the measurement does not settle #

Three limits, and the third is the one a reader should carry.

It says nothing about whether a copied drawing looks wrong. Nineteen millimetres of pavement inside one cell of a floor is a geometric statement about which scene the drawing is a projection of, and whether any viewer could tell is a question this collection does not answer and does not have the instruments for.

It is a measurement of one pavement at one field of view. The error scales with how strongly the plane recedes, so a shallower floor in a narrower picture reads far less, and a steeply raking floor in a wide one reads far more. The exponent is general — it comes from the order of the Taylor expansion and not from the scene — while the coefficient in front of it is not.

And it does not license the conclusion that hand copying is unreliable. The grid method’s corners are exact, its interior error falls as the square of the cell, and its worst reading here is under half a per cent of the picture’s width. Judged as an approximation it is a good one. Judged as what it is usually presented as — a way of removing judgement from a transfer — it is a method that removes ninety per cent of the judgement and hides the rest.

Exact where it is checked, silent where it is not #

That last sentence is the shape this belongs to, and it is a shape the collection has now met twice from different directions.

Two rules for one pavement takes a pair of taught recipes that their author asserted agree, executes both from the same ground line and the same free parameter, and finds them agreeing to a fraction of a pixel — and then finds them parting by twenty-four pixels when each is executed from the numbers its own wording invites. The recipes are exact. What is not specified is a parameter, and the wordings imply different values for it.

The grid method is the same defect one level down. Its construction is exact and its specification stops at the cell boundary, so what happens inside is not wrong so much as unstated, and every copyist supplies the same reasonable guess. A method that is exact where it is checked and silent where it is not will pass every check it proposes for itself, indefinitely, and the silence is where the whole of its error lives.

Two other essays in the collection turn on the same distinction. What a straightedge reaches on a receding line finds that repeated construction reaches exactly the rational points of a receding line and no others, so a method can be perfectly exact and still unable to reach most of what it is asked for. The centre of the picture is not the centre of the paper finds an exact construction fed a false premise about the sheet it is drawn on, returning a wrong answer with no residual to show for it.

The habit these suggest is a small one and it is not scepticism about taught methods, most of which are very good. It is to ask, of any construction, which of its steps produce a residual and which produce none — and then to look hard at the second kind, because a step that cannot report an error is a step whose error nobody has ever seen.