The slip that leaves no trace

Of the four ways a hand enters a perspective drawing, three leave something a reader can find and one leaves nothing at all. The one that leaves nothing is the one that changes what the drawing means.

The construction, and its single mark #

The distance-point method has one step a person performs by choosing rather than by following. The orthogonals are drawn from the braccia marks to the centric point, which the panel already fixes. Then one further mark goes on the horizon, at the viewing distance from the centric point, and the diagonal from the near corner to it cuts every orthogonal at a tile corner.

The distance point is the viewing distance establishes what that mark is: the vanishing point of the tiles’ diagonals, which run at forty-five degrees to the picture plane, and a direction at forty-five degrees has its vanishing point exactly the focal length from the principal point. So the mark is not a mnemonic — it is the panel’s own viewing distance, drawn as a length on the page, and the only place in the classical literature where it appears as something a hand touches.

Put the mark somewhere else and the construction runs exactly as before. The diagonal still crosses every orthogonal, the crossings are still in order, and the transversals still march toward the horizon without reaching it.

What moves, and by how much #

A slip of twenty-four pixels in placing the distance point on a panel whose viewing distance is five hundred and twenty moves the worst transversal by about two and a half pixels. That is a slip of under five per cent in one number, and it is plainly visible in the drawing: at the scale these figures are drawn, two and a half pixels is a line width and a half.

Drag that figure and the pavement breathes. The transversals bunch when the point comes in and spread when it goes out, which is exactly what a reader would expect a mistake to look like.

That two and a half pixels is not a measured curiosity either. Writing $\rho = u/f$ for the angular width of a braccio and $r$ for the drop from horizon to ground line, a transversal sits at $r/(1+k\rho)$ below the horizon, and a fractional change $s/f$ in the viewing distance moves it by

$$\frac{r\,k\rho}{(1+k\rho)^{2}}\cdot\frac{s}{f}.$$

That is largest at $k\rho = 1$ — the tile row whose distance along the ground equals the eye’s distance from the panel — where the factor is exactly $1/4$. So the worst displacement is

$$\frac{r}{4}\cdot\frac{s}{f},$$

a quarter of the panel’s own horizon-to-ground drop times the fractional slip: 24 pixels in 520 on a drop of about 220 gives 2.5, which is the figure quoted. And the transversal that moves most is always the same one geometrically, wherever the slip is made — which is a small thing to know and a useful one, since it says where on a suspect drawing to look.

And the reading does not move at all #

Now run the reader’s test on the slipped drawing. Four consecutive transversals, their cross-ratio, against the four thirds that four evenly spaced points must give.

The departure is three parts in ten thousand billion, which is the arithmetic floor of the machine and not a small error. It is not that the test is insensitive. It is that there is nothing to be sensitive to.

The reason is short and it is the whole essay. A drawing made with the distance point at distance d is the exact perspective projection of a square pavement seen with focal length d. Move the mark and d changes; the drawing is then the exact perspective projection of a square pavement seen with a different focal length. Both are correct drawings. The slip did not take the drawing off the set of correct perspectives — it slid it along that set.

Every test a reader can run on a single picture is a test of whether the picture is on that set. So every such test returns the same verdict for both.

What the reader loses #

The drawing is still correct and it is correct from somewhere else, and that somewhere is the number the whole site is about.

A picture’s viewing distance is its focal length scaled to the width it is displayed at. Shown a hundred and sixty millimetres wide, the panel these figures use is correct from about twelve centimetres. Slip the distance point ninety pixels outward and the same panel, drawn identically as far as any reading goes, is correct from fourteen — a sixth further away, for a mark nobody can find.

The same expression prices the loss. The viewing distance changes by $s/f$ — 17 per cent for a ninety-pixel slip on a 520-pixel panel, which is the sixth the paragraph above quotes — while the worst mark on the page moves by a quarter of that fraction of the drop. Measured in each quantity’s own natural fraction, the mark moves by $(1/4)(s/f)$ of the pavement’s depth while the viewing distance moves by $s/f$ of itself — so the picture’s meaning changes four times faster than the picture’s appearance does. A seventeen per cent error in where the reader must stand shows up as a four per cent shift of one transversal, and four per cent of a pavement’s depth is inside the width of a drawn line.

That factor of four is the whole reason the error is invisible rather than merely small, and it is a constant: it does not depend on the panel, the braccia, the drop or the focal length. Every distance-point drawing understates its own mistake by exactly four.

The point to stand at is where this collection computes the number in the first place, and standing in the wrong place is what a reader sees when they are not at it. Neither of those essays could say what this one says: that the number is not merely usually unknown, it is unrecoverable in principle from a picture that does not state it.

The control, which is the point #

An invisible error is only interesting beside a visible one of the same size, and the comparison has to be fair. So both procedures are handed the same misplacement of the same mark on the same page: twenty-four pixels.

Alberti’s section has its hand step on the braccia marks along the section’s ground line. Twenty-four pixels of slip there moves the worst transversal by 2.74 pixels — within five per cent of what the distance point’s slip moved it. The two errors are the same size in the drawing.

And the cross-ratio departs by eight parts in a hundred, which is not a floor, is not a rounding, and is visible to anybody who does the arithmetic.

The two figures are drawn with the same machinery on the same panel, and the first is empty. That is the finding stated as a picture.

Why this is not a defect of the method #

It would be easy to read the above as an argument against the distance point, and it is not one.

The construction is exact. Its one free mark is a genuine free parameter of the situation, not of the recipe: any correct perspective of a square pavement has a viewing distance, the pavement’s proportions do not determine it, and a draughtsman has to choose it somehow. Alberti chose it with a sentence about how far away a man would stand. The distance point chooses it with a mark. A camera chooses it with a focal length.

What is unusual about the distance point is that it makes the choice visible on the drawing surface — the mark is there, on the horizon, at a measurable distance from the centric point — and then the finished panel is cropped and the mark is outside it. The one procedure that draws the free parameter is the one whose drawings least often contain it.

What would have to be true to recover it #

The number is recoverable from a picture that contains something else, and it is worth being precise about what that something else is, because two of the three routes are available on ordinary drawings.

Three vanishing points. Recovering the camera from the picture it drew takes three mutually perpendicular directions and returns the focal length exactly. A pavement supplies one direction and the picture plane’s normal is not one of the others, so a floor alone is not enough — but a floor with a wall standing on it and a box on the floor usually is.

A known shape. If the tiles are known to be square, the diagonal’s vanishing point is determined and it is the distance point, so the number comes straight back. That is why the assumption is worth so much and why the proportion is the assumption treats it as the load-bearing statement it is.

A circle. One conic calibrates the camera recovers the focal length from a single circle whose image is a conic, which is available in more paintings than three vanishing points are.

None of the three is a reading of the pavement’s spacing. Every one of them is a second thing in the picture, and that is the general shape: the free parameter is recoverable from redundancy elsewhere and never from the family it was used to construct.

The two families a panel actually contains #

The argument above is about the transversals, and a panel has a second family which is worth following to the end because it settles what “no trace” means.

The orthogonals — the lines running away from the reader — are drawn to the centric point before the distance point is placed. They do not move when it slips. So the slipped panel has exactly the orthogonals it was always going to have and transversals belonging to a different focal length, and a reader might reasonably expect the two halves to disagree.

They do not, and the reason is that the orthogonals carry no scale. Every correct perspective of a pavement, at every viewing distance, has orthogonals converging on the same centric point at the same angles, because those angles are fixed by where the braccia were marked along the ground line and by nothing else. The orthogonals are a statement about the width of the tiles; the transversals are a statement about their depth; and the viewing distance is the ratio between the two.

Which is why the assumption that the tiles are square is worth so much. Assert it and the two families become one statement, the ratio is pinned, and the viewing distance falls out. Withhold it — and no drawing states it — and the two families are independent, the slip moves one and not the other, and the drawing is a perfectly consistent picture of a floor whose tiles are longer than they are wide.

What the drawing would need to carry #

It is worth asking what a panel would have to contain for the slip to leave a trace, because the answer is short and it is not a matter of degree.

It would have to contain something whose image depends on the viewing distance while the pavement’s does not. A vertical of known height standing on the floor does not: carrying a height across the room transfers it with the horizon alone and never asks for a focal length. A second pavement at a different scale does not, for the same reason the first does not. A row of columns does not.

What does is something that fixes a third direction. Two vanishing points on the horizon and one off it give a triangle whose orthocentre is the principal point and whose sides give the focal length, and that is the whole of recovering the camera from the picture it drew. A pavement supplies one direction along the floor and one across it, both on the horizon; the third has to come from something standing up, and it has to be a box or a wall rather than a post, because a post supplies a direction the pavement’s two already determine.

So the practical statement is unusually clean. A panel showing a floor and nothing that stands on it at an angle to the floor’s own grid does not determine its viewing distance, whatever else it shows. A great many do show such a thing; the point is that it is the thing doing the work, and not the pavement everybody looks at.

The slider, and what it is for #

The figure at the top of this essay has a slider on the slip, and it is worth saying what dragging it is meant to demonstrate, because it is the opposite of the usual purpose.

Most sliders here show a quantity responding. This one shows a quantity not responding: the drawing changes visibly, frame by frame, and the reading printed underneath does not. The pavement bunches and spreads across four line widths and the cross-ratio holds at the machine’s floor throughout.

That is a hard thing to show and an easy thing to assert, which is why it is a figure rather than a sentence. A reader who has watched a drawing move through a dozen configurations while its only checkable property stands still has a better grip on what “invariant” means than any amount of algebra supplies — and this collection’s habit is that a claim which can be turned into a crank should be.

The same shape in the other fields #

This is not the only place in the collection where an error maps the correct answers onto themselves, and the others are worth naming because the family is more common than it looks.

A stereo pair’s scale. The one thing a single view cannot give is size, and a pair recovers everything except the baseline’s length — so scaling the whole scene and the whole rig together leaves every picture identical. The error that changes the answer is the one no picture can see.

A parallel drawing’s depth. The drawing does not say which corner is nearer is a reversal that leaves every drawn line where it was, so it is not a small error either — it is a different solid with the same drawing.

And the absorbed second eye. A design that lands in two rooms shows a picture drawn from two centres that is exactly a picture drawn from one, of a sheared room. Same structure again: the wrong thing is invisible because the wrong thing produces a right thing.

What all four have in common is that the invisible error is a transformation the reading is invariant under, rather than a small quantity. That is worth carrying as a habit, because it says where to look: not for a more sensitive test, but for a second measurement that is not invariant under the same thing.

What a reader should conclude #

Three statements, in decreasing order of how sure they can be.

A panel’s viewing distance cannot be read off its pavement. Certain, and it follows from one line of algebra rather than from a measurement.

A claim that a panel was drawn at a particular viewing distance is a claim about something outside the pavement. A wall, a circle, a stated proportion, a document. If the claim names none of those, it has no support.

And the classical constructions are not distinguishable by their errors here. The distance point’s characteristic error is not a characteristic error, because it is not an error in anything a picture records — so a reading that reports “no departure” has ruled out three procedures and one of them not at all.

The short version #

A misplaced distance point does not make a worse perspective. It makes an exact perspective of a different room, seen from a different distance, and no reading of the finished picture will ever find it — while the same slip on the other classical construction, moving the drawing by the same amount, is caught immediately.

So the error the whole viewing field is about is precisely the error a picture is incapable of recording, and the viewing distance a panel implies is a fact about the panel’s other contents rather than about its floor.