Alberti draws a pavement, and chooses where the reader stands

In 1435 Leon Battista Alberti wrote down how to draw a tiled floor receding into a picture. The method is short enough to fit on a page and it is the first correct construction of a perspective in the European record. It has been reproduced in every book on the subject since, usually as a historical illustration, and almost never as something to be checked.

It survives being checked. Run as a construction — with rays drawn from an eye to marks on a ground line, and intersections taken where those rays cross a picture plane — it produces transversals that agree with a pinhole camera of the same focal length to 6 × 10⁻¹⁴ pixels. There is no approximation in it anywhere.

The construction, as a construction #

The panel gets a horizontal line called the ground line along its bottom edge, divided into equal braccia — the arm’s-length unit the whole scheme is built in. A point is chosen on the picture at the height of the eye and called the centric point. Lines are run from each braccio mark to the centric point, and those are the orthogonals: the images of the floor’s joints running away from the viewer.

That much is easy and it is not the difficult half. What remains is where the transversals go — the joints running across, which get closer together as they recede — and getting those right is the whole problem. Placing them by eye is what the century before Alberti did, and it is wrong in a way that has its own essay.

Alberti’s answer is to draw a second, smaller diagram beside the panel: a lateral section. In it, the ground is a horizontal line, the panel is a vertical line at the distance the eye is from it, and the eye is a point at its own height above the ground. The braccia are marked off along the ground, receding away from the panel. A ray is drawn from the eye to each mark, and where the ray crosses the panel line is the height of that transversal. The heights are carried across to the picture, and the pavement is complete.

The reconstruction here draws exactly that. The crossings come out of intersect(lineThrough(eye, mark), panelLine) — the same two functions this site uses on every drawn picture — rather than out of the closed form the construction turns out to satisfy. That distinction is the point of running it at all. Evaluating the formula and comparing it with a camera would be checking arithmetic; drawing the rays and taking the intersections is checking that the construction is the projection.

Three routes, one answer #

The closed form does exist, and having it is what lets a third route into the comparison. Working in picture units where the eye is f from the panel, the k-th transversal sits at

$$y_k = h + \frac{r}{1 + k\,u/f},$$

where h is the horizon’s height in the picture, r is the drop from horizon to ground line, and u is the width of a braccio along the ground line.

Two things about that formula are worth noticing before it is used.

The eye’s height above the ground has vanished. So has the panel’s distance in world units. Only their ratios to picture quantities survive — r is the image of the eye height, and f is the image of the viewing distance. That is the scale ambiguity of single views, arriving from the construction side: the panel knows the angles and none of the sizes.

And there is exactly one free parameter left, which is f.

The three routes — the lateral section, the distance-point construction, and a pinhole camera built with focal length f and the panel’s centric point as its principal point — are all run and required to agree. The worst disagreement over all three, across all seven transversals, is 5.7 × 10⁻¹⁴ px.

That is a stronger statement than it looks. The three were separated by five hundred years and by the whole idea of a focal length. Alberti had no notion of one; the distance-point method is a workshop shortcut with no theory attached; the camera is a nineteenth-century object with a twentieth-century formalisation. They produce the same numbers because they are the same operation, and the agreement is what says so.

The two things the transversals must not do #

Running a construction rather than a formula makes two failures available that the formula hides, and both are asserted rather than assumed.

The first is that the transversals must march upward and never turn round. Each successive one is closer to the horizon than the last, because each is further away. That sounds automatic and it is not: an implementation that took the section’s rays from the wrong side, or that marked the braccia toward the panel rather than away from it, produces a sequence that reverses somewhere in the middle. The result still looks like a floor — a floor with one strange band in it — and nothing in the drawing says which band is wrong.

The second is that no transversal ever reaches the horizon. The k-th one sits r/(1 + k·u/f) below it, which is positive for every finite k. The floor approaches the horizon and never touches it, and the last drawn transversal in the figure sits a measurable distance below it — a distance the construction reports rather than a gap the illustrator left.

That second one is the projective content of the construction, stated as a check. The horizon is the image of the ground plane’s line at infinity, so a transversal landing on it would be a floor joint infinitely far away, which is not a joint. A pavement drawn by eye can and does put its last band on the horizon, because the bands are getting close together and the illustrator runs out of room. A pavement drawn by construction cannot, and the assertion is what makes the difference visible.

Both checks cost a line each. Neither would fire on a correct implementation, which is the usual and unhelpful state of an assertion — so both are run against deliberately broken input in the site’s gate, along with a pavement whose ground line is above its horizon, which is refused outright.

The parameter nobody names #

Now the uncomfortable part, and the reason this essay is on this site rather than in a history.

The construction is correct at every value of f. Put the eye close to the panel and the pavement recedes steeply; put it far away and the pavement flattens out. Both are perfect perspectives. Both are pictures of a floor of equal square braccia. They are pictures of that floor from different places, and nothing in the drawing distinguishes them.

So the construction has a free parameter, and the parameter is the distance from the eye to the panel — which, scaled to the width the picture is actually displayed at, is the distance the reader must stand at for the picture to be a correct projection of anything.

What do the sources say about it?

Alberti says the eye should be no nearer than the height of the man depicted in the picture. That is a sentence about proportion, offered as a matter of decorum, and it fixes nothing: it is a lower bound, in units of a figure that has not been drawn yet.

The distance-point method fixes it by where the distance point is put on the horizon, and the workshop tradition puts it “off to one side, comfortably” — often outside the panel, sometimes marked on the studio wall.

Neither names what is being chosen. A painter following either recipe selects the viewing distance of the finished picture, in the same act as selecting the composition, and no step of either method says so.

The panel below makes the choice explicit. At 260 picture-pixels of viewing distance the pavement spans 71° and the figure is correct from 6 cm when it is shown 160 mm wide; at 1040 it spans 20° and is correct from 24 cm. Same floor, same braccia, same ground line, four times the distance.

What the picture would have to say #

There is a version of this complaint that is unfair and it is worth setting aside first.

Nobody in 1435 could have computed a viewing distance in centimetres, because the arithmetic requires knowing the physical width the picture is displayed at and relating it to an angle — which is easy once a focal length is a familiar object and is not obvious before. Alberti had the geometry and not the vocabulary.

But the geometry is enough. The distance from the eye to the panel is a length in the section drawing, in the same braccia as everything else, and the section is right there beside the panel. Alberti could have written “the eye stands three braccia from the panel” and a viewer of the finished picture could have stood three braccia from it. The information exists in the method; it is simply not carried forward into the object.

That is the specific loss and it is not a small one. A fresco on a wall is correct from one point, and it is often possible to work out where — the vanishing points are recoverable from the painted architecture, and the recovery is exactly what this site’s foundation phase does. Doing it to real paintings is a small and rewarding literature, and it repeatedly finds viewpoints that are peculiar: a metre above the floor, or well outside the room, or in a doorway.

The construction chose those points. Nobody recorded the choice, so it has to be recovered from the picture five hundred years later.

What the braccio is doing #

There is a unit in this construction and it is not a decoration. Alberti divides the ground line into braccia — arm-lengths, the ordinary building measure of fifteenth-century Florence — and the choice does two things at once.

The first is that it makes the floor a measuring instrument inside the picture. A figure standing on the third transversal is three braccia into the room; a doorway spanning two orthogonals is two braccia wide. The pavement is a ruler laid into the scene, and every subsequent object in the painting can be sized against it. That is why so many quattrocento interiors have a tiled floor whether or not the room would have had one: the floor is the scale, and a room without one has no way to say how big anything in it is.

The second is subtler and it is the same point this site’s metrology field keeps arriving at. The braccio is a length supplied from outside the picture. The construction determines every ratio in the pavement — that this band is that fraction of that one, that the third transversal is here — and it determines no size at all. Assigning the ground-line division a braccio is exactly the act of supplying the one reference length that turns a set of ratios into a set of measurements.

So a fifteenth-century painter constructing a floor and a modern investigator measuring a height from a security still are doing the same two-part operation: take the projective structure from the picture, and import one length from the world. The painter imports it going in and the investigator imports it coming out, and neither can do without it.

There is a third thing the unit does that is worth flagging as a hazard rather than a benefit. Because the braccio is a real unit, it is easy to read the construction as though the picture knew about real sizes — as though drawing a six-braccio floor made the panel a picture of a six-braccio room. It does not. It makes the panel a picture of a room six units across, from a viewpoint measured in the same units, and the identification of that unit with an arm-length is a label. The same drawing is a picture of a cathedral floor seen from far away, exactly.

What “correct” meant in 1435 #

It is easy to read the agreement above as the fifteenth century anticipating a modern result, and that is not quite the right description of what happened.

Alberti was not approximating a projection. He had the projection: the section drawing is a pinhole camera, drawn in cross-section, with the eye as the pinhole and the panel as the film. What he did not have was a way to say so, because the vocabulary — focal length, principal point, projection through a centre — took another four hundred years to assemble, and because there was no independent object to compare the construction against.

That second absence is the important one. The construction produces a pavement because the construction was designed to. Nothing in the method asks whether the drawn pavement depicts squares, and nothing could, because the only available check was whether it looked right and looking right is what the previous century’s constant-ratio pavements also did.

So “correct” in 1435 meant derived from a defensible geometrical argument, and it was the strongest thing available. “Correct” here means agrees with an independent computation to fourteen digits, and the independence is what the fifteenth century could not have. Both are real standards and the second is not merely a sharper version of the first — it is a different kind of claim, available only once there are two ways to get the same number.

Why the section is the honest half #

Of the two diagrams the method uses, the section is the one that gets dropped in modern retellings, and dropping it is what makes the method look like a recipe.

In the panel, everything is a construction line: the centric point is placed, the ground line is divided, the orthogonals are drawn. Nothing in the panel is a length in the world.

In the section, the eye is at a height and a distance, both of them real, both measurable with a ruler on the drawing. The section is where the picture is tied to a physical arrangement of a viewer and a surface, and it is the only place in the classical apparatus where that connection is drawn rather than assumed.

This site’s whole premise is that the connection is the interesting part — that a picture is a projection through a centre, and the centre is a fact about the picture that can be computed and stated. Alberti drew that centre. He drew it in the diagram that gets cropped out of the reproduction.

What survives, and what does not #

The summary is unusually clean for a historical method.

Exact: the construction, in every particular. The transversals are the projected transversals, to fourteen digits, by three independent routes. There is no sense in which the fifteenth century had an approximation that later work refined.

Free, and unnamed: the viewing distance. The method requires a value, produces a correct picture for any value, and provides no way for the finished picture to report which value was used.

Absent: any check. The construction produces a pavement of squares because it was built to. Nothing in the method asks what the drawing depicts, which is why the century of constant-ratio pavements before it could not tell that they were wrong, and why the two-point cube that every modern tutorial teaches is still not measured. Constructing correctly and being able to tell whether one has are different capabilities, and the second one is what this site is about.