Visual Plane
every picture states the point it is correct from
Constructing a view

Brunelleschi drilled a hole in his panel

The first perspective demonstration in the European record came with its viewing point enforced — a hole through the back of the panel, a mirror held out in front, and one place to stand. That distance is computable from the panel's size and the angle the Baptistery subtends, and the answer lands squarely on the arrangement the account describes.
15 min read 6 figures The viewer is in the geometryTaught, and never measuredProjected, not constructed

Some time around 1425, and probably a decade before Alberti wrote anything down, Filippo Brunelleschi painted a small panel of the Florence Baptistery as seen from just inside the central door of the cathedral opposite. Then he did something nobody had done before and almost nobody has done since: he drilled a hole through it.

The account comes from Antonio Manetti, writing decades later. The viewer held the panel with the painted side away, put an eye to the hole at the back, and held a mirror out in front with the other hand. The painting appeared in the mirror. Lower the mirror and the real Baptistery appeared in its place. Raise it and the painting returned.

Every part of that arrangement has one purpose. It puts the eye at one point and keeps it there.

Brunelleschi's panel, and where the eye had to beThe Baptistery is about 25.6 m across and the door it was painted from about 53 m away, so it subtends 27.2°. On a 290 mm panel that it fills 100% of, the picture is correct from 60 cm — which is a mirror at arm's length, or half of one, depending on the single number nobody knows.the piazza, in planthe Baptistery, 25.6 m53 m27.2°the cathedral doorthe panel, and the eye it needsthe Baptistery fills 100%the eye is 60 cm back — off this sheet290 mm panel · 27.2° of Baptisterycorrect from 60 cm · 27° across
Fig. 1 Left: the piazza in plan, to scale — the Baptistery about 25.6 m across, the cathedral door about 53 m from it, and the angle that subtends. Right: the panel, and the eye position its geometry demands. The slider is the one quantity nobody knows.

The arithmetic

Two numbers determine the answer and one of them is unknown.

The angle the Baptistery subtends from the spot Manetti describes. The building is an octagon roughly 25.6 m across; the standing place is roughly 53 m from it, a few paces inside the cathedral door. That is 2·arctan(12.8/53) = 27.1°.

The size of the panel. Manetti gives half a braccio square, which in the Florentine braccio of about 58 cm is 29 cm.

The fraction of the panel the Baptistery filled. Nobody knows. Manetti says the panel showed the Baptistery and as much of the piazza either side as the eye takes in, which is a description and not a measurement.

Given the three, the correct viewing distance is the focal length: half the Baptistery’s width on the panel, divided by the tangent of half the angle it subtends. If it filled the whole width, that is 14.5 cm / tan(13.55°) = 60 cm. If it filled half the width, 30 cm.

What those two numbers are

They are, respectively, arm’s length and half of it.

Manetti’s arrangement has the panel at arm’s length in one hand and the mirror at half arm’s length in the other, with the eye at the hole. The light path runs from the panel to the mirror and back to the eye, so the optical distance from the eye to the painted surface is roughly twice the mirror distance — which for a mirror at 30 cm is 60 cm.

So the two ends of the computed range are the two distances the described procedure actually produces. If the Baptistery filled the panel, the correct distance is 60 cm and the mirror at half arm’s length delivers exactly that. If it filled half the panel, the correct distance is 30 cm and the mirror at arm’s length delivers that.

The result should not be over-sold. It does not prove Manetti’s account, and it does not settle the panel’s dimensions — the calculation has three inputs and two of them are estimates from a text written from memory decades after the event. What it does is check the arrangement for consistency, and the arrangement is consistent: a panel of that size, of that scene, from that spot, has to be looked at from somewhere between a quarter and two-thirds of a metre away, and the mirror procedure puts the eye there.

An arrangement that had come out at four metres, or at four centimetres, would have said something. This one says the geometry and the procedure fit.

Where the reader has to be for a 40° picture to be correctShown 160 mm wide, this picture is a correct projection only from 22 cm away. Drawn to scale.the picture, 160 mm wide15 cm57°the eyefocal length 635 px15 cm at 160 mm wide
Fig. 2 The relationship the calculation runs on, drawn generally. A picture of a stated width made at a stated angular field is correct from one distance, and the distance falls fast as the field widens.

Why the hole is the invention

The panel is lost, so no claim about its execution can be checked. The arrangement is the part that survives, and the arrangement is the interesting object.

Every picture ever made is correct from one point. That is this site’s premise and it follows from a picture being a projection through a centre. What is unusual about a picture is not having such a point — they all do — but being shown from it, and there is no mechanism in the ordinary business of hanging a painting on a wall that puts a viewer anywhere in particular.

Brunelleschi built the mechanism. A hole is an aperture the eye has to be at; a mirror at a fixed reach fixes the path length; a panel held in the other hand fixes the rest. The whole apparatus is a device for removing the viewer’s freedom, and it is the only such device in the history of the medium that was built deliberately for a picture rather than for an instrument.

It also does something a modern reader may miss. The demonstration’s force came from comparison: lower the mirror and the real building is in the same place, at the same size, with the same edges. That comparison is only possible if the eye is at the correct point, because from anywhere else the painted building and the real one differ in size and in the convergence of every line. The hole is not a curiosity attached to the demonstration; it is what makes the demonstration a demonstration.

How far back the picture is correct from, against how wide it isA 24° picture is correct from 38 cm and a 90° picture from 8 cm — closer than most people can focus.020406020406080horizontal field of view (degrees)correct viewing distance for a 160 mm wide picture (cm)long lens — 38 cma normal photograph — 22 cmwide — 13 cmvery wide — 8 cmsame picture width throughoutthe only variable is the angle
Fig. 3 The curve the Baptistery arithmetic sits on. A 27° picture is correct from a comfortable distance; a very wide one from closer than an eye can focus. Brunelleschi’s panel is at the easy end, which is part of why the demonstration could work at all.

The mirror is doing two things

Manetti’s arrangement is usually described as though the mirror were a convenience — a way to see the front of a panel whose back the eye is pressed against. It is that, and it is also doing something the account does not comment on, which is worth separating out because it is the reason the demonstration works at all.

The first job is the one everyone notices: with the eye at the hole on the painted side’s reverse, the painting is facing away, and a mirror is the only way to see it.

The second job is that the mirror doubles the path length. Light leaves the painted surface, travels to the mirror, and comes back. The eye is at the far end of a path twice the mirror’s distance, so a mirror held at 30 cm puts the painting optically at 60 cm — and that is what makes a comfortable arm’s-length arrangement deliver the distance a 29 cm panel of a 27° scene actually needs. Holding the panel at 60 cm and looking at it directly would need an arm nobody has.

The third thing, which is a consequence rather than a job, is that the mirror reverses handedness. The image in the mirror is left-right flipped relative to the panel, so for the demonstration to work — for the painted Baptistery to coincide with the real one when the mirror drops — the panel itself must have been painted mirror-reversed. That is a real constraint on the lost object and it is the kind of detail that gets left out of a summary.

This site has already measured the handedness reversal, in the essay on mirrors as second cameras: a reflection computed by reflecting the scene and one computed by reflecting the camera disagree by 906 pixels and agree to 6 × 10⁻¹⁴ once one image axis is reversed. Which axis appears reversed depends only on how the up vector is carried, which is the geometrical answer to why a mirror swaps left and right and not up and down. The same fact, applied here, says Brunelleschi painted his panel backwards on purpose.

A box over a reflecting floor, with the reflection computed twiceReflecting the scene and reflecting the camera disagree by 315 px and agree to 0e+0 px once one image axis is reversed — which is what a mirror reversing handedness looks like in numbers.grey: the reflectiontwo routes, agreeing to 0e+0 px after one flip
Fig. 4 The mirror, as geometry. A reflection is the view from a camera on the far side of the glass, and the handedness reversal that follows is why the panel must have been painted backwards for the demonstration to land.

What the arrangement measures

Seen in modern terms, the panel and hole together constitute an instrument for testing whether a picture is a correct projection — and it is a better instrument than anything the following four centuries produced.

An ordinary perspective drawing can be checked by construction: the orthogonals should converge, the transversals should follow the rule, the vanishing points should be where they belong. All of that checks the drawing against the method. If the method is wrong, or if it has a free parameter that was set by eye, the check passes anyway. This site’s wrong field is about constructions that survive that kind of checking and do not survive being asked what they depict.

The mirror test checks the drawing against the world. It is a null test: the painted edges either coincide with the real ones or they do not, at every point in the frame simultaneously, and there is nowhere for an error to hide. A picture that passed it was demonstrably a projection of the thing, from the place the hole put the eye.

That is a residual, in the sense this site uses the word — a quantity that can be non-zero and is reported. Brunelleschi’s version of it was visual and unquantified, and it was still the only measurement of a picture’s correctness anybody made until photography.

How wrong the inputs are allowed to be

An arithmetic with three estimated inputs deserves a sensitivity analysis rather than a shrug, and this one is short because every dependence is simple.

The panel width enters linearly. A braccio of 55 cm instead of 58 moves the answer by 5%. Nothing about the conclusion turns on it.

The standing distance enters through the subtended angle, and for a small angle it is close to linear too: 53 m instead of 50 m changes the angle by 6% and the distance by the same. The account says a few paces inside the door, which is not precise, and it does not need to be.

The Baptistery’s width likewise, and it is the best-known of the three: the building still stands.

The fraction of the panel it filled enters linearly and ranges over a factor of two or three. It is the only input whose uncertainty is large enough to matter, and it is why the answer is quoted as a range instead of a number.

Put together, the first three contribute perhaps 10% between them and the fourth contributes a factor of two. So the calculation has one real unknown and three quibbles, which is a much better position than “three estimates from a text” sounds like — and it means the conclusion is not sensitive to the details of Manetti’s arithmetic at all, only to the one thing his description does not pin down.

The other unknown, and why it is not fatal

The fraction of the panel the Baptistery occupied is the calculation’s soft input, and the slider on the figure exists so that a reader can watch what it does rather than take a number on trust.

The relationship is exactly proportional. Halve the fraction and the correct distance halves; the whole answer is linear in it. That is a comfortable kind of dependence — no threshold, no blow-up, no region where a small error in the estimate produces a large error in the answer.

It also means the calculation cannot be used to recover the fraction from anything, which is worth saying plainly. If some independent evidence fixed the viewing distance, the fraction would follow; there is none. The panel is gone, the account is second-hand, and the geometry connects three quantities of which one is known well, one is known roughly and one is not known at all.

What survives is a range, an arrangement that produces distances inside it, and a proportionality that a reader can check by moving a slider. That is the honest form of this kind of historical arithmetic and it is not much less than the strongest version — the striking part was never a particular number, it was that a fifteenth-century apparatus and a modern focal-length calculation land in the same place at all.

Alberti's construction, with the section that fixes the depthsLeft: the panel, six braccia across, its transversals found where the section's rays cross the picture plane. Right: the section, with the eye at its true distance. The transversals agree with a pinhole camera of the same focal length to 6e-14 px.the panelhorizon — the centric point's heightthe section — the eye, the panel, the ground530 px — the viewing distancethree routes agree to 1e-13 pxsection, distance point, and a pinhole camera
Fig. 5 The next decade’s version of the same quantity. Alberti’s lateral section puts the eye at a stated distance from the panel and draws it. Brunelleschi’s hole enforces it. Between them the two make the viewing distance visible and then compulsory, and everything since has done neither.

The silvered sky

One detail of the account is not geometry and is worth recording because it is the same instinct as the hole.

Manetti says the sky in the panel was not painted. Where the sky would have been, Brunelleschi laid burnished silver, so that the real sky and the real clouds were reflected in it and moved as the wind moved them.

Read as decoration that is a charming touch. Read as an instrument it is something else: the one part of the scene a projection cannot get right is the one part he refused to project. A painted sky is a fixed pattern that will not coincide with the real sky when the mirror drops, and it would have been the only place in the frame where the comparison visibly failed. Replacing it with a mirror makes the comparison succeed everywhere, by removing the thing that could not match.

That is either a very careful piece of instrument design or a very lucky aesthetic choice, and there is no way to tell which. What can be said is that it is consistent with everything else about the arrangement. The hole removes the viewer’s freedom of position; the mirror sets the path length; the silver removes the one region where the null test could not be null. Every element does the same kind of work.

It is also the first appearance on this site of a decision that recurs constantly in its own machinery: when a construction cannot support a claim about some part of a figure, the honest move is to refuse to draw that part rather than to draw something plausible. A generator here that cannot place a label inside the frame drops the label; a camera given a point behind the eye returns nothing. A camera given a point behind the eye returns nothing. Brunelleschi did not paint the sky.

A box drawn from a known camera, and the camera recovered from the drawingThree vanishing points found from the twelve drawn edges alone give back the focal length to 4e-15 relative.recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across
Fig. 6 What can still be done to a picture whose viewpoint nobody recorded. Three bundles of drawn edges give three vanishing points, the vanishing points give the focal length, and the focal length gives the distance the picture is correct from.

What was lost afterwards

Nothing like the hole was ever built again for a picture, and it is worth asking why, because the answer is not that it was a bad idea.

Pictures got larger. A 29 cm panel viewed from 60 cm is a picture that one person looks at, held in two hands; an altarpiece is a picture that a congregation looks at, from wherever they happen to be. As soon as a picture is architectural, enforcing a viewpoint means enforcing where people stand, and nobody was going to do that.

Pictures also got wider. The Baptistery panel at 27° is a narrow, long-lens view whose correct distance is comfortable. The later tradition of grand interiors and sweeping piazzas works at fields of view where the correct distance is well inside the panel’s own width, and a picture that has to be viewed from half its width away cannot be viewed properly by anybody in a room with it.

And the loss was not entirely felt. A viewer at the wrong distance sees a picture that is a correct projection of a different scene — a room stretched or compressed in depth — and does not usually notice, because the visual system is not measuring and there is no comparison available. That is a fact about seeing rather than about projection, and it is the reason five centuries of pictures were made without anybody minding.

But the number never went away. It is computable from any picture whose field of view can be recovered, it is stated on every figure on this site, and it was drilled through a panel in Florence before anybody had a word for it.

Where this goes next

the picture, 160 mm wide22 cm40°the eyefocal length 948 px22 cm at 160 mm wide The point you have to stand at A perspective picture is a projection through a centre, and scaling that centre's distance to the width the picture is actually shown at gives a distance in centimetres. Shown 160 mm wide, a 40° picture is correct from 22 cm and a 90° one from 8 cm. Nobody stands there, and that single fact explains most of what gets called distortion. the panelhorizon — the centric point's heightthe section — the eye, the panel, the ground470 px — the viewing distancethree routes agree to 6e-14 pxsection, distance point, and a pinhole camera Alberti draws a pavement, and chooses where the reader stands The costruzione legittima of 1435 is exact. Run as a construction — with drawn rays and drawn intersections rather than the formula it turns out to satisfy — it agrees with a pinhole camera to six parts in a hundred trillion of a pixel. And it has one free parameter that the recipe never names, which is the distance from the eye to the panel. centric pointdistance point, 12 px off the sheet →430 pxcorrect from 10 cm at 160 mm wide47° across The distance point is the viewing distance, drawn There is exactly one place in the whole classical apparatus where the distance from the eye to the picture appears as a length on the page, and it is the offset from the centric point to the distance point. Everything this site exists to compute was drawn in the fifteenth century, on the horizon, and nobody said what it was.