The point you have to stand at

Every perspective picture was made by projecting a scene through a single point onto a surface. That point is not a fiction and it is not lost: it is recoverable from the picture, and once the picture is given a physical size, it becomes a place in the room where the picture hangs.

The arithmetic is one line. The focal length is a distance in the units of the picture; scale the picture to the width it is actually shown at and the focal length scales with it; the scaled focal length is the distance from the eye to the picture.

$$d = f \cdot \frac{W_{\text{display}}}{W_{\text{image}}}$$

Nothing about the scene enters. The correct viewpoint is a property of the picture and its printed size, not of what it depicts. A photograph of a mountain range and a photograph of a matchbox, taken with the same lens and printed the same size, are correct from the same distance.

The numbers #

For a picture shown 160 mm wide, which is roughly what a figure occupies on a laptop:

The last row is the one that matters. A 90° picture, shown at a comfortable size, is a correct projection only from eight centimetres — closer than most adults can focus. There is no viewing position from which such a picture is geometrically right and also legible.

That is not a defect in the picture. It is what a 90° projection is. The picture contains a 90° cone of the world compressed onto a flat surface 160 mm across, and the only way to see that cone at its true angular size is to put the eye where the cone’s apex is.

What happens at the wrong distance #

Almost every picture is viewed from too far away, and the consequence is systematic rather than random.

Viewing a picture from further than the correct distance means the picture subtends a smaller angle at the eye than the scene did at the camera. Every drawn angle is therefore reduced, and the reduction is not uniform — it compresses the periphery more than the centre. The perceived effect is that depth is exaggerated: objects seem more strongly separated in depth than they were, near things loom, and receding lines seem to plunge.

Viewing from closer than the correct distance does the opposite, and flattens.

The size of the effect is the ratio of the actual viewing distance to the correct one. A 40° picture correct from 22 cm, viewed at the ordinary reading distance of 40 cm, is being seen at 1.8× the right distance, which is a substantial exaggeration and is entirely invisible as such.

The reason it usually does not matter #

It is worth being clear that this is a geometric statement and not a claim about experience.

Pictorial perception is very tolerant of the wrong viewing distance. The visual system has strong assumptions about the shapes of familiar objects, and it applies them: a face known to be roughly spherical is perceived as roughly spherical almost regardless of what the projection did to it. This is sometimes called compensation and it is remarkably robust.

There are also competing cues saying the picture is flat — binocular disparity, accommodation, the visible frame and surface — and those cues are correct. A viewer who registers the picture as a flat marked surface is not being fooled and does not need to be at the correct point for anything.

So the honest statement is narrow: the geometry is correct from one point, and perception mostly does not care. Where perception does start to care is when the mismatch gets large, which is what makes wide-angle photographs of faces controversial and is entirely explained by the same arithmetic.

Where it does matter #

Four cases where the correct viewing distance is a working constraint rather than a curiosity.

Trompe-l’œil and quadratura. A ceiling painted to look like an opening in the roof only works from one point, and the point is marked on the floor of every serious example. Pozzo’s ceiling in Sant’Ignazio has a disc set into the pavement showing where to stand, and the illusion collapses conspicuously when the viewer moves off it — which is not a failure of the painting but a demonstration that the painting is doing exactly what this essay describes.

Anamorphosis. The extreme case, where the correct viewpoint is chosen so far from the natural one that the picture is illegible from anywhere else. The construction is the same projection with an unusual centre, and it is the clearest demonstration available that a picture is correct from a point.

Cinema. A film shot on a 50 mm lens and projected onto a large screen is correct from a specific row, and cinemas are designed around that. The strong feeling of being in a scene shot on a wide lens and viewed from close to the screen is the geometry working; the flatness of the same shot on a phone is the geometry failing.

Architectural rendering. A visualisation of a building that is correct from six centimetres reads as exaggerated, and this is the single commonest fault in the genre. The fix is a longer lens, which moves the correct point back to where people actually stand.

Why the arithmetic works #

The derivation is short enough to give, because it explains why the scene drops out.

A pinhole camera maps a world direction making angle θ with the optical axis to a picture point at distance f·tan θ from the principal point. Rearranged: a picture point at distance r from the principal point represents a direction at angle arctan(r/f).

Now put an eye at distance d from the printed picture. The point at distance R from the principal point subtends angle arctan(R/d) at the eye.

For the eye to see each point in the direction the camera recorded it, those two angles must match for every point. Since R = r·(W_display/W_image), matching requires d = f·(W_display/W_image) exactly, and no other value works for more than one point at a time.

The scene never appears in the derivation because the condition is about directions, not about what is in them.

What this site does about it #

Every figure here states the distance it is correct from, in its caption strip, alongside the display width the calculation assumed.

The assumption has to be stated because it cannot be measured from the server. 160 mm is roughly a figure’s width on a laptop; on a phone it might be 70 mm, which halves every distance quoted; on a projector it might be 2 m, which multiplies them by twelve. The arithmetic is given every time so it can be redone, and burying the assumption in a constant would make the claim look more certain than it is.

The figures are also drawn at fields of view that keep the correct distance plausible — mostly 40 to 46°, correct from 20 to 25 cm, which is closer than a reader will be but not absurdly so. That is a deliberate constraint on the site’s own figures, and the wide-angle figures that break it break it on purpose and say so.

The thing worth carrying away #

Perspective is usually taught as a way of making a convincing picture. It is more precisely a way of making a picture that is correct from a point, and the point is as much part of the specification as the horizon or the vanishing points.

Every complaint about perspective drawing that sounds aesthetic — this looks exaggerated, this feels flat, this seems to loom — turns out to be about the relationship between the drawing’s correct point and the point the viewer actually occupies. That relationship is a single number, it is computable from the drawing, and once it is computed the complaint stops being a matter of taste and becomes a design decision that somebody made, usually without knowing they were making it.

Placing two vanishing points comfortably on a sheet of paper is choosing a viewing distance of a few centimetres. Nobody writes that down, and it is what the drawing then says.

The station point, which is the same thing in older language #

Classical treatments have this quantity and call it the station point: the position of the observer’s eye, brought into the drawing as a point in the plan.

The classical construction places it explicitly. Alberti’s method draws a separate side elevation with the eye at its true distance from the picture plane and reads the depths off that; the distance point method puts two points on the horizon at exactly the eye’s distance either side of the centre, and uses them to lay out depths. In both, the viewing distance is a length on the paper, drawn, and impossible to leave out.

What changed is not the geometry but the workflow. Once vanishing points could simply be placed on the sheet where the composition wanted them, the station point stopped being drawn, and the last thing in the construction that mentioned the viewer disappeared. That is the step this site’s figures put back.

There is a nice consistency check available between the two vocabularies. The distance from a vanishing point to its measuring point equals the distance from that vanishing point to the station point, which equals the focal length. So a drawing whose measuring points are marked reveals its own focal length with a ruler, and the recovery from three vanishing points must give the same answer. Two routes, agreeing, on a question that most treatments do not pose at all.

Two eyes, and why this is not undermined by having them #

An objection worth answering, because it is the first one anyone raises: people have two eyes, and a picture drawn for one centre of projection is being looked at by two.

It does not undermine anything, for a reason that cuts the other way from what the objection expects. Binocular vision gives depth information about the picture surface, which is flat, and that information is correct. Both eyes see the same flat marked sheet at the same distance, and they report a flat sheet.

So the binocular cues do not fight the perspective — they contradict it, correctly, and the visual system resolves the contradiction by treating the picture as a picture. That is why pictures do not usually feel like windows, and why the trompe-l’œil cases that do feel like windows go to such lengths to suppress the competing cues: a single viewing aperture, no visible frame, controlled lighting, and a marked spot on the floor.

The practical upshot is that the correct viewing distance matters most exactly when the other cues have been suppressed, and matters least when the picture is obviously a picture. It also means the geometry’s authority stops at the picture’s edge: the geometry says where the eye must be for the projection to be exact, and says nothing about whether a viewer will experience anything in particular there.

A quick way to find it for any picture #

For anyone wanting the number for a photograph in hand, there is a shortcut that needs no metadata.

If the picture contains anything rectangular, its vanishing points give the focal length in pixels. Divide by the image width in pixels, multiply by the width the picture is being displayed at, and that is the distance.

If the focal length is known in the photographic sense — 50 mm, 24 mm — the conversion is easier still: the correct viewing distance is the focal length multiplied by the enlargement from the sensor to the print. A 50 mm lens on a 36 mm-wide sensor, printed 360 mm wide, is a tenfold enlargement, so the print is correct from 500 mm. That is close to the ordinary distance for viewing a print of that size, which is not a coincidence: the “normal” lens is defined as the one whose pictures are correct from a comfortable viewing distance, and everything wider or longer departs from it in one direction or the other.

Renaissance practice, which took it literally #

The early treatments of perspective are much more explicit about the viewing point than modern ones, and the reason is that they were arguing for the construction rather than teaching it.

Brunelleschi’s demonstration of about 1425 is the extreme case. He painted the Florence Baptistery on a small panel, drilled a hole through the panel at the vanishing point, and had the viewer look through the hole from the back at a mirror held in front — which fixed the eye at exactly one position and one distance, removed binocular disparity by using one eye, and hid the panel’s edges.

Everything about that apparatus is a device for putting the eye at the correct point and suppressing the cues that say the picture is flat. It works, by all accounts spectacularly, and it is a controlled experiment rather than a picture.

Alberti’s construction of 1435 keeps the distance explicit by drawing a side elevation with the eye at its true remove. Later methods let the distance be implied by where the vanishing points are placed, and eventually stop mentioning it at all — which is how a free parameter enters the standard construction without anyone deciding to put one there.

The distance point, which is the same number again #

Classical practice has a construction that makes the viewing distance a mark on the paper, and it is worth knowing because it is the most direct way to read a drawing’s implied distance.

For a one-point construction, mark two distance points on the horizon, one either side of the centric point, each at a distance equal to the eye’s distance from the picture plane. Lines from a division on the ground line to a distance point cut the receding edge at the correct depth — the measuring-point construction in its simplest form.

The reverse reading is the useful one. Given someone else’s one-point drawing with a tiled floor in it, the diagonals of the tiles converge at the distance point, so measuring from the centric point to that convergence gives the viewing distance directly, in units of the drawing.

Doing that to a good many pictures produces a consistent finding: painters chose viewing distances of one to two times the picture’s width, which corresponds to fields of view of about 30° to 50° — comfortably in the range where a viewer standing in a gallery is near enough to correct. The apparently exaggerated perspectives are rarer than their reputation, and where they occur they are usually deliberate.