Standing in the wrong place

The site’s premise is that every picture states the point it is correct from. The point to stand at computes it: the focal length scaled to the width the picture is actually shown at, which for a 40° picture 160 mm wide is 22 cm from the page.

Nobody stands at 22 cm. Readers hold a page at forty centimetres, sit two metres from a screen, and walk past a painting at whatever distance the room allows. So the interesting question is not where the correct point is but what the error costs, and the answer is unexpectedly clean.

The derivation, which is four lines #

A picture made with focal length $f$ puts a world point $(X, Y, Z)$ at $(fX/Z,\ fY/Z)$ from the principal point.

A reader at distance $d$ from the picture receives that mark along the direction $(fX/(dZ),\ fY/(dZ),\ 1)$ — the mark’s offset divided by the reader’s own distance, which is the reader’s own focal length.

The reader now has a direction and needs a scale. Any anchor supplies one: the ground plane at their assumed eye height, a known object, the floor the depicted room stands on. Whatever the anchor, it fixes a factor $t$, and imposing the same anchor as the true scene gives $t = dZ/f$. So the reconstructed point is

$$X' = X, \qquad Y' = Y, \qquad Z' = Z\cdot\frac{d}{f}$$

Depth is stretched by the ratio of the two distances, and nothing else changes. A reader at twice the correct distance sees a depicted room twice as deep; one at half the distance sees it squashed to half.

The part that makes it invisible #

Now the consequence that matters, and the figure asserts it rather than stating it.

Take those stretched points and re-project them through a camera whose focal length is the reader’s actual distance. They land on the marks already on the paper, to 6e-14 px.

Of course they do — that is what the derivation says. But it is worth having as a measurement rather than as an inference, because the conclusion is strong: nothing in the picture changes. No line moves, no angle changes, no proportion of anything drawn is different. The reader is not looking at a distorted picture; they are looking at the same picture, and reconstructing a different scene from it.

So there is no visual cue. The error cannot be detected from the picture at any level of attention, by anyone, ever. It can only be detected by knowing something about what the picture shows — that the object in it was a cube, that the room was square, that the person was of ordinary height.

That is the whole reason the wrong viewing distance is universal and unremarked. Every other error in a picture announces itself somehow; this one has no signature at all.

The size of it, in ordinary conditions #

Some arithmetic on the cases a reader will actually meet.

A 46° picture at 160 mm wide is correct from 18.8 cm — closer than anyone reads a book. At 40 cm the depicted scene is 2.12 times too deep.

Widen the lens and it gets worse, because a wider picture is correct from closer. A 90° picture 160 mm wide is correct from 8 cm and read at 40 cm depicts a scene five times too deep. Narrow it and it improves: a 24° picture is correct from 38 cm, which is almost exactly reading distance, and reads nearly true.

That last observation is worth holding onto, because it explains something about how pictures are actually made. A long lens produces a picture that reads correctly at an ordinary distance, and a wide one does not, and photographers have known the perceptual half of that for a century without the arithmetic. The “compression” of a telephoto and the “exaggerated depth” of a wide-angle are not properties of the lenses; they are the reader standing in the wrong place by different amounts.

Why the anchor does not matter #

One step in the derivation deserves scrutiny, because it looks like a place where an assumption is smuggled in.

The reader receives a direction from each mark, and a direction alone does not fix a point — a whole ray of candidate points sits behind every mark. Something has to choose along that ray, and the derivation chose by “imposing the same anchor as the true scene”.

The reason that is not a smuggled assumption is that every anchor gives the same answer up to one overall scale, and the overall scale is not what this essay is about.

Suppose the reader anchors on the ground plane at their own assumed eye height; the factor comes out as $t = dZ/f$. Suppose instead they anchor on a known object’s width; the factor comes out proportional to the same thing. Suppose they anchor on nothing at all and simply assume the nearest object is a metre away; same again. In every case the reconstruction is the true scene with $Z$ multiplied by $d/f$ and then everything multiplied by whatever the anchor decided.

A uniform scaling of the whole reconstruction is exactly the one thing a single view cannot supply, and it is not an error introduced by standing in the wrong place — it is missing from the picture whatever the reader does. What standing in the wrong place adds is the anisotropic part: depth relative to width, which is a ratio and therefore not affected by the overall scale at all.

So the honest statement of the result is about a ratio: the depicted scene’s depth-to-width ratio is multiplied by $d/f$. That claim has no anchor in it, and it is the claim the figure measures.

What the reader actually perceives #

An honest limit, and this site states it whenever it comes up.

Everything above is geometry: it says what scene the picture is a correct projection of, from the reader’s actual position. It does not say what the reader perceives, and the two are not the same.

Real viewers are remarkably tolerant. Binocular vision, the flatness cues from the picture’s own surface, the frame, the texture of the paper — all of them tell the visual system that it is looking at a flat thing, and it compensates. A photograph read from three times its correct distance does not look three times too deep to most people; it looks approximately right, and the compensation is a large and well-studied perceptual effect.

So this essay’s claim is precise and limited: the picture is a correct projection of a scene stretched in depth by $d/f$, and no viewer standing there is receiving the geometry of the original scene. Whether they notice, and what they experience, is a question about seeing rather than about projection, and the site borrows no authority from the geometry for it.

Where the geometry does bite is anywhere the picture is being measured rather than looked at — and anywhere the compensation fails, which is chiefly when the picture is large, when it fills the field, or when the viewer has no cue that it is flat.

The one case where it is not invisible #

There is a configuration in which the wrong viewing distance does announce itself, and it is worth having because it is the exception that makes the rule precise.

If the picture contains an object whose true proportions the reader already knows — a cube, a circle, a human figure, a familiar building — then the reconstruction can be checked against that knowledge, and the discrepancy is exactly the factor.

That is why the figure draws a cube. The picture of a cube from the wrong distance is a picture of a box, and a reader who knows it was a cube can read the factor off the box’s proportions. The information was in the reader’s head rather than in the picture, which is precisely the condition the invisibility claim allows for.

It is also the mechanism behind the one visual cue people do reliably notice. A face photographed from very close and read at ordinary distance reads as stretched forward — the nose too large, the ears too small — and everybody notices, because everybody knows the proportions of a face to a fraction of a per cent. The same picture of a rock formation reads as fine.

So the effect is invisible in general and glaringly visible on faces, and the difference is entirely how well the reader knows the subject’s true shape. That is a satisfying place for the geometry to hand over to perception: it says exactly what information the picture withholds, and leaves what a viewer does with their own knowledge alone.

What it explains #

Three things that are usually explained some other way.

The trompe-l’œil. A ceiling painted in perspective, viewed from the point the painter constructed it for, reads as architecture continuing upward. Viewed from anywhere else it reads as a painting. The whole technique is the exploitation of a single station point, and its fragility is exactly the factor above.

The theatre set. A forced-perspective set is built to be correct from one seat, and everyone else in the theatre sees a stretched or squashed version. Set designers know which seat, and choose it.

And the panoramic photograph nobody can read. A very wide picture printed small is correct from a few centimetres, which is closer than the eye can focus, so there is no distance at which it can be read correctly. That is not a defect of the print; it is the flat picture surface running away as the field widens, meeting the limits of a human eye.

The third one is the interesting case because it is not fixable by standing anywhere. The picture is correct from a point the reader cannot occupy, so there is no correct reading of it at that size, and the only repairs are to print it larger or to cast it onto a different surface.

Why this is not marginal distortion #

It is worth separating this essay’s effect from the one wide-angle is not distortion is about, because both are about wide pictures and the reader’s position and they are quite different.

Marginal stretch is a fact about the picture: a sphere at the edge of a wide frame is drawn as an ellipse. It is present in the ink and measurable from the picture alone, and it vanishes when the picture is viewed from its station point, because the ellipse then subtends a circular cone.

Depth scaling is not present in the ink at all. It is a fact about the reconstruction, and it is unmeasurable from the picture, and it vanishes from the same station point.

They share the cure and nothing else. Standing at the right place fixes both, which is why the site’s premise is one sentence rather than two — but the two failures are of different kinds, and only one of them is visible.

The pair is also a neat argument for the site’s habit of computing the station point at all. Two distinct things go wrong when the reader is elsewhere, one visible and one not, and the single number that describes where to stand covers both.

The measurement, done properly #

The figure’s assertion has two clauses and the second is the one that took an attempt to get right.

The obvious way to measure the stretch is the reconstructed solid’s own depth-to-width ratio: draw a cube, look at what is reconstructed, and report how much deeper than wide it is. That is wrong, and it was the first version.

The camera is oblique to the box, so the box’s extents along the camera’s own $x$ and $z$ axes are not equal even before anything is stretched. Comparing them with each other reports the obliquity as though it were the effect, and gives 2.41 where the factor is 2.12.

The fix is to compare each axis with its own true extent: depth against true depth, width against true width. Depth comes back at exactly the factor and width at exactly 1, which is the claim the derivation makes and the previous version was not testing.

That is a small instance of a habit worth having. When a claim is “this axis changed and that one did not”, the measurement has to look at each axis separately against itself. A ratio between two different axes conflates the effect with everything else that distinguishes them.

Two readers, one picture #

A last consequence worth drawing out, because it is the one that affects how pictures are shown rather than how they are made.

Two people looking at the same picture from different distances are looking at two different scenes. Not two impressions of one scene — two scenes, differing by a depth scaling, and the arithmetic says by exactly how much.

That is a strange thing to be true of an object that is supposed to be a record. A photograph is often described as evidence of what was there, and in a precise sense it is: the picture determines the scene up to a scale, and the recovery gets the camera back out of it. But the scene a viewer takes from it is determined by where their eye is, and nothing about the print announces the right place.

The practical version of that is a suggestion this site has been making implicitly since its first field and may as well make explicitly: a printed picture could carry its station point the way a map carries its scale bar. One line — correct from 22 cm at this size — costs nothing and turns an unstated assumption into a stated one.

Every figure on this site does exactly that, which is the site’s one non-negotiable piece of furniture, and it is worth noticing how unusual it is. No photograph in any book carries it; no gallery label states it; no museum caption mentions it. The number exists, it is computable from the picture and its size, and it is nowhere written down.

The rule the taught cone of vision is #

There is a well-known piece of advice in every book on perspective: keep the subject within a cone of about 60°. It is presented as a rule about drawing, and this essay is most of the argument that it is not.

A 60° picture is correct from 0.866 of its own width. The rule is therefore an instruction to the reader — stand at 0.87 widths — dressed as an instruction to the illustrator, and readers of books do not obey it.

That is the subject of the next essay, where the rule’s own arithmetic is followed through and the marginal stretch it is nominally about turns out to be exactly zero from the station point, measured two ways.