The screen sets the distance

Every figure on this site prints a strip saying what distance it is correct from, and every one of those strips carries a hedge: at 160 mm wide. The site cannot know how wide its figures really are on a reader’s screen, so it names an assumption and gives the arithmetic so a reader can redo it.

That hedge has been honest and it has also been a way of not finishing the sentence. This essay finishes it. A photograph’s viewing distance is not conditional on anything unknowable — it is the product of three lengths, two of which are properties of the camera and one of which is a property of the display:

$$d = f \cdot \frac{W_d}{w}$$

focal length, times display width, over sensor width. Every symbol is a millimetre and none of them is a guess.

The arithmetic, and why it is this and not something else #

The claim rests on one observation, which this site’s camera has carried since the first commit: the focal length, expressed in the picture’s own units, is the viewing distance in those units.

The reason is a similar triangle and nothing more. The picture was formed by rays converging on a point $f$ behind the sensor. Enlarge the sensor’s image to width $W_d$ and the whole triangle scales by $W_d/w$, taking the vertex with it. Put an eye at the scaled vertex and the rays reaching it from the enlarged picture make exactly the angles the original rays made — so the picture subtends what the scene subtended, and it is a projection of the scene from that point.

Nothing about the scene enters. That is worth repeating because it is the counter-intuitive half and it is the half the site’s whole viewing field is built on: the correct viewing distance is a property of the picture and its size, not of what it depicts. A photograph of a mountain range and a photograph of a coin taken with the same lens on the same camera and printed the same size are correct from the same distance.

Five displays, one lens #

Take a 50 mm lens on full frame, which is 39.6° across and is about as ordinary a photograph as exists.

The “people sit at” column is the softest thing in this essay and it says so: those are typical distances rather than anybody’s measurement, and the arithmetic is given so a reader can substitute their own. Everything else in the table is exact.

Three things fall out of it.

The phone is the worst by a long way. A photograph held at arm’s length on a phone is being read from nearly four times its station distance. Which means — by the viewing field’s central result — that its depicted depth is stretched by 3.7, and nothing in the picture reveals it.

The cinema is the best, at 0.84, and that is not an accident: cinema seating and lens choice have been negotiated against each other for a century by people watching the result. The medium that had the most opportunity to converge has converged.

And the monitor is on the other side of one. At 0.78 a picture is being read from closer than its station point, which compresses depth rather than stretching it. So the errors are not all in one direction and cannot be corrected by a single habit.

What being at the wrong distance does, exactly #

This is already established and it is worth restating because it is what makes the table mean something.

The viewing field’s measurement is that reading a picture from distance $d$ when its station point is at $f$ produces a perceived scene in which

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

Depth is stretched by $d/f$ and nothing else changes. And the consequence that makes it invisible: those stretched points, re-projected by the reader’s own eye from where the reader actually is, land on exactly the marks already on the paper. Nothing in the picture changes, so nothing in the picture can betray the error.

So the ratio column in the table is not an abstract mismatch. It is a depth factor. A photograph on a phone depicts a room 3.7 times deeper than the one photographed, and the only way to notice is to know the true proportions of something in it.

The second projection #

The framing worth taking away from this field is that a photograph is projected twice.

The first projection is the camera’s: scene to sensor, through a point, at a focal length. This site has measured that projection in fifteen different ways.

The second is the display’s: sensor to room, through the reader’s eye, at a distance the reader chose without knowing there was a choice. It has all the same properties — a centre, a station point, a field of view — and the only difference is that nobody specified it.

Every property this site has established about the first applies to the second. The station point is computable. Standing away from it stretches depth. A wide picture has a near station point. A projection of a projection is a projection, so the composite of the two is itself a projection — with a focal length that is the product of the two ratios, and a station point that is the reader’s actual position rather than either of the intended ones.

That composite is what a viewer is actually looking at, and it is exactly the scene the picture depicts to them: the real scene, with its depth multiplied by the ratio of the two projections’ distances.

Why nobody is standing there, and why it is not carelessness #

It would be easy to read the table as a list of mistakes. It is not, and the reason each display sits where it does is worth spelling out, because in every case there is a constraint that has nothing to do with perspective.

A phone is held where an arm reaches and where text is legible. Neither has any relation to the field of view of whatever photograph is on it, and both are much more strongly constrained than the picture is.

A monitor is placed where a keyboard puts it. Which is decided by the depth of a desk.

A television’s distance is set by the room. Furniture, not optics.

Only the cinema had a design process in which the picture’s geometry was one of the inputs, and it is the one that comes closest to one. That is a genuine result and it is the strongest evidence in the table that the geometry is doing something a viewer can feel — because if it were not, no amount of negotiation would have converged on it.

So the mismatch is not a failure of attention. It is what happens when a quantity that nobody has computed competes with several that everybody has.

The chain has a fourth link nobody controls #

The three-link chain above is honest about the photograph and quiet about the file, and the file is where a fourth ratio hides.

A photograph is cropped, resized, letterboxed, fitted to a page, scaled by a browser and shown in a window. Every one of those is a change to the width the picture is displayed at, relative to the width of the picture’s own frame — and each one multiplies the correct viewing distance by the same factor.

Cropping is the interesting case, because it changes the width of the picture while leaving the width of the display alone. A photograph cropped to half its width and shown at the same size on the same screen has had its effective focal length doubled: the remaining half now covers the whole display, so the station distance doubles too.

That is worth stating because it is the reverse of the intuition. A crop makes a picture correct from further away, not closer, even though the crop shows less of the scene. The angle has narrowed and the display width has not, so the ratio has gone up.

And it means the station point of an image on a page is essentially never the station point of the photograph that was taken. Somebody cropped it to fit a layout; somebody else chose a column width; a browser scaled it to a viewport. Four decisions, none of them geometric, each of them multiplying the answer.

Which is why this site’s figures print the strip they do. The strip is honest about exactly this link — at 160 mm wide — because the width a figure is shown at is the one number in the chain that neither the author nor the reader can pin down.

The one instrument that gets it right by construction #

There is an exception and it is the oldest one in the subject.

Brunelleschi’s demonstration was a small painted panel, a peephole drilled through it at the vanishing point, and a mirror. The viewer looked through the panel from behind, at its reflection, with one eye, at the distance the arrangement forced. This site has already computed that distance: the geometry of the panel and the mirror fixes it, and the viewer has no freedom at all.

That is the only viewing arrangement in the whole history of the subject that enforces the station point rather than stating it. Every other one — a painting on a wall, a print in a book, a photograph on a screen — leaves the reader to choose, and readers choose by ergonomics.

The peephole is usually described as a demonstration of perspective’s correctness. It is more precisely a demonstration that perspective’s correctness is conditional, and a piece of apparatus for meeting the condition.

The 50 mm convention, priced #

There is a piece of received wisdom this chain lets the site price, and the result is more interesting than either of the usual verdicts.

A “normal” lens is conventionally the one whose focal length equals the sensor’s diagonal — 43 mm on full frame, rounded to 50 by history and by the ease of making one. The justification usually offered is that it “matches the perspective of the human eye”, which is not a claim anybody can make precise, since the eye’s field is around 200° and no lens is.

The chain gives a version that is precise. A 43 mm lens on full frame is 45.7° across, and a picture 45.7° across is correct from 1.19 of its own width. So a print or a screen viewed from a little over one picture-width away is being read from its station point.

And one-and-a-bit picture widths is very close to how people actually look at things: a photograph held in the hand, a page at reading distance, a laptop on a desk. The 27-inch monitor in the table above sits at 1.09 widths.

So the convention is defensible and its stated reason is wrong, in exactly the pattern the sixty-degree cone of vision turned out to follow. It is not about the eye’s field of view. It is about the distance at which people hold things, and a lens equal to the diagonal is the one whose pictures are correct from there.

That reading also explains the exception. Cinema does not use it — the seats are two to four screen widths back, which wants a much narrower angle — and cinema is, as the table shows, the medium closest to getting the geometry right. Two different conventions, two different viewing distances, and both defensible for the same underlying reason.

What this changes about the rest of the site #

One thing, and it is a change of status rather than of any number.

Every figure on this site prints “correct from 40 cm, at 160 mm wide”. That strip is not a hedge to be apologised for — it is the correct form of the claim, because the strip is attached to a picture and a picture’s station point genuinely is conditional on the width it is shown at. The hedge is the honest statement.

What this essay adds is that for a photograph the conditionality can be discharged, because the display width is knowable in a way the browser’s layout is not, and the two ends of the chain are physical lengths rather than abstract ones.

And it adds a way to read every one of those strips. A reader who knows that a figure is 160 mm wide on their screen and that they are 55 cm away can compute the ratio, and the ratio is the factor by which the figure’s depicted depth is stretched for them personally. That has been computable from the strip all along; this is the essay that says to do it.