What perspective gave up

Every field before this one on this site has measured a departure from the pinhole. Water, glass, a real lens, a curved surface, a taught construction, a second camera — all of them are described as differences from a projection through a centre, and the pinhole is the thing they are differences from.

This is the essay that puts a number on what the pinhole itself costs.

Why this is not a rhetorical exercise #

There is an obvious objection to an essay of this shape: any system can be made to look costly by measuring it on properties another system has, and doing it to perspective at the end of a field about other systems is a set-up.

Three things distinguish this from that.

The four quantities were not chosen here. Each was measured in an earlier essay for its own reasons: the midpoint drift is the parallel field’s founding measurement from three phases ago, the page budget and the area uniformity come out of this phase’s own libraries, and the coverage spread is the removed-roof measurement. None was introduced to make a point about perspective.

Every one has a zero or an exact value on the other side. A cost measured against something that also varies is a comparison of two variances; these are measured against 0.000, 1.000000000000 and 10.0%, all exact. That is what makes them prices rather than differences.

And the thing bought is stated in the same units of specificity as the things sold. The section after the list is not an apology; it is four capabilities that no other system in this field has, each of them the foundation of a field on this site.

A trade stated with only one side priced is an argument. This one has both sides priced, and the reader is left to decide which side is worth more for which picture — which is the only honest place to leave it.

The four #

The ratio along a receding line: 15.6% of the segment. The midpoint of a world segment whose ends lie 11.1 m and 21.0 m from the eye images 15.6% of the drawn length away from the midpoint of its image. Under every parallel system on this site the same number is zero, exactly. This is the site’s own measure test and it is the exact difference between the two families.

The depth range in the last tenth of the page: 69%. Of a depth range from 3 m to 60 m, sixty-nine per cent lands in the tenth of the page nearest the far end. A linear depth map puts exactly ten per cent there, which is the control. So a perspective picture spends nine tenths of its page on the nearest third of what it shows, and the imbalance grows without limit as the scene deepens.

A square metre, near against far: 53% larger. Across the three-room building this field measures, the same square metre of floor images 1.535× larger in one room than in another under the eye that frames the building. Under a parallel system the ratio is 1.000000000000, to arithmetic noise.

The floor of the far room: 6.2 points less. The framing eye reaches 49.4% of the near and far rooms’ floors and 55.6% of the middle one. A parallel system reaches the same share of every room, with a spread of exactly zero.

Every one of those is computed by machinery that existed before this field did, on the same scenes, with the same samples.

What it bought #

The list above is not a case against perspective and would be a dishonest one if presented as such, because the thing perspective bought with it is not a small thing.

A picture that is a projection of the scene from one point. That single property is the source of everything the rest of this site computes. The viewing distance exists because there is a point. The camera is recoverable from the picture because the picture is a projection from one. A height is recoverable from a photograph because the cross-ratio survives, and the cross-ratio survives because the rays meet. Two views determine a scene because each is a projection from a point and the two points have a relationship.

None of that is available in any other system in this field. A scroll has no camera to recover. An oblique drawing has no viewing distance. An aspective composite has no viewpoint for two of them to be compared from.

And a picture that is evidence. This is the property worth stating last because it is the one that is easiest to take for granted. A photograph constrains the scene whether or not anybody was careful, because it was made by light rather than by decisions. A scale drawing constrains the scene as far as its maker did and no further. Everything in the metrology field — a height from a photograph, a plan of a ground, a façade flattened — depends on the picture being a measurement of the world rather than a statement about it.

So the trade is: exactness of measure, page budget, uniformity and comparable coverage, in exchange for a station point, recoverability and evidential force. Stated that way it is obvious that neither side is the right answer in general, and the whole of this field has been an argument for stating it that way.

Each cost, traced to the same source #

The four numbers are not four independent debits. Every one of them comes out of the same divide, and it is worth showing the chain because it explains why they cannot be paid off separately.

A projection through a centre at distance $z$ divides by $z$. From that one operation:

The ratio goes, because the division is nonlinear in depth — the midpoint of a segment does not divide by the same number as its endpoints.

The page budget goes, because $1/z$ compresses everything beyond a distance into a band, and the band is the horizon.

Area uniformity goes, because the area scale is $1/z^2$ and $z$ varies over the scene.

And comparable coverage goes, because the angle at which a sightline clears an obstacle depends on where the obstacle is relative to the eye, which is again a function of $z$.

One operation, four consequences. That is why no amount of care with a camera recovers any of them, and why the only way to get them is to send $z$ to infinity — which is precisely what the other systems do, and which costs the station point because a centre at infinity is not a place.

The trade is therefore a single trade with four visible faces, not a menu. A system either divides by depth or it does not.

The four prices are one price, and it has a dial #

Tracing the four costs to the divide says they cannot be paid off separately. It does not say how large any of them is, and the answer to that is a single dimensionless number that every one of them is a function of.

Let the scene’s near and far ends sit at depths $z_{1}$ and $z_{2}$ from the eye, and write

$$\rho = \frac{z_{2}-z_{1}}{\bar{z}}, \qquad \bar{z} = \tfrac{1}{2}(z_{1}+z_{2}),$$

the depth range measured against the distance the scene stands at. It runs from 0, for a scene infinitely far away or infinitely shallow, up to 2, where the near end is at the eye.

The measure cost is exactly $\rho/4$. The midpoint drift derived in the field’s first essay is $(z_{2}-z_{1})/2(z_{1}+z_{2})$, which is that expression rearranged. The test segment used throughout this field sits at $\rho = 0.62$, and $0.62/4 = 15.6\%$ — the first of the four numbers, from the dial alone.

The area cost is $2\rho$ to first order, since the ratio of near to far area scale is $(z_{2}/z_{1})^{2}$, which expands to $1 + 2\rho + O(\rho^{2})$. The three-room building’s 1.535 corresponds to $\rho = 0.21$; the second-order term is a fifth of the effect at that depth, which is why the expansion is quoted as first order rather than as an equality.

And the page budget and the coverage spread are the same shape, both rising from zero at $\rho = 0$ and both monotone in it, though neither has as clean a closed form.

Three things follow, and the first is the one that reorganises the field.

Perspective and the parallel systems are not two families but one, parameterised. At $\rho = 0$ a pinhole is a parallel projection — measure exact, area uniform, page budget flat, coverage equal — and every column of the table agrees with the parallel rows. The dichotomy the whole field has been comparing across is the two ends of a dial, and the intermediate values are occupied by every photograph ever taken of a distant subject. That is the precise version of the claim that neither family is primitive.

The maximum cost is bounded and small-sounding. At $\rho = 2$ the measure cost is exactly one half, and it cannot exceed that for any scene, any lens or any framing. The four prices are not unbounded quantities that happen to be modest here; they have a ceiling that the geometry fixes.

And the dial is set by the subject, not by the photographer. Focal length does not appear in $\rho$, nor does the principal point, nor the framing. A long lens on a deep scene from far away gives a small $\rho$ because it is far away, not because the lens is long. So a picture’s costs on all four counts are decided by where the camera stands relative to the depth of what it is looking at — which is the same quantity, arrived at from the other side, that decides whether a viewer in the wrong place sees anything wrong at all.

The thing that is not a trade #

One asymmetry does not fit the pattern, and it is worth being clear that it is real.

Perspective is the only system here whose picture has a correct viewing position, and that is a cost as well as a benefit. The picture is right from one point and wrong from everywhere else, by an amount this site measures — a viewer at twice the correct distance sees a scene stretched in depth by a computable factor, and almost every reader of almost every perspective picture is in the wrong place.

The other systems in this field are not correct from anywhere, which means they are not incorrect from anywhere either. A fukinuki yatai interior looks the same from every position a reader might take, and there is no position from which it is more right. For a picture five metres long that is a genuine advantage, and it is the advantage the handscroll’s whole geometry is organised around.

So the station point is not straightforwardly an asset. It is an asset for a picture the size of a page viewed by somebody who can be positioned, and a liability for a picture the size of a room. The systems in this field are, without exception, systems for large pictures.

What the site itself gave up to be built this way #

There is a version of this essay that turns the question on the site rather than on the projection, and it belongs here because the same trade shows up one level higher.

This site computes the geometry of pictures. Every figure is projected from a stated camera, every claim is given a test it could fail, and the whole apparatus rests on a picture being a projection through a centre — which is why the two fields that broke that assumption, water and glass and now the scroll, each needed a new piece of machinery and a new entry in a gate.

What the site gave up for that exactness is everything about pictures that is not geometric. It says nothing about how a picture is perceived, nothing about what a convention meant to the people who used it, nothing about why any tradition drew as it did. This field has been careful about that line and has come close to it repeatedly — the observation about lecterns, the observation about hieratic scale, the observation about which subjects each system suits — and in each case the honest form was the same: the geometry establishes the negative, and the positive is somebody else’s question.

That is the same shape as the trade this essay is about. Exactness in one direction, bought by declining to say anything in the others. A site that also discussed meaning would have less of both.

What a reader should take from the four numbers #

Not that perspective is expensive. The four costs are large on the quantities they measure and every one of them is invisible in ordinary use — a viewer of a perspective picture does not notice that the midpoint has moved, does not notice that most of the depth is in a band, and does not notice that the far room is drawn smaller. Nothing here is a defect a viewer experiences.

What the numbers establish is that the costs are definite. They are not a matter of taste, they do not depend on the picture being badly made, and they cannot be reduced by care. A projection through a centre destroys measure by an amount computable from the scene and the focal length, and the systems that keep measure do so by an amount computable to be exactly zero.

A trade with both sides priced is a different object from a trade with one side asserted, and pricing the second side is the whole of what this field added.

Why this field exists at all #

A closing statement of the case, because a field that spends five essays refusing to rank things should say what it was for.

The systems measured here are usually described by what they lack. No vanishing point, no diminution, no consistent viewpoint, no horizon. Each of those descriptions is true and each is a subtraction, and a subtraction cannot be a description of a thing — it is a description of a difference from another thing.

What the measurements show is that every one of those absences is the other half of a presence, and the presences are exact:

• No station point, and the ratio along the roll survives exactly.
• No diminution, and the depth scale is uniform to twelve decimal places.
• No single view, and every part of a figure reaches the picture at full extent, where the best single view keeps 58% of each.
• No rectification, and a ruler on the paper measures the ground.
• No consistent perspective across the picture, and every room reports on identical terms.

Those are five conventions from five traditions with no contact between them, and each of them turns out to be an exact answer to a problem a projection through a centre cannot solve. That is the finding, and it is not a defence of anything — it is the result of applying the same battery to nine systems and reading the answers off.

And what the site does with this #

One practical consequence, which is the reason this field belongs on this site rather than beside it.

Every figure on this site prints the distance it is correct from, and that strip is the site’s one non-negotiable piece of furniture. This field’s figures do not print it — the scroll’s print a miss, and the parallel systems’ print nothing at all — and the gate that decides which figures may do which is now a list with two kinds of entry on it: figures whose light bends, and figures whose eye moves.

That is a small piece of machinery and it carries the field’s whole argument. A site built around one number has to be able to say this picture has no such number, and to prove it, or the number becomes decoration the moment anything outside the assumption is drawn.