A picture with no size–distance signal

Persian and Indian miniature painting, Japanese e-maki, Chinese architectural painting and European medieval manuscript illumination all share a property that gets described as “no perspective” and is worth stating exactly instead. Objects in them do not get smaller with distance. A figure at the top of the picture is the same size as a figure at the bottom, and a building at the back is the same size as one at the front.

That is a measurable claim about a drawing system and it has a slope attached.

The claim, stated so it can be wrong #

Before measuring it, the claim needs a form that could fail, because “no perspective” is not one.

The testable version is: the drawn size of an object of fixed real size does not depend on its distance. That is a statement about a function — the derivative of drawn size with respect to distance is zero — and it can be checked at every distance rather than argued about.

Two things it does not say, and both are ways the claim is often overstated.

It does not say the picture has no depth. A system with no diminution can and does show depth, by position on the page, and the depth it shows is on a uniform scale rather than a compressed one.

And it does not say objects are never drawn at different sizes. They are, constantly, and in these traditions the size often means something else entirely. What the claim says is that size does not track distance, which leaves it free to track something else.

Zero is a stronger statement than small #

The slope of the oblique system’s size–distance curve is exactly zero, and the assertion in assertNoDiminutionMeansNoHorizon requires it to be zero to twelve decimal places rather than merely small. That distinction is the site’s habit and it earns its keep here.

A system whose diminution is small would still carry a depth signal, weakly. A reader comparing two figures could in principle recover which is further, with poor precision. A system whose diminution is zero carries no such signal at all: two identical figures at any two depths are drawn identically, and nothing whatever in their drawn sizes distinguishes them.

That is a categorical difference and it forces the rest of the system. Something has to say which figure is further away, and in a system with no size cue the only thing left is position on the page.

What position on the page can do #

The oblique map sends a ground point at depth $Z$ to a page height proportional to $Z$. Not to a compressed function of $Z$, not to something that saturates — to $Z$ itself, times a constant.

Two consequences, and the second is the one that never gets stated.

The depth scale is uniform. A metre of depth is the same number of millimetres of page wherever it is. A reader can compare the gap between the first and second courtyard with the gap between the ninth and tenth by measuring, and the comparison is exact. In a perspective picture the same two gaps are drawn at wildly different sizes and comparing them requires undoing the projection.

And there is no horizon. The image of a ground point rises linearly with depth and keeps rising: there is no height it approaches. So the picture has no line at which the ground plane’s image accumulates, which means it has no image of the line at infinity — because a parallel projection sends points at infinity to points at infinity rather than to a finite line.

A system with no diminution has no horizon, and those are not two facts. They are one fact, seen from the two ends of the same map.

The horizon, priced #

Sixty-nine per cent against ten. That is the sharpest single number in this field and it is worth turning over.

A perspective picture of a landscape spends nine tenths of its page on the nearest third of what it depicts and one tenth on the remaining two thirds. As the depth range grows the imbalance grows without limit: extend the scene to the horizon and everything past a certain distance shares a few pixels.

This is normally described as a virtue — the compression is what makes a perspective picture feel deep — and it is one, for a picture of a view. It is a serious defect for a picture of a journey, which is what these systems are mostly used for. A procession along a road, a river with settlements along it, the rooms of a palace one after another: in every one of those the subject is extended in depth and the interest is evenly spread through it, and a system that spends 90% of the page on the first third is spending it in the wrong place.

So the choice between diminution and no diminution is not a choice between sophistication and naivety. It is a choice about where the page goes, and the right answer depends on whether the subject’s interest is concentrated near the viewer or spread through the depth.

Sixty-nine per cent is one case of a very short law #

The band measurement is quoted at one depth range and it generalises to an expression with a single variable in it, which is worth having because the variable is the only thing about a scene that matters here.

Let $\rho = Z_{\text{far}}/Z_{\text{near}}$ be the depth range as a ratio. A pinhole puts a ground point’s image at a height proportional to $1/Z$, so the strip runs uniformly in inverse depth, and the share of the depth range that lands in the top tenth of the strip works out to

$$\frac{\rho}{\rho + 9}.$$

The figure’s scene runs 3 m to 60, so $\rho = 20$ and the share is $20/29 = 69.0\%$ — the measured number, from one fraction. For a top $1/k$ of the page the expression is $\rho/(\rho + k - 1)$, and for a linear depth map it is $1/k$ at every $\rho$, which is the control the figure draws.

Four readings, and the last is the one that reframes the comparison.

Half the depth range lands in the top tenth at $\rho = 9$. A scene whose far end is nine times its near end already spends half its depth on a tenth of its page — and nine to one is an ordinary room, not a landscape.

The share rises to one. As $\rho$ grows, $\rho/(\rho+9) \to 1$: a scene extending to the horizon puts all of its depth in the top tenth, exactly, in the limit. So the compression is not severe at large ranges, it is total.

And it is mild at small ones, which is the same crossing the two-centre conventions find from a different direction: a picture without much depth in it cannot be told apart from its alternatives, whatever the alternatives are, because depth is what every one of these measurements is reading. At $\rho = 2$ the share is 18%, against the linear map’s 10% — a factor of under two, and a picture of something twice as deep as it is far away is barely compressed at all. Which says exactly when the two systems are interchangeable: a subject with a small depth ratio can be drawn either way and a reader will not be able to tell.

So the choice is set by one number about the subject. Not by the tradition, not by the period, and not by whether a horizon is wanted: by $\rho$. A courtyard viewed from its own doorway has $\rho$ near two and either system serves; a river running to the horizon has $\rho$ in the hundreds and only one of them shows the far end at all.

That is the quantitative form of this essay’s own claim that the choice depends on where the subject’s interest lies. It also explains the correlation the field keeps noticing between conventions and subjects: a scroll’s journey and a mansion’s rooms in a row are exactly the subjects with large $\rho$, and a single interior is the subject with small $\rho$ where a horizon costs nothing. The traditions are not disagreeing about geometry; they are drawing different values of one ratio.

What a reader loses, and it is real #

The honest counterweight, and there are two parts to it.

Absolute depth is not recoverable, at all. The height of a figure on the page gives its depth only if the depth scale is known, and the depth scale is a convention the painter chose. In a perspective picture the depth scale is determined — by the horizon, the eye height and the focal length — so a photograph of a road gives its length, up to one reference. A miniature does not.

And the vertical axis is doing two jobs at once. Height on the page carries depth and it carries height above the ground, and nothing in the picture separates them. A figure drawn higher up may be further away or may be standing on something. Perspective separates the two with the horizon: above the horizon is above eye level, below it is below, and that is a fact about the picture rather than about the scene. A system without a horizon has no such divider and must resolve the ambiguity by other means — overlap, a drawn ground line, a change of scale for figures of different status — none of which is geometric.

That second point is the real cost and it is why these systems look flat to a viewer trained on perspective. It is not that the depth is missing; it is that the depth and the height are being carried on one axis and the picture provides no key.

The horizon is not the only thing that goes #

A perspective picture’s horizon does three jobs and losing it loses all three, which is worth enumerating because only the first is usually mentioned.

It bounds the depth range, which is the compression already measured.

It marks eye level. The horizon sits at the eye’s own height exactly, so anything drawn above it has its base above the viewer’s eyes and anything below it does not. That is a fact about the picture obtainable by looking, and no system without a horizon has an equivalent.

And it locates the vanishing points of every horizontal direction. A perspective picture’s horizon is the image of the ground plane’s line at infinity, so every horizontal family’s vanishing point is on it — which is what makes a horizon findable from the picture and what makes the camera recoverable.

The third is the one whose loss matters most for this site. A system with no horizon has no vanishing points, so there is nothing in the picture to fit a camera to, and the whole round trip that this site is organised around simply has no input.

That is not a small loss and it should be stated as a loss rather than folded into the trade. A perspective picture reports the camera that made it; these do not, and cannot, because there was no camera.

A note on what “no perspective” would mean #

It is worth disposing of the phrase, since it is the one these systems usually get.

A picture with no perspective would be a picture with no rule relating the world to the page — marks placed by judgement, with no map. Nothing in this field is that. Every system here is a map, applied consistently, with properties that can be computed and checked, and the checking is what the last two essays of this phase do at scale.

What these systems lack is not perspective in the sense of a rule. It is the divide by depth, which is one particular rule with one particular set of consequences, four of which the final essay prices.

The measurement, and its control #

depthResponse samples two hundred depths and computes, for each, the drawn size of a fixed object and the page height of a ground point, under both systems. Three assertions follow and the third is the control.

The oblique size slope is zero to twelve decimal places.

The pinhole’s is negative and large — 4.06 px per metre averaged over the range, with the size falling 20.0× from end to end.

And the oblique depth map is linear, checked by comparing the local scale at the near end of the sweep with the scale at the far end and requiring the ratio to be 1 to twelve decimals. That check is what separates the system being described from a different one that also holds size constant while bunching depth — which would be a legitimate drawing system and is not this one.

The tenth-of-the-page figures come with their own control built into the definition: under a linear map, a tenth of the page necessarily carries a tenth of the range, so the 10.0% the oblique system reports is not a measurement of the system so much as a confirmation that the quantity is being computed correctly. The 69.0% next to it is the measurement.

What else has to change when diminution goes #

A drawing system is not one rule; it is a set of rules that have to fit each other, and removing diminution forces three other things.

Overlap has to carry near-and-far. With no size cue and an ambiguous vertical, the most reliable depth statement left is that one thing hides another. So systems without diminution lean heavily on overlap, and they arrange scenes so that overlaps are unambiguous — figures offset rather than directly behind one another, buildings stepped rather than aligned. That is a compositional consequence of a geometric choice.

Ground lines have to be drawn. A figure’s depth is its foot’s height on the page, so the ground has to be visible under every figure for the reading to work. Systems without diminution draw a great deal of ground: terraces, carpets, bands of hillside, each with figures standing on them and their feet showing.

And scale has to be freed for other work. This is the consequence with the largest visible effect. In a perspective picture, a figure’s drawn size is spoken for — it means distance — so it cannot be used to mean anything else without conflict. In a system with no diminution, size is available, and it gets used: for importance, for rank, for narrative weight. A ruler drawn larger than a servant is not a mistake about distance in a system where distance is not carried by size.

That third point is worth stating carefully, because it is a place where the geometry does have something to say about the pictures and it is easy to overstate. The geometry does not explain why any tradition used size for status. What it establishes is that the channel was free — and in a perspective picture it is not, which is why European painting after the fifteenth century largely abandoned hieratic scale and had to convey importance by other means.

Where the convention and the subject meet #

One closing observation, and it is the same one the handscroll essays keep arriving at.

The systems that give up diminution are the ones used for subjects extended in depth with interest spread evenly through them: journeys, processions, palaces, gardens, battles seen from above. The systems that keep it are used for subjects concentrated near the viewer: an altarpiece, a portrait, a room seen from a doorway.

That correlation is not evidence that anybody reasoned about page budgets. It is evidence that drawing systems are shaped by what they are used to draw, which is the least surprising claim in this field and the one that most reliably gets left out of the account. A convention is not an opinion about how the world looks. It is a rule that has been kept because it worked on the things it was used for, and measuring what it preserves is a way of finding out what those things were.