A centre and a measure are exclusive

Every field on this site has been asking one system at a time what it keeps. This is the table, and its first property is a decision rather than a result: the pinhole is a row in it.

That is the whole editorial position of this field, put into a data structure so that it cannot slip out of the prose. A comparison in which perspective is the standard and everything else is a departure has already answered the question it claims to be asking.

Why a table at all #

Five fields on this site have each measured one system carefully, and a table is a coarser instrument than any of them. It is worth saying why it is worth building.

Because the individual measurements are not comparable unless they are made the same way. A scroll’s centre miss is in metres, a refracted picture’s in millimetres, a parallel system’s is infinite. Those three numbers say the same thing and look like three different findings until one function computes all of them.

Because an absence is only visible against a pattern. That no system has both a centre and true measure is not a statement any single system’s essay could make; it needs the whole set, and it needs the set to include systems from outside the family the pattern was noticed in.

And because the alternative is an ordering. The account these conventions usually get is a ranking with perspective at the top, and a ranking is what a table refuses by construction: a table has columns, and a column is a question, and a question can have a different winner from the one before it.

The five questions #

Does it have a centre? Take the system’s own rays and fit a common point. A projection through a centre gives the point to the last bit; a parallel system’s rays are all one direction and the solver refuses, which is the answer at infinity rather than an error; a scroll’s rays miss by metres.

Does it keep true measure? midpointDrift — the site’s own test since the parallel field: does the image of a segment’s midpoint land on the midpoint of its image?

Does size fall with distance? The drawn height of a fixed object at 6 m and at 48 m.

Is its depth range bounded? Does the image of a ground point converge to a limit as the depth grows without bound — a horizon — or does it run off the page?

Does it keep straight lines straight? The sag of a receding line from its own chord.

None of these is a summary of a book. Each is a computation on the system’s map, and a system added to lib/scene.js or lib/cultures.js appears in this table described correctly without the figure being touched.

The exclusion #

Read down the first two columns. Exactly one row is filled in the first — perspective — and that row is empty in the second. No system has both a centre of projection and true measure.

That is not a coincidence of which systems happen to be listed, and it is not deep either: a projection through a centre divides by depth, and dividing by depth is what destroys the ratio in which a point divides a segment. The exclusion is the parallel field’s finding stated over a wider set, and the wider set is what makes it worth restating — the two systems this phase added, a scroll and an aspective composite, are outside the parallel/perspective dichotomy entirely and neither of them breaks it.

The scroll is the interesting case. It has half of each: measure along the roll and a divide across it. It therefore appears in the table as having neither — no centre, because the whole picture’s rays do not meet, and no measure, because the test is applied to a receding segment. That is a fair summary and it is also the table’s one lossy cell, which the row’s own essays exist to unpack.

Why the exclusion is arithmetic rather than taste #

The exclusion has a one-line proof and it is worth writing out, because a table that merely observes a pattern is weaker than one that observes a pattern with a reason.

A projection through a centre at distance $z$ sends a world point to $f\,x/z$. For two points at depths $z_1$ and $z_2$, the image of their midpoint is $f(x_1 + x_2)/(z_1 + z_2)$ and the midpoint of their images is $\tfrac{1}{2}(f x_1/z_1 + f x_2/z_2)$. Those agree for all $x$ only when $z_1 = z_2$.

So a centre at a finite distance destroys measure on any segment with depth in it, exactly. Move the centre to infinity and the divide disappears along with the diminution, the convergence and the station point, all at once.

That is the whole content of the first two columns, and it is why the pattern is a theorem rather than an observation about which systems happen to be in the table. What the table adds is the scope: the two systems this phase built are outside the parallel/perspective dichotomy the proof is usually stated in, and the exclusion holds for them too, for the same reason applied to a stranger object.

The refusal, which is what makes it a measurement #

An exclusion that cannot be failed is a restatement of the code. So the gate hands assertNoSystemKeepsBoth a system that does not exist — a fabricated row with a centre and true measure — and requires it to be rejected. It is. And it hands it a table with nothing in one half, which would make the exclusion vacuous, and requires that to be rejected too.

The second refusal runs on the boundary itself, and it is the curved field’s method repeated. That field found that nothing is both straight and conformal, and then showed that loosening conformal enough admits the plane — which is what says the exclusion is about the definitions rather than about the sample.

Here the loosening is the tolerance on midpoint drift.

What the boundary being at 15.6% means #

The step is not at $10^{-16}$ and it is not at 100%. It is at the drift a real perspective picture of a real segment actually produces, which is the honest place for it to be and is worth reading carefully.

It means the exclusion is not the statement that perspective’s measure error is astronomically large. It is the statement that the error is a definite, measurable quantity — 15.6% of the drawn segment, for the test segment this table uses — and that calling perspective measure-preserving requires a tolerance above it.

It also means the exclusion is scene-dependent in a way the table does not show. A perspective picture of a shallow scene has a small drift; a picture of a deep one has a large one. In the limit of a very distant, very shallow subject a pinhole becomes a parallel projection and does preserve measure, to any tolerance whatever — which is the same continuity the removed roof essay found when its search for the best eye walked to the far corner of the grid.

So the honest form of the exclusion is: a projection through a centre at a finite distance from a scene of finite depth cannot preserve measure, and every term in that is doing work.

Where the 15.6% comes from, and what it can never exceed #

The step’s position is quoted as a measurement, which it is, and it also has a closed form. Deriving it is worth the paragraph, because the form says which of the arrangement’s numbers the boundary depends on — and the answer is almost none of them.

Put the segment’s ends at depths $z_{1}$ and $z_{2}$ from the eye, at any lateral positions whatever. The image of the world midpoint is $f(x_{1}+x_{2})/(z_{1}+z_{2})$; the midpoint of the images is $\tfrac{1}{2}(fx_{1}/z_{1} + fx_{2}/z_{2})$; and subtracting one from the other, the lateral positions factor straight out. What is left is that the drift vector is the drawn segment itself, scaled:

$$\frac{\text{drift}}{\text{span}} = \frac{|z_{2}-z_{1}|}{2\,(z_{1}+z_{2})}.$$

The focal length is absent. The principal point is absent. Where the segment sits in the frame is absent, and so is which way it runs. Only the two depths enter, and only through the ratio of their difference to their sum.

That is checkable against the number the figure reports. The test segment runs from world $z = 4$ to $z = 14$, and the camera stands seven metres back of the origin, so the depths that matter are 11.06 m and 21.05 m rather than 4 and 14 — a distinction the drift is sensitive to and the world coordinates hide. The expression then gives $9.99/64.21 = 15.56\%$, which is the step the sweep finds. One line of algebra, no camera parameters, and the boundary the whole loosening figure is about.

Three consequences follow, and the third is the one worth carrying.

The scene-dependence has a law rather than a direction. A segment from 100 m to 110 m drifts by 2.4%; from 1,000 m to 1,010 m, by 0.25%. Writing the ratio as $r = z_{2}/z_{1}$ makes it $(r-1)/2(r+1)$, so a scene whose near and far ends differ by less than 4% of their distance drifts under one per cent — which is the precise version of the claim that a distant shallow subject photographs like a parallel projection.

The drift is bounded above by one half. As $r$ grows without limit the expression rises to $\tfrac{1}{2}$ and never reaches it. So no pinhole picture of anything can drift a midpoint by more than half the drawn length of the segment it sits on, however deep the scene or however close the near end. The sweep in the figure runs nine decades of tolerance and could have run ninety: past 50% every system on the table is admitted, for every scene, and the column stops distinguishing anything at all.

And that fixes what the loosening actually measures. The exclusion is not a claim that can be loosened arbitrarily far in an interesting way — it has a ceiling at 50% and a floor at zero, and the whole informative range of the tolerance lies between them. Where a particular system’s step falls inside that range is a fact about the scene it was tested on, not about the system. The honest reading of the figure is therefore that it locates the boundary for this segment, and that the boundary for any other is one arctangent-free division away. That is a considerably stronger result than a single step at 15.6%, and it is the kind the field’s closing essay asks for: a column whose value can be predicted for a case nobody has drawn.

The columns nobody looks at #

The last three columns are less discussed and each contains a small surprise.

Diminution nearly tracks the centre and does not quite. Every row with a centre diminishes, because diminution is the divide by depth and a centre is what forces the divide. The converse fails, on exactly one row: the handscroll diminishes and has no centre, because across its roll it is an ordinary perspective and along the roll the eye moves, so it divides by depth everywhere while having no single point that all its rays pass through. The implication runs one way. That is worth knowing because it says the trade is not symmetric — a station point buys convergence and size-with-distance, and a system can buy those without paying for a station point, which is precisely what the scroll does and what makes it the row worth reading twice.

Bounded depth does not track anything. Perspective is bounded, which is what a horizon is. Elevation is also bounded, for a completely different reason — it has no depth axis at all, so infinite depth lands nowhere rather than at a limit. The handscroll is bounded across its roll and unbounded along it, and the table reports the across-roll answer. Three rows, three unrelated mechanisms, one column: this is the cell where the table is least informative and it is left in because the alternative is to leave the question out.

Straight lines separates one row from all the others, and it is the scroll — the row the next essay uses to argue that a column with one dissenter is barely a column — for a reason that has nothing to do with the surface. Every other row here draws on a flat plane with an ideal projection, so straightness is free; the scroll’s eye moves.

How the cells are actually computed #

It is worth being concrete about what “read out of the system’s own map” means, because it is the difference between this table and every other comparison of drawing systems.

measureSystem takes two things: a function from a world point to a page point, and — optionally — a function from a page point to the ray it stands for. It knows nothing else about the system. No name, no family, no declared properties.

The centre comes from handing fifty-two rays to closestPointToRays, the solver lib/refraction.js wrote to ask whether a picture through water has a station point. A parallel system’s rays are all one direction, so the solve is singular and the solver refuses — which is caught and recorded as an infinite miss, because refused and not measured are different answers and the exclusion counts rows.

Measure comes from midpointDrift, lib/scene.js’s function, on a segment from 4 m to 14 m.

Diminution is the drawn height of a 1.7 m object at 6 m and at 48 m.

Bounded depth compares the image of a ground point at $10^8$ m with one at $10^{10}$ m — two decades apart, so a linear depth map cannot pretend to converge.

Straightness samples a receding line and measures the sag from its own chord.

Every one of those is a function this site already had, written for a different field, applied to a new argument. That is the property that makes the table a measurement: a cell cannot be wrong in a way that a figure would hide, because no cell was written down.

What the table cannot say #

Three limits, stated because a table invites more confidence than it earns.

It has one row per system and some systems are two things. The scroll is the clearest case; its row is a summary of two different answers and the summary is the less flattering of them.

It cannot represent a composite. An aspective figure is not one map applied to a scene; it is several maps applied to different parts of one object. Each of its parts is an orthographic projection and appears in the last row; the assembly is not a row at all, because the table’s unit is a map and the assembly is not one.

And the questions were chosen. Five properties, picked because this site had already built machinery to measure them, and a different five would produce a different table. The next essay is about that — what it takes for a comparison like this to be a result rather than a construction, and what test the choice of columns has to pass.

What would break it #

A useful last question for any table is what would have to be true for it to be wrong, and here there are three answers of decreasing severity.

A system with both a centre and true measure would break the exclusion outright. None exists, and the arithmetic above says why none can: a finite centre divides by depth and the division is what destroys the ratio. A counterexample would have to be a projection through a finite point that somehow does not divide, which is a contradiction rather than an unexplored possibility.

A system this table cannot represent would limit its scope, and one already exists — the aspective composite, which is several maps rather than one. That is a real limitation and it is stated rather than hidden. The table’s claim is about maps, and a composite is not one.

And a badly chosen column set would make the pattern uninformative, which is the failure the next essay is about and the one that actually applies. The exclusion survives it, because the exclusion is a theorem about two of the columns and not an observation about all five.

So the table is wrong in no way that has been found, incomplete in one way that is named, and weaker than it looks in a way the next essay measures.