Visual Plane
every picture states the point it is correct from
What each system gave up

A centre and a measure are exclusive

Eight drawing systems, measured on five questions, with the pinhole as a row rather than the header. Exactly one has a centre of projection and it is exactly the one with no true measure — and loosening the measure test by a hair lets it in, which is what says the boundary is real.
17 min read 9 figures What a projection destroysFitted, not assumed

Worth reading first: Parallel projection is not primitive perspective · A scroll is a camera that moves · What the removed roof buys.

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.

What each drawing system preservesEvery cell is read out of the system's own map rather than typed: the centre is fitted from its rays, measure is the site's own midpoint test, diminution is the drawn size of a fixed object at two depths, bounded depth asks whether infinite depth lands in finite page, and straightness is the sag of a receding line. Only perspective has a centre, and it is the one system with no true measure. No row has both.a centretrue measurediminutionbounded depthstraight linesperspectivehandscrollisometricdimetrictrimetriccavaliercabinetelevationfilled means the system keeps itmeasured from each system's own projectionno row is filled in both of the first two
Fig. 1 Eight systems on five questions. Every cell is read out of the system’s own map rather than typed: a centre is fitted from its rays by the same solver the refraction field uses, measure is the site’s midpoint test, diminution is the drawn size of a fixed object at two depths, bounded depth asks whether infinite depth lands in finite page, and straightness is the sag of a receding line. Exactly one row has a centre and it is the one row with no measure.

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 it is the only row 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 zz sends a world point to f x/zf\,x/z. For two points at depths z1z_1 and z2z_2, the image of their midpoint is f(x1+x2)/(z1+z2)f(x_1 + x_2)/(z_1 + z_2) and the midpoint of their images is 12(fx1/z1+fx2/z2)\tfrac{1}{2}(f x_1/z_1 + f x_2/z_2). Those agree for all xx only when z1=z2z_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 midpoint of one segment, under both familiesThe parallel projection places it exactly halfway (0e+0 px out). The perspective projection places it 22 px away from halfway, 7% of the drawn length.halfway along the drawn linethe actual midpointcorrect from 26 cm, at 160 mm wide22 px apart
Fig. 2 The exclusion’s mechanism, on one segment under both families. A parallel projection sends the midpoint to the midpoint and a perspective one does not, and the gap grows with the depth range. Everything in the first two columns of the table above is this figure, applied to eight systems.

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.

How loose the test has to be before perspective preserves measureThe count of systems the table calls measure-preserving, against the tolerance. It sits at 6 across nine decades and then steps to 8 when the tolerance passes 15.6% — the drift a real pinhole picture actually produces. The exclusion in the table above is a statement about that boundary, and this is where the boundary is.02468-8-6-4-20log₁₀ of the tolerance on midpoint driftsystems counted as preserving measureperspective admitted at 15.6%the exclusion, sweptthe boundary is measured, not chosen
Fig. 3 How loose the test has to be before perspective preserves measure. The count of measure-preserving systems sits flat across nine decades of tolerance and then steps up when the tolerance passes 15.6% — the drift a real pinhole picture actually produces on the test segment. The exclusion above is a statement about that boundary, and this is where the boundary is.

What the boundary being at 15.6% means

The step is not at 10−1610^{-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 a segment running from 4 m to 14 m at this focal length — 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.

The columns nobody looks at

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

Diminution tracks the centre exactly on this table — the one row with a centre is the one row that diminishes — and that is not a definitional link. Diminution is a consequence of the divide, which is the same divide that destroys measure, so all three columns are one column wearing three labels. That is worth knowing because it says the trade is one trade and not three: a system gets convergence, size-with-distance and a station point together, or it gets none of them.

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 — 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.

What each drawing system preservesEvery cell is read out of the system's own map rather than typed: the centre is fitted from its rays, measure is the site's own midpoint test, diminution is the drawn size of a fixed object at two depths, bounded depth asks whether infinite depth lands in finite page, and straightness is the sag of a receding line. Only perspective has a centre, and it is the one system with no true measure. No row has both.a centretrue measurediminutionbounded depthstraight linesperspectivehandscrollisometricdimetrictrimetriccavaliercabinetelevationfilled means the system keeps itmeasured from each system's own projectionno row is filled in both of the first two
Fig. 4 The same table with the scroll highlighted. It is the only row that answers no in the first two columns and the only row that bends a straight line, and its three unusual answers all come from the same fact: the eye moves while drawing. A system can fail to be a projection in more than one way, and this table has two of them in it.

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 10810^8 m with one at 101010^{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.

The rays of a refracted picture, continued into the waterEvery ray leaves the pinhole, bends at the surface and carries on. Fitted to a common point they miss it by 9.9 mm — the circle is that miss drawn at the figure's own scale. With the water removed the same fit misses by 0e+0 m.the water surfacethe pinholethe rays miss by 9.9 mmno single viewpoint — the rays miss by 9.9 mmdry control: 0e+0 m
Fig. 5 The first column’s solver, in the field it was written for. Rays continued into the water are fitted to a common point and the miss is reported; the control is the same fit with the water removed. The table above hands the same function eight different bundles, and the parallel systems’ refusal is one of its answers rather than an error.

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 each parallel system does to the three axesThe smallest axis scale is plotted; the note says how many of the three coincide and whether the system is an orthographic projection or an oblique construction. cavalier and isometric both make all three the same — but isometric is the only orthographic one that does, at 0.8165, and the oblique route gets there by drawing depth at true length.elevation0.0000two equal, orthographiccabinet0.5000two equal, obliquecavalier1.0000all three equal, obliquedimetric0.4714all three different, orthographicisometric0.8165all three equal, orthographic ←trimetric0.5479all three different, orthographicsmallest of the three axis scalesmeasured from each projection
Fig. 6 The narrower comparison this table generalises, from the parallel field: six systems on the axis scales alone. It measures one property very precisely; the table above measures five properties less precisely across eight systems, including two that field had no way to describe. Both are the same operation at different scopes.

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.

How loose the test has to be before perspective preserves measureThe count of systems the table calls measure-preserving, against the tolerance. It sits at 6 across nine decades and then steps to 8 when the tolerance passes 15.6% — the drift a real pinhole picture actually produces. The exclusion in the table above is a statement about that boundary, and this is where the boundary is.02468-8-6-4-20log₁₀ of the tolerance on midpoint driftsystems counted as preserving measureperspective admitted at 15.6%the exclusion, sweptthe boundary is measured, not chosen
Fig. 7 The one place the exclusion has a boundary rather than a proof: the tolerance at which perspective is admitted as measure-preserving. Below 15.6% it is excluded and above it is not, and the step is at the drift a real picture produces. A theorem with a tolerance in it is a theorem about that tolerance, and this is where it is.

The table’s two neighbours on this site

Straight, or conformal — the corner that is emptySix picture surfaces plotted by how much they bend a straight line (across) against how far they are from preserving shape (up). The plane sits on the left edge and the top; stereographic on the bottom and the right. Nothing sits in the bottom-left corner, and by Beltrami's theorem nothing can.straight AND conformal — empty10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰10¹10²bend of a straight line, as a fraction of its own chorddeparture from conformal (degrees, or % of anisotropy)planecylinderstereographicequidistantequal-areaequirect.sampled over a 120° fanlower left would be a surface with no cost
Fig. 8 The curved field’s version: picture surfaces on two properties, with the corner empty and a refusal that shows the boundary is about the definitions. This table is the same operation with five properties and eight systems.
The axis scales a pitch of 35.3° can reachSweeping the yaw at a fixed pitch traces one curve, not a region: the identity leaves only two of the three scales free. At this pitch the curve passes through the point where x and z are equal, which is isometric — 0.816497 against √(2/3) = 0.816497.00.2500.5000.75010.7000.8000.9001scale of the x axisscale of the z axis, at this pitchx = z at 0.8355y is fixed at 0.7771 by the pitch aloneevery point on the curve sums to 2 within 7e-16
Fig. 9 And the parallel field’s: which triples of axis scales an orthographic projection can produce, established by sampling rather than by assertion. A comparison with no columns to choose, and therefore no column-choice problem — which is the subject of the next essay.

Where this goes next

plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces No picture surface keeps everything A picture has to be cast onto something, and every candidate surface destroys something different. Six of them are measured here on the same three properties, and the corner of the plot where a surface pays nothing is empty — not because nobody has thought of one, but because a theorem says there is none. elevationx 1.000y 1.000z 0.000cavalierx 1.000y 1.000z 1.000cabinetx 1.000y 1.000z 0.500isometricx 0.816y 0.816z 0.816dimetricx 0.943y 0.943z 0.471axis scales measured from the drawingall 5 preserve midpoints Parallel projection is not primitive perspective Isometric and oblique drawing are not what people used before they worked perspective out. They are a different answer to a different question, and the difference is one measurable quantity — a parallel projection preserves the ratio in which a point divides a segment, and a perspective projection destroys it by 7% of the segment's drawn length at a comfortable depth, rising to 13% over the range the slider covers. the ratio along a receding line15.6%0% parallelthe depth range in the last tenth69.0%10% lineara square metre, near against far53.5% larger0% parallelthe floor of the far room6.2 points0 parallelwhat it costs, and what the other systems have insteadmeasured on this site's own machinerythe price of a station point What perspective gave up The field ends by turning its own battery on the system it has been comparing everything against. Four quantities a pinhole destroys that the other systems keep, each measured on this site's own machinery, and each the price of the one thing perspective has and they do not.

What links here

Computed from the collection, not written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

centre of projectionDemonstrationDiminutionDrawing systemHorizonMidpointOblique projectionParallel projectionPushbroomStation point