The miniature convention wants its floor drawn from overhead and its figures drawn from in front, and the two optical axes it asks for are exactly ninety degrees apart. Read as a picture with two centres rather than as a picture with none, the arrangement stops being a contradiction and becomes a quantity: the rays of the composite miss their own best point by more than a metre, and the absorbed reading is a floor that leans.
A carpet is drawn as a true rectangle, seen from directly above. The people standing on it are drawn at full height, seen from in front. Both halves are correct pictures of what they show, and no camera in the room produces the pair.
Fig. 1 The convention and the contradiction. A camera on the carpet’s normal images it as a true square to the arithmetic floor; the same camera images a standing figure not as a shortened mark but as a mark whose length is its distance from the point under the eye.
That was measured here as a refusal: the two requirements are individually satisfiable and jointly impossible, and the two optical axes they ask for are exactly ninety degrees apart. This rung asks what the picture is instead.
The instrument is the one built for pictures with more than one centre. Give every drawn mark the world point it is a mark of, join the two, and ask for the point all the lines pass through.
Fig. 2 The convention’s own arrangement, run through it. Carrying the second eye up toward overhead — which is what the ninety degrees asks for — the composite’s rays miss their best point by more than a metre.
So the refusal has a size. That is worth more than the refusal, because a size can be compared: the same solver on a picture through water gives millimetres, on a handscroll gives metres of track, and on a curved mirror ball gives millimetres of the world.
Fig. 3 Millimetres, for a picture made through a refracting interface.Fig. 4 And metres, for a picture drawn by an eye travelling along a track. Same solver, same units, comparable numbers.
Put on that axis, the two-ground convention is at the far end. It is not a small departure from a projection; it is one of the largest this collection has measured, and it is deliberate.
The convention is not attempting a photograph. What it buys is stated exactly and it is worth restating, because the size of the miss is the price and this is the goods.
A floor a reader can measure with one ruler. Drawn from overhead, the carpet is a true rectangle at a uniform scale, so its pattern is readable, its proportions are true, and anything laid on it can be located by measurement rather than by inference.
Fig. 5 The same purchase in a different convention, measured: what a removed roof buys is exactly this, and the price is the same price.
And figures whose heights are comparable. Drawn from in front, a standing figure has a drawn height that means something. Under the overhead camera it has none — the plan view does not shorten a figure, it replaces its height with its distance from the point under the eye, which is a peculiar and severe exchange.
Fig. 6 That exchange, computed. A figure standing under the eye images to nothing at all whatever its height, and two identical figures at different radii are drawn different lengths.
So each half of the picture is doing something a single camera cannot do while also doing the other, and the convention is a decision about which two things to have rather than a failure to have three.
Fig. 7 The whole comparison on one battery. A centre and a true measure are exclusive, and every system in the table gives up one of them.
The description matters as much as the number, and it is a change of frame rather than a new result.
Said as “this picture has no centre of projection”, the convention is a negative fact and sits with the refracted picture and the scroll — three arrangements in which something went wrong with a projection.
Said as “this picture has two centres”, it is a positive fact with a structure. There are two eyes, they are in stated places, each half is an ordinary projection from one of them, and the whole is a composite. Nothing failed; something was assembled.
Fig. 8 The positive description drawn: two eyes, one sheet of glass, and each part of the room drawn by whichever eye owns it.
The second framing is the one that leads anywhere, because it supports the question the next section asks — what single-eye picture is this the same as — and the first framing does not.
Fig. 9 And it puts the convention among the others rather than among the optical failures. Each of these is a stated map from a world to a page; two of them turn out to be composites.
Take the second eye away and the picture is a projection from the first eye of a sheared scene. What has the shear done here?
The second eye is displaced almost entirely upward — that is what “overhead” means — so the shear runs vertically and grows with depth. The absorbed reading is therefore a floor that is not horizontal: it leans away from the viewer, rising as it recedes, and the figures standing on it stand at angles.
Fig. 10 The absorbed reading of an upward displacement. The map is the same map as for a sideways one and the direction is different, so the room comes out tipped rather than skewed.Fig. 11 Both rooms together. Parallel walls are still parallel and every midpoint is still a midpoint, because the map is affine; the corners are out of square by tens of degrees.
That is a satisfying answer because it is the answer this collection already gave to a related question. An inverse perspective is a leaning plane finds that a divergent construction is a correct picture of a plane that leans. The two-ground convention is the same kind of statement one dimension larger: it is a correct picture of a room whose floor leans.
Fig. 12 The earlier result. A drawing that reads as a mistake is a correct picture of a nameable arrangement, and naming the arrangement is what turns the criticism into a description.
The tilted-plane reading, and why it is not quite enough #
There is a tempting shortcut here, and it fails in an instructive way.
Since the absorbed reading is a leaning floor, why not say the convention simply draws a tilted ground plane from one eye and be done with it?
Because the figures. A leaning floor drawn from one eye foreshortens the figures standing on it, and the convention does not foreshorten them — it draws them at full height. A single camera looking at a tilted floor gets the floor’s proportions right in exchange for the figures, which is the trade it was refusing to make.
Fig. 13 The convention at a shallower tilt, where the two demands conflict less and the floor is less fully measurable. The trade is continuous and the endpoints are the two single-camera answers.
So the absorbed reading tips the floor and shears the figures, and the figures come out leaning with it. A room in which the floor slopes and everything standing on it leans by the same amount is a perfectly consistent room, and it is not the room anybody had in mind.
Fig. 14 Which is what the cost measurement says. The absorbed scene keeps flatness and midpoints exactly and loses angles by tens of degrees, and the leaning figures are those lost angles.
The convention’s whole purchase is that the floor can be measured, so it is worth asking whether the absorbed reading keeps that.
It does, and for a reason that is now familiar: the absorbing map is affine, so parallel stays parallel and every ratio along a line survives. The carpet in the absorbed room is still a parallelogram with its pattern in the right proportions along each of its two directions; what has changed is that it is no longer a rectangle and no longer horizontal.
Fig. 15 The quantities the map keeps and loses. Midpoints exactly, flatness exactly, angles not at all.
So a reader measuring along the carpet’s own directions gets true ratios in the absorbed reading as well as in the convention’s own reading, and a reader measuring an angle gets a different answer from each. That is a clean statement of what the convention delivers: ratios along the floor’s own directions, and nothing about angles or about where anything stands relative to anything else.
Fig. 16 The quantity that survives, in its simplest form. A parallel projection keeps a midpoint and a central one does not, and an affine map of space keeps it before any projection happens.
Ninety degrees is what the convention asks for, and it is worth asking whether anything about the geometry singles it out.
Nothing does. The miss grows smoothly as the second eye is carried up, with no feature at any angle, and ninety is where it ends because that is where “from directly above” is. What ninety degrees does supply is the extreme of the trade: a plan view is the one direction in which the floor’s proportions are exactly true and a standing figure’s height is exactly unavailable.
Fig. 17 The sweep, with no feature anywhere along it. The convention’s ninety degrees is the end of the range rather than a distinguished point in it.Fig. 18 And the general version of the same trade, from the aspective convention: a single viewing direction keeps at most the square root of what several directions keep between them, so the reason for taking more than one is arithmetic rather than stylistic.
That is a mildly deflating finding and it is the honest one. The right angle is a choice about what the picture is for, and the geometry prices it without preferring it.
Where the two eyes have to be, for this to be the convention #
One more thing the instrument can settle: how far apart the two eyes have to be before the picture reads as the convention rather than as a slightly odd photograph.
The answer is not a distance in metres, because what matters is the angle between the two optical axes rather than the gap between the eyes. A second eye a stride to the side produces a picture with a modest seam; a second eye directly overhead produces the convention. And “directly overhead” is a limit rather than a place: as the second eye rises, the floor’s drawn shape approaches a true rectangle and the figures’ drawn heights approach the plan view’s replacement of height by radius.
Fig. 19 The convention nearer its limit, where the floor is more nearly a true rectangle and the trade is more nearly complete.Fig. 20 And well short of it, where the picture reads as an ordinary steep view of a floor with people on it. Nothing discontinuous happens between the two; the convention is one end of a continuum.
That is a useful thing to know about every convention this collection models. None of them is a discrete alternative to perspective. Each is a point on a range whose other end is an ordinary projection, and the numbers move smoothly all the way along.
Fig. 21 Which is why the comparison is a battery of measurements rather than a list of categories. Every entry is a number and every number has a limit at one end of its own range.
It says nothing about what any tradition meant, why it drew this way, what it was for, or how it was taught. The convention is named here as the source of a rule being modelled, and everything computed is a computation on a map from a world to a page. This collection has no standing on the rest and does not take any.
It says nothing about how a viewer reads the result. The miss is a fact about rays.
And it does not say the composite is unreadable or wrong. The absorbed reading is a consistent room and every part of the picture is a correct projection of something; what is not correct is the relation between the parts, which is exactly the quantity the shear moves.
Fig. 22 The battery again, which asks what each system keeps rather than how far each falls short of a photograph. Nothing in it has a column for correctness.
The absorbed reading is not one room. Composing the absorbing map with anything that already holds the picture plane and the first eye still gives another map, so what the convention is a picture of is a one-parameter family of rooms rather than a room.
Fig. 23 Three members of that family, more than a metre and a half apart at the widest, all three drawing the identical picture from the one remaining eye.
That is worth stating because it forecloses a reading that would otherwise be tempting: that the convention depicts a leaning floor, in the sense that a particular tilt could be read off it. No tilt can. Every member of the family has a different floor at a different angle, and the picture is silent about which.
Fig. 24 The nearest relative in this collection: two parallel views leave a one-parameter relief, and every member of it draws the same two pictures.
So the honest sentence is that the convention is a picture of some room with a leaning floor, from one eye, and that nothing in it names which. Which is the standing answer to every question this collection asks about a single picture, arriving at a convention rather than at a photograph.
Two things this measurement is silent on, and both are worth naming because a number invites over-reading.
It cannot say how many eyes. The composite here is modelled with two, and a picture drawn with four, or with a continuum along an arc, produces a miss of the same kind. The miss measures how far the rays are from meeting, not how many bundles they came in.
Fig. 25 Two of these are composites and the instrument reads both the same way. Distinguishing them needs the parts to be identified, which is a decision about the picture rather than a measurement of it.
And it cannot say where the eyes were. The fit returns one best point and its residual, and the two actual centres are nowhere in the output. Recovering them would need the assignment of marks to eyes — which mark was drawn by which — and that assignment is exactly what a composite picture does not record.
Fig. 26 The same limitation on a scroll, where the eye’s track is known by construction and cannot be read back out of the picture’s own rays.
So the number is a measure of departure and not a reconstruction. That is the usual position for a residual, and stating it stops the metre from being read as a claim about the arrangement rather than about the disagreement.
A convention that no camera reproduces is not a picture of nothing. It is a picture of something else, and the something else is reachable — which turns a refusal into a description and a description into a price.
The measurement here went in three steps and each was available only after the one before it. First a refusal: no camera has both grounds. Then a size: the composite’s rays miss by more than a metre, on the same instrument that measures water and scrolls. Then a description: the picture is a projection of a room whose floor leans and whose figures lean with it, from one eye, exactly.
The third step is the one worth taking away, because it is the one that makes the convention comparable with the systems that do have a centre. A drawing system with no centre and a drawing system with a leaning floor are two descriptions of one map, and the second one can be put in the same table as everything else.