Two grounds, and what the second one costs

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.

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 same arrangement, with an instrument on it #

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.

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.

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.

What the second ground buys #

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.

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.

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.

Two centres, not none #

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.

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.

What the absorbed reading is #

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.

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.

How far the floor leans, and what the limit is #

“Out of square by tens of degrees” can be made exact, because the absorbing shear has a rate and the rate has a closed form. The shear’s own rung derives it: with the two eyes separated by $|\mathbf{s}|$ and standing $d$ in front of the glass, the shear’s rate is $|\mathbf{s}|/d$ and the departure from square is

$$\arctan\frac{|\mathbf{s}|}{d},$$

the angle the two eyes subtend at the picture plane. Here the separation is vertical, so what the angle measures is how far the absorbed floor leans away from horizontal.

That turns the convention into a one-parameter family with two nameable ends.

At $|\mathbf{s}| = 0$ the floor is level and the picture is an ordinary single-eye perspective. Nothing has been assembled and nothing is owed.

As the second eye rises the floor tips, by the arctangent, and the figures standing on it lean with it. A second eye one glass-distance up gives 45°; two glass-distances, 63.4°; ten, 84.3°.

And in the overhead limit the floor stands vertical. The 90° that the previous rung finds between the two required optical axes is $|\mathbf{s}|/d \to \infty$ on this scale, and the arctangent’s limit is a right angle. The absorbed reading of a fully realised two-ground picture is a room whose floor has been stood up into the picture plane.

Which is the convention’s own account of itself, arrived at from the other direction. A plan and an elevation on one sheet is exactly a floor rotated into the wall’s plane, and the shear says the absorbed reading is that rotation carried out on the room rather than on the paper. The two descriptions — “a composite with two centres” and “one eye looking at a room whose floor has been tipped upright” — are the same picture, and the second is the one a draughtsman would recognise as a presentation drawing.

It also prices the convention’s position on the field’s own scale. The miss the solver reports grows with $|\mathbf{s}|$, so a picture that leans its floor a little misses by a little and one that takes the convention seriously misses by more than a metre — which is why this arrangement sits at the far end of a scale whose other entries are millimetres. A scroll reaches metres by moving its eye continuously; this reaches them by moving one eye a long way once.

Two further readings are worth extracting, and the second is a caution.

The overhead limit is unreachable rather than merely extreme. An infinite separation is not a picture anybody draws, so a real two-ground painting sits somewhere finite on the family and its absorbed floor leans by a definite amount short of vertical. What the limit supplies is the direction the convention is heading in, and the observation that its idealised form is a presentation drawing rather than a picture at all — which is the honest description of what a removed roof is doing too, one convention over.

The lean is the whole cost. A shear is affine, so the absorbed room keeps every parallel, every midpoint and every ratio along every line; what it spends is squareness, and the arctangent is the entire bill. That is a much cheaper price than a parallel floor under a perspective room pays, where the absorbing map is projective and the ratios go too — so of the field’s two composites, the one that looks more extreme is the one with the tidier alternative reading.

And the angle is not a property of the drawing. It depends on $d$, the distance from the eyes to the glass, which is a coordinate choice rather than something a reader can see. The same marks with the picture plane taken further out describe a gentler lean, and with it nearer, a steeper one. So a reader asking “how tilted is this floor” is asking a question with a one-parameter family of answers, and the parameter is where the glass is taken to be — which is precisely why counting the eyes needs the room rather than the sheet.

The one quantity that survives that freedom is the ordering: whatever the glass’s position, the absorbed floor leans away from the viewer rather than toward, and it leans more for a higher second eye. So the convention’s direction is a fact about the picture and its magnitude is not, which is the same split an inverse perspective records for a splayed construction — the sign of the lean is intrinsic and the size of it is a statement about an assumed camera.

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.

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.

The carpet’s own measurability, checked #

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.

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.

The right angle, and whether it is special #

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.

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.

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.

What this does not say #

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.

What is left free, and it is the usual thing #

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.

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.

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.

What the instrument cannot say about it #

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.

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.

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.

The transferable form #

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.