Several traditions draw the floor from one place and the people on it from another. This site has said no single camera produces both. What such a picture actually is has a measurement attached: give the rays their world points and ask for the one place they all pass through, and at a stride of separation the best answer misses them by six tenths of a metre.
A Persian miniature draws the carpet as though from above and the figures standing on it as though from in front. This site has already measured what that asks for: two optical axes exactly ninety degrees apart, and no single camera has both.
Fig. 1 The convention, and the contradiction in it. A camera on the carpet’s normal images it as a true square and images a standing figure as a mark whose length is its distance from the point under the eye.
That is a refusal. It says what such a picture is not. It does not say what it is, and the answer turns out to be an ordinary object with a number attached.
The first thing needed is a place to draw on that is not either eye’s coordinate system.
Everywhere else on this site a picture is a camera’s pixels. That will not do here, because two eyes cannot share a set of pixels that belongs to one of them. So the picture plane is put in the world: a real plane in the room, at a stated distance along the axis, and every centre draws onto it by the same rule — join the centre to the world point and see where the join crosses the glass.
Fig. 2 The arrangement. The rectangle is the picture plane, a plane in the room rather than anybody’s pixels, and both eyes draw onto it.
The check that makes this comparable with the rest of the site is that drawing from the camera’s own eye reproduces the camera’s own projection routine to the last bit, at two different depths of the plane — which also says the depth of the plane is a free choice that changes nothing, as the plane is a choice requires.
Fig. 3 The result that entitles the picture plane to be moved. One eye and two picture planes give a homography of the picture for any scene whatever, so where the glass is put is a decision about coordinates.
Now draw a room. The floor and the near box from the first eye; the two far boxes from a second eye a stride away.
Fig. 4 The result. The faint outlines are where the first eye would have put the far boxes; the drawn ones are where the second eye put them, and the two answers about one world point differ by up to nine hundred pixels.Fig. 5 The two eyes closer together. The seam narrows and the picture looks more and more like a photograph.Fig. 6 And at zero separation, which is the control. One eye, an ordinary photograph, and the figure prints a viewing distance rather than saying it has none.
The last of those three is the reason the other two mean anything. A generator that produced a two-centre picture at every setting could not distinguish its own defect from its own machinery; one whose slider passes through the ordinary case can.
Here is the question worth asking: how far is this picture from having a centre?
It has an answer, and the answer uses machinery this site already had. Take every drawn mark together with the world point it is a mark of, and join the two — that is a line in space. A picture with a centre is a picture whose lines all pass through one point. So fit the best point and report how far it misses.
Fig. 7 The miss, against how far apart the two eyes are. At a stride and a half of separation the best point misses the rays by nine tenths of a metre on average, and the miss falls to nothing as the eyes come together.
That solver is closestPointToRays, and it is on its fourth object. It measures how far a refracted picture is from having a centre; how far a handscroll is; how far a curved mirror’s bundle is; and now this. Using the same solver is deliberate, because it makes the millimetres comparable across four different reasons for a picture to have no viewpoint.
Fig. 8 Its first use: rays continued into water, missing their own least-squares centre by millimetres. A refracted picture is not a projection of anything from anywhere, measured the same way.Fig. 9 And its third: a handscroll’s section, whose rays miss by metres because the eye translates. Same solver, different reason, comparable number.
Three things move it, and separating them is what makes it a measurement rather than a number.
The separation of the eyes, which is the plot above and is nearly linear over the range drawn. Doubling how far apart the two centres are roughly doubles the miss.
The depth spread of the scene. The miss is a parallax, and parallax needs depth: a scene entirely on one plane produces no miss at all whatever the two eyes do, because a plane seen from two points differs by a homography of the picture and a homography is a perfectly good picture from one point.
Fig. 10 The reason. Two eyes looking at one plane differ by a map of the picture, so a flat scene cannot record how many eyes drew it.
And which points are handed to which eye. The measurement is not a property of the two centres alone; it is a property of the composite, and a division that gives the second eye only far-away things produces a smaller miss than one that splits the scene near the camera.
Fig. 11 The second eye moved mostly upward rather than sideways. The miss is a different size because the displacement is along a different direction relative to the scene’s own depth.
The second of those is the one to carry, because it is a genuine limitation on the whole enterprise: a picture of a flat thing cannot be caught at having two eyes, however far apart they are. Every measurement here needs depth in the scene to work with.
Fig. 12 The same fact from the panorama side: rotating a camera about anything but its entrance pupil leaves parallax between frames, and the parallax depends on the depth of what was photographed. No depth, no parallax, nothing to detect.
The other three are ways for a picture to lose its centre by accident. The medium bends the light; the eye moves during the exposure or along the roll; the reflector curves. In every one of them somebody wanted a projection and the physics did not supply one.
This one is different. The picture has no centre because it was drawn from two on purpose, and the traditions that do it are doing it for a reason — a floor that can be measured with one ruler, a figure whose parts are each seen from the direction that identifies them.
Fig. 13 The five systems this site models, each a stated map from a world to a page. Two of them are two-centre pictures once they are looked at this way.Fig. 14 And what each buys with what it gives up, on one battery. A centre and a true measure are exclusive, and a system that wants both takes two centres.
The measurement is therefore not a criticism. It is a size: the miss is how much geometry the convention is spending, and a convention that spends a metre of it is making a bigger claim than one that spends a centimetre.
Fig. 15 The miniature’s own arrangement run through the instrument. Carrying the second eye up toward overhead — which is what the convention’s ninety degrees asks for — the composite’s rays miss their best point by more than a metre.
Nothing above needs the number two. The construction takes a list of parts, each with its own centre, and draws every part from its own — so a picture assembled from a dozen viewpoints is the same object with a longer list.
That matters because it puts two of the five systems this site models on one axis rather than in two categories.
A handscroll is drawn by an eye that translates continuously, imaging one column at a time. That is not two centres; it is a centre for every column, which is a continuum of them. Read through this instrument it is the limiting case of the composite picture, and the miss it produces has a closed form — the spread of the eye’s own track — which is exactly what a continuum of centres spread along a line should give.
Fig. 16 The scroll’s arrangement in plan. Every column is drawn from a different point of a track, and the track’s own length is the miss.Fig. 17 And the consequence in the picture: the midpoint survives along the roll and not across it, because there is one eye per column and no eye for the whole.
An aspective figure is the other extreme: a small number of centres, each owning a named part, chosen so that each part is seen from the direction that identifies it.
Fig. 18 The aspective arrangement measured as an optimisation. Any single viewing direction keeps at most the square root of what several directions keep between them, which is the reason for taking more than one.
So the four-eyed miniature, the continuous scroll and the composite figure are three points on one scale — how many centres, and how far apart — and this instrument reads all three.
There is a natural expectation that a two-centre picture has a visible join, and it is half right.
The disagreement between the two eyes about one world point is real and can be large — nine hundred pixels here. But it is only visible where both eyes drew something. In the picture above each world point is drawn once, by whichever eye owns it, so there is no double image anywhere and nothing on the sheet is inconsistent with itself.
Fig. 19 A wide separation. The far boxes sit in the wrong place relative to the floor and neither of them is drawn twice, so nothing in the picture contradicts anything else in it.
What does show is a relational wrongness: the boxes stand somewhere the floor does not put them. That is the same complaint a viewer has about the miniature — the figures do not stand on the carpet in a way the carpet agrees with — and it is a complaint about the arrangement rather than about any mark.
Fig. 20 A seam of the other kind, for contrast: one drawn on purpose where two receiving surfaces meet, which is a fact about the receiver rather than about the number of eyes.
One more thing has to be checked before any of the numbers above are quotable, and it is the check every use of this solver on this site carries.
With one eye, the fit has to land exactly. The same points, the same picture plane, the same least-squares routine, with both halves drawn from the same centre: the rays meet to eight parts in ten thousand million million of a metre, which is the arithmetic floor.
Fig. 21 The control drawn rather than described. One eye, an ordinary photograph, and a fit that returns the eye.
Without that, the metres above would be indistinguishable from the solver’s own noise on a badly conditioned bundle, which is not a hypothetical worry — the rays of a narrow-angle picture are nearly parallel, and a least-squares point fitted to nearly parallel rays is exactly the kind of quantity that comes back large for no reason.
Fig. 22 And the shape of that worry in the two-view case, where two rays that should meet miss by millimetres for reasons that have nothing to do with the geometry being measured.
It says nothing about why a tradition would do this. That is a question about pictures and people and this site’s ruling on the conventions field is explicit: what is modelled here is a map from a world to a page, and every claim survives with no painting, no period and no object named.
It says nothing about how a viewer reads such a picture. The miss is a fact about rays; whether a person notices is a fact about seeing.
And it does not yet say what the picture is. A number for how far from having a centre it is, is not the same as an account of what it is a picture of — and the answer to that is the next rung, and it is more surprising than the measurement.
Worth separating before the next rung, because the two are easy to run together.
Established. A picture drawn from two centres has no centre of its own, and how far it is from having one is a number in metres of the world. The number needs the scene — the rays are joins of drawn marks to world points — and it needs depth in the scene, and it is zero when the two centres coincide.
Not established. That such a picture is wrong, or that it is not a projection of anything at all. Those are different claims and the second one is false, which is the finding of the next rung and is the reason this one stops here.
Fig. 23 The battery this site measures a drawing system on. Nothing in it asks whether a system is right; every entry asks what it keeps.
The distinction matters because “no centre” reads as a defect and is not one. A parallel projection has no centre either — its centre is at infinity, which is to say it has none in the room — and nobody regards an isometric drawing as a failed photograph.
Fig. 24 The system with no centre that nobody complains about. What it gives up is diminution and what it buys is a drawing a ruler can be used on.
One design decision underpins every measurement here and it is worth defending, because the obvious alternative fails in a way that would not have been visible.
The obvious alternative is to draw both halves in one camera’s pixel coordinates: project the near half with camera A, project the far half with camera B, and put both sets of pixels on the same canvas.
That produces a picture, and it is the wrong picture. Camera B’s pixels are its own — its principal point is at the centre of its frame, its focal length scales its image — so pasting them onto camera A’s canvas silently applies a similarity that has nothing to do with the geometry. The result depends on the two cameras’ frame sizes rather than on where the two eyes were.
Fig. 25 The quantity that would have leaked in. A focal length is not an angle until a frame of stated width is named, so two cameras’ pixels are not comparable without one.
Putting the plane in the world removes the question. There is one sheet of glass, both eyes draw on it, and the pixel coordinates are read off it afterwards by one map — camera A’s, chosen arbitrarily, and the choice affects nothing because it is applied to the whole picture at once.
Fig. 26 The result that says the choice is free: one eye and two picture planes give a homography of the picture, so which plane is used is a change of coordinates rather than a change of picture.
That is the same reason this collection insists on projecting from a stated camera rather than constructing by eye, applied one level up: the coordinates a measurement is made in have to be a choice that provably does not enter the answer.
A refusal becomes a measurement the moment the same instrument that fails is asked how badly. “No camera does this” and “the rays miss by 0.63 m” are different kinds of statement, and only the second can be compared with anything.
This site has made the first kind of statement about five drawing systems and about three optical arrangements. The second kind is what let the scroll’s miss be compared with the water’s, and it is what lets a convention’s departure from a projection be put on the same axis as a physical one.
The next question follows straight from having a number: a miss of a metre is a metre of something. The rays are not passing through one point, so they are passing through a family of points, and asking what that family is turns the whole arrangement into an object with a name.