A picture with two eyes in it

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.

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.

A picture plane that belongs to nobody #

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.

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.

The picture #

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.

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.

The measurement #

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.

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.

What the miss depends on #

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.

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.

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.

The miss is bounded by half the separation #

The plot is described as nearly linear in the separation, and the linearity has a coefficient and a ceiling, both of which follow from what the solver is actually being asked.

The bundle is a mixture: some rays pass through the first eye and the rest through the second. Any candidate point $P$ on the segment joining the two centres sits a distance $t|\mathbf{s}|$ from one and $(1-t)|\mathbf{s}|$ from the other, and its perpendicular distance to a ray through a centre is that distance times the sine of the angle between the ray and the segment. Minimising over $t$ with the two halves equally represented puts $P$ at the midpoint and gives

$$\text{miss} \approx \frac{|\mathbf{s}|}{2}\,\langle\sin\vartheta\rangle_{\text{rms}},$$

with $\vartheta$ the angle each ray makes with the line joining the two eyes.

So the miss is linear in the separation, exactly, and it can never exceed half of it. The sine is at most one, so a picture drawn from two eyes a stride apart cannot miss its own best centre by more than half a stride however the scene is arranged. That is a ceiling rather than a tendency, and it is worth having because a number quoted without one invites the reading that a badly assembled picture could miss by anything.

The reported nine tenths of a metre at a stride and a half of separation sits close to that ceiling, which says something about the arrangement: the rays are mostly perpendicular to the eyes’ displacement, which is what happens when two eyes are displaced sideways and the room is ahead of them. Displace the second eye along the line of sight instead — forward rather than across — and the sines collapse toward zero and so does the miss, however far the eye moves.

Three readings follow, and they sharpen the section above rather than replacing it.

The bound is what makes the refusal a small claim. A miss that cannot exceed half the separation is a miss whose size is fixed by an arrangement rather than by a defect, so the honest reading of the number is not “this picture is badly broken” but “this picture was drawn from two places that far apart” — which is the shear’s own reading arrived at before the shear is constructed, and the reason the next rung can absorb it at all.

The angular factor is what “which points go to which eye” is measuring. Giving the second eye only things far off to one side raises the mean sine; giving it only things straight ahead lowers it. The split matters through the geometry of the rays it selects, and not otherwise.

And the separation direction matters as much as its size. A second eye a metre above and a second eye a metre to the side are the same $|\mathbf{s}|$ and different $\langle\sin\vartheta\rangle$, which is why the figure with the eye moved upward reports a different number for the same displacement. On this collection’s own conventions that is the difference between a two-ground carpet, whose second eye rises, and a two-centre panel whose eyes are level.

And it places the four objects on one axis honestly. The miss for this arrangement is bounded by the eyes’ own separation; a scroll’s is bounded by the spread of its track, which is the same statement with a continuum of eyes rather than two; a curved mirror’s is bounded by the spread of its virtual sources; and a refracted picture’s is a fixed fraction of the standoff. Four objects, four bounds, and in every case the miss is half the spread of wherever the rays actually came from. That is not a coincidence — it is what a least-squares fit to a bundle of rays leaving a distribution of points does, and it says the solver is measuring the spread of the origins rather than anything about the scene.

Which is the reason the same solver serves all four, and the reason the numbers are comparable. What it cannot report is why the origins are spread, and the four reasons are the four essays.

Which of the four reasons this is #

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.

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.

Two centres, or a hundred #

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.

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.

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.

The seam, and where it is not #

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.

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.

The instrument’s own control #

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.

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.

What this does not say #

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.

What has been established, and what has not #

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.

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.

Why the picture plane had to be in the world #

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.

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.

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.

The transferable form #

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.

That family has a name and a geometry of its own, and it is where two rays that do not meet takes the same arrangement next: a miss measured in metres is the first sign that the two bundles belong to two centres rather than one, and the shape of the miss says which.