A picture drawn from two eyes is a picture drawn from one, of a different room. The map that puts the second eye away holds the picture plane still point by point and carries one centre onto the other, and the member of its family that matters turns out to be affine — a shear along the line joining the eyes, growing with depth, which is the same operation an oblique drawing performs.
The previous rung ends with a number: a picture drawn from two eyes has rays that miss their own best point by six tenths of a metre. That is a measurement of a failure.
This one is about the failure not being one.
Fig. 1 The thick outlines are the far boxes as the second eye drew them. The thin ones are the first eye’s picture of a moved copy of those boxes, and the two lie on top of each other to three parts in a million million of a pixel.
There is a projective map of space with two properties, and it has three lines of algebra in it.
It holds the picture plane still, point by point. Not “maps the plane to itself” — every point of the glass is fixed exactly.
And it carries the second eye onto the first.
T=I+μOπT
is an identity plus one outer product. Any point of the plane satisfies πTX=0 and is therefore fixed entry for entry; O on the line joining the two centres and the right scalar μ does the rest.
Then, for every world point,
draw(C2,X)=draw(C1,TX)
and the reason is two lines. The line C2X meets the glass somewhere; T fixes that meeting point and carries C2 to C1, so it carries the whole line C2X to the line C1(TX), and both lines meet the glass at the same place. The drawn point is the same.
Fig. 2 Both rooms, from somewhere else. The thick boxes are where the far half really is; the thin ones are where the map puts them, and the first eye’s picture of the second set is the second eye’s picture of the first set, mark for mark.
So the two-eyed picture is a one-eyed picture of a different scene. Nothing in the picture can object, because the picture is identical.
Fig. 3 A wider separation. Without the map the same first eye would have drawn those boxes fifteen hundred pixels away; with it, the agreement is at the arithmetic floor.
Which is why the picture prints a viewing distance #
This site’s one non-negotiable piece of furniture is the strip that says what distance a figure is correct from, and its second half is the strip that says a figure has none. The previous rung’s figures print the second — a two-centre picture is a projection of that room from nowhere.
The figures here print the first, and that is the argument rather than the furniture.
Fig. 4 The two-centre picture, which says it has no viewpoint and names the metres by which its rays miss.Fig. 5 The same picture, once the second eye has been absorbed into the scene. It is a projection from one point again, so it states the distance it is correct from — and every mark is where it was.
Two figures, identical marks, opposite strips. The strip is not a property of the drawing; it is a property of the drawing together with what it is taken to be a picture of.
The family of maps that absorb the second eye is one-parameter — composing with anything that fixes the plane and the first eye gives another — and one member is distinguished: the one that becomes the identity as the two eyes come together.
That member’s centre is a point at infinity. So the map is affine, and what it does is a shear: every point displaced along the line joining the two eyes, by an amount proportional to its depth beyond the glass.
Fig. 6 The shear at a smaller separation. The displacement direction is the same at every point and the amount grows linearly with depth, which is what a shear is.
That was not the plan. The construction was written as a projective homology of space and the check that it was affine — that the bottom row of the four-by-four is untouched — came back true, which is a stronger statement than the construction was aiming at.
It also joins two things that looked unrelated. Oblique is a shear shows that the oblique drawing systems are shears of space along the viewing direction, and the shear is the whole system. Putting a second eye away is the same operation with a different amount in it.
Fig. 7 The oblique systems, whose entire content is a shear along the viewing direction. The map that absorbs a second eye is a member of the same family.Fig. 8 And another appearance of the same object: a splayed construction read as a shear rather than as a mistake.
An affine map keeps parallelism, midpoints, and the ratio in which a point divides a segment. It does not keep angles or lengths.
So the room the picture is equally a picture of is a room with parallel walls still parallel, with the middle of every wall still in the middle, and with its corners out of square.
Fig. 9 The three quantities, swept. A right angle comes out up to forty-nine degrees from square; the midpoint of every edge is still the midpoint to the arithmetic floor; and the control — another member of the family, which is projective rather than affine — moves a midpoint by nearly seven per cent.
The control is the half that makes “affine” a measurement rather than an assertion. A different member of the same family absorbs the same eye and draws the identical picture, and it moves midpoints. So keeping them is a property of the member rather than of the problem, and the member had to be identified before the claim could be made.
Fig. 10 The quantity being watched. A parallel projection keeps a midpoint and a projection through a centre does not, and an affine map of space keeps it in the world before any projection happens.
A shear has a rate, and it is worth having because it says how much of a distortion the absorption costs before any angle is measured.
The displacement is proportional to depth beyond the picture plane, so the constant of proportionality is a pure number: how far a point moves, per metre of depth, in units of the separation between the eyes. Measured on the arrangement above it comes to 1.14 per metre — so a box three metres beyond the glass is displaced three and a half times the distance between the two eyes.
Fig. 11 The same shear at a wider separation. The rate is a property of where the glass is rather than of the separation; what the separation sets is the direction and the overall size.
Two consequences follow and both are worth having.
Deep scenes are distorted more. The absorbed reading of a room whose far wall is thirty metres away moves that wall ten times as far as it moves something three metres away, so a picture with a lot of depth in it is a picture whose second-eye reading is a very strange room.
And the glass’s position sets the rate. Move the picture plane further from the eye and every depth beyond it is smaller, so the shear is gentler — which is a reminder that the picture plane is a coordinate choice and that any quantity depending on it is a quantity about the coordinates.
Fig. 12 The result that licenses moving the glass. One eye and two picture planes give a homography of the picture for any scene, so the drawing is unchanged and the numbers describing the map are not.
The one number that does not depend on the glass is the picture, which is the point.
Fig. 13 Three members drawn together, more than a metre and a half apart at the widest, all three drawing the identical picture from the one remaining eye to eight parts in ten million million of a pixel.
So the honest statement is not that a two-eyed picture is a picture of one particular other room. It is that it is a picture of a one-parameter family of them, and the picture has nothing whatever to say about which.
That is this site’s usual shape of answer and it arrives here in a usual place. Every recovery leaves something free; the useful thing is what the free thing is, and here it is a one-parameter group of maps of space fixing a plane and a point.
Fig. 14 The nearest relative, from the parallel field: two parallel views leave a one-parameter relief, and walking along it changes every depth and no drawn mark.Fig. 15 And the general form, from the manyviews field: seven directions no quantity of pictures fixes, walked along to show that they are free rather than merely small.
An inverse perspective is a leaning plane finds that a divergent construction — lines spreading as they recede, which reads as a mistake — is a correct picture of a plane that leans. A drawing that looked like a failure turned out to be a projection of something else.
Fig. 16 That earlier result. The construction is not wrong; it is right about a different arrangement, and the arrangement is nameable.
This is that statement generalised from a plane to a scene and from one construction to any number of centres. A picture drawn by a rule other than projection from a point is, very often, a projection from a point of something else — and the something else is reachable by a map with a name.
Fig. 17 The same move in the parallel field: one drawn projection and the family of solids consistent with it, which is a shear family along the projection’s own kernel.
The map takes one second eye away. A picture assembled from four centres needs three maps, one per extra eye, and each of them fixes the same glass and carries its own centre onto the first.
The parts are then each transformed by their own map, so the absorbed scene is a scene assembled from differently sheared copies — which is a perfectly good scene, made of perfectly good solids, whose parts stand in the wrong relation to each other.
Fig. 18 The arrangement several conventions actually use: a floor and the things on it drawn by different rules. Read this way each part is a sheared copy and the composite is one picture from one eye.
That is a satisfying reading of the aspective figure, which draws each part from the direction that identifies it. Absorbed, it is a single projection of a body whose head has been sheared one way, whose shoulders another, and whose feet a third — all of them still solid, still flat-faced, still with parallel edges parallel, and jointed wrongly.
Fig. 19 The aspective arrangement’s own measurement, which counts how much of each part reaches the picture. The absorbed reading says what the composite is a picture of; this says what it was for.
It does not say the absorbed room is the room. It is emphatically not: its corners are out of square by tens of degrees, and if the scene contained something whose shape is known, the absorbed reading has the wrong shape.
It does not say the second eye is undetectable. It says the picture cannot detect it, which is the next rung and needs the distinction stated carefully.
Fig. 20 The distinction in one pair of numbers. Told the room, the fit refuses; told the room the picture is consistent with, the same rays meet exactly.
And it says nothing about a centre at infinity being the same case. It nearly is, and the difference is instructive: absorbing a parallel projection into a perspective one has to bring a point in from infinity, which no affine map does, so that case has no shear in it at all. It is the subject of a rung further along.
There is a reason the construction could not have been anything else, and it is short enough to be worth stating.
Two pictures are the same picture when every mark is in the same place. A transformation of space changes a picture unless it leaves every drawn mark alone. A drawn mark is where a ray crosses the glass. So a transformation that is to leave a picture alone has to leave every point of the glass alone — and a projective map of space fixing a plane pointwise is exactly a homology or an elation with that plane as axis, which is a two-parameter family before the second condition is imposed and a one-parameter family after.
Fig. 21 The plane version of the same census: a map of a plane sorted by what it holds still. The maps here are its three-dimensional relatives.
So the whole construction was forced. There was no design decision except which member of the family to call natural, and even that was settled by asking which one becomes the identity when there is nothing to absorb.
Fig. 22 The census in use, where three constructions this site had treated separately turn out to be three members of one family sorted by one ratio.
The two eyes are not symmetric in the construction #
A small asymmetry in the construction is worth pointing at, because it looks like an arbitrary choice and is not quite one.
The map carries the second centre onto the first. It could have carried the first onto the second instead, which would leave a different scene and the same picture, drawn from the other eye. So which eye survives is a choice, and there is nothing in the picture to prefer one.
Fig. 23 A small separation, where the two absorbed readings are nearly the same. As the eyes move apart the two readings diverge, and neither is more correct than the other.
What does distinguish them is which half of the picture is left undistorted. Absorb the second eye and the first eye’s half is untouched while the second’s half is sheared; absorb the first instead and it is the other way round. So a reader who trusts one half — because it contains something whose shape is known, say — should keep that half’s eye.
Fig. 24 And the reason a reader might trust one half over the other: a known shape in that half is what turns the picture into a measurement, and the half containing it is the one to leave undistorted.
That is a genuinely practical rule and it is the only place in these three rungs where the geometry has a preference. Everywhere else the family is symmetric and the picture is silent.
When a construction produces a picture no camera makes, ask what scene a camera would have to be looking at to make the same picture. The answer is usually reachable by a map that fixes the picture, and the map is the description the construction was missing.
The map that does it here is not exotic: an identity plus one outer product, whose fixed set is the sheet of glass. And the fixed set being the picture is the whole reason the absorption is undetectable — a transformation that holds the picture still cannot change the picture, and that sentence is a tautology and a proof at the same time.
The essays written alongside this one found the same shape three times over. A map of a line holds two points still, or one, or none, and which decides how a drawn row behaves. A quadric’s outline is the section by the one plane the eye’s polarity holds still. And a second eye is absorbed by the map that holds the picture still. In each of them the useful object was the fixed set, and in each of them it had been sitting there unnamed.