The second eye is a shear

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.

The map #

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.

$$T = I + \mu\, \mathbf{O}\, \boldsymbol{\pi}^{\mathsf T}$$

is an identity plus one outer product. Any point of the plane satisfies $\boldsymbol{\pi}^{\mathsf T}\mathbf{X} = 0$ and is therefore fixed entry for entry; $\mathbf{O}$ on the line joining the two centres and the right scalar $\mu$ does the rest.

Then, for every world point,

$$\operatorname{draw}(C_2, X) = \operatorname{draw}(C_1, T X)$$

and the reason is two lines. The line $C_2X$ meets the glass somewhere; $T$ fixes that meeting point and carries $C_2$ to $C_1$, so it carries the whole line $C_2X$ to the line $C_1(TX)$, and both lines meet the glass at the same place. The drawn point is the same.

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.

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.

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.

What the map turns out to be #

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.

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.

The rate is the separation over the standoff #

The shear’s rate is measured below and quoted as 1.14 per metre, and it has a closed form that is worth having because it collapses every number in this essay into one ratio.

The map has to carry the second eye onto the first. Put the eyes at distance $d$ in front of the glass and let $\mathbf{s}$ be the vector from one to the other. A shear displaces a point by $k\,z\,\hat{\mathbf{s}}$ with $z$ its depth beyond the glass, so at the eyes’ own depth $z = -d$ the displacement must be $-\mathbf{s}$, which gives

$$k = \frac{|\mathbf{s}|}{d}.$$

The rate is the separation of the two eyes divided by their distance to the picture plane — a dimensionless number, and the same dimensionless group everything else in this collection turns on. The arrangement above has 1.3 m of separation and returns 1.14 per metre, so its glass stands 1.14 m in front of the eyes.

That immediately explains the essay’s other measurement. A shear tilts the depth direction by $\arctan k$, so a right angle between a depth edge and a cross edge comes out that far from square:

$$\arctan 1.14 = 48.7°,$$

against the forty-nine degrees the sweep reports. The two numbers are one number, and the identification is exact rather than approximate.

Read geometrically, $\arctan(|\mathbf{s}|/d)$ is the angle the two eyes subtend at the picture plane. So the statement is as compact as it can be made: the absorbed room’s corners are out of square by exactly the angle the two eyes subtend at the glass. A draughtsman who moves their head by a tenth of the distance to the paper has drawn a room whose corners are 5.7° out; one who moves by half of it has drawn a room 26.6° out; and the two figures are the same arctangent.

Four consequences follow, and the last two are about where this rung sits.

The angle is the whole of the cost. A shear is affine, so lengths along the glass and every ratio along every line survive it untouched; what it spends is squareness, and the arctangent is the entire bill. That is a far cheaper price than the projective alternative the control below exhibits, which spends midpoints as well — and it is why the affine member of the family is the one worth singling out.

The distortion saturates. As the separation grows the angle tends to 90° and never reaches it, so however far apart the two eyes are placed the absorbed room’s corners flatten toward a plane and never invert. There is no separation at which the alternative reading becomes impossible — only ones at which it becomes strange.

And it is linear for small separations. At $|\mathbf{s}| \ll d$ the angle is $|\mathbf{s}|/d$ radians, so a draughtsman shifting slightly has drawn a room out of square in proportion. That is the quantitative form of what the previous rung reports as a miss in metres: the miss says the picture is not a projection, and the arctangent says how odd the scene has to be for it to be one.

The ratio is the thing to count, not the eyes. Two eyes half a metre apart in front of a wall panel and two eyes half a metre apart in front of a hand-held sheet are completely different arrangements, because $d$ differs by a factor of ten and the angle by nearly as much. Which is why counting the eyes needs the room — the number of centres is not a property of the drawing, and neither is the size of what having two of them costs.

And it places the neighbouring conventions on one scale. A carpet drawn with its people upright needs optical axes ninety degrees apart, which on this scale is an infinite separation — the limit the arctangent approaches and never reaches, and the reason that convention cannot be absorbed into a sheared room at all. Two grounds and a parallel floor under a perspective room sit at finite angles on the same axis and can be. So the field’s conventions sort by one number, and the number is an angle a ruler on the plan can measure.

The same expression also says what a reader cannot recover. The rate is $|\mathbf{s}|/d$, so a picture that has been absorbed into one eye’s reading carries the ratio and neither quantity separately — the separation of the two eyes and their distance to the glass are not individually in there, which is the scale ambiguity arriving in the one place this field might have hoped to escape it.

What a shear keeps #

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.

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.

The shear’s own number #

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.

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.

The one number that does not depend on the glass is the picture, which is the point.

The family, which is the honest answer #

The absorbed scene is not one scene.

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.

The result this generalises #

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.

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.

Two eyes, or several #

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.

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.

What this does not say #

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.

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.

Why the fixed plane had to be the picture #

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.

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.

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.

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.

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.

The transferable form #

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.