A parallel floor under a perspective room

A great many drawing traditions do the same thing with a floor. The tiles do not converge — the pattern runs to the top of the picture at the same width it has at the bottom — and the people standing on the floor are drawn smaller when they are further away.

That is a parallel projection glued to a perspective one, and it is a picture with a centre at infinity in one half and a centre in the room in the other.

A centre at infinity is a centre #

The construction that absorbs a second eye takes the centre as a homogeneous point, so a direction is an ordinary input rather than a special case. Nothing in the algebra changes: an identity plus one outer product, a picture plane held still point by point, and the second centre carried onto the first.

So the composite has the same description as the previous ones: it is a one-eyed picture of a scene that has been moved by a map fixing the glass. What differs is the map.

And this time there is no shear #

The family of absorbing maps has one member that goes to the identity as the two centres come together. When both centres are finite, that member’s own centre is a point at infinity, which makes it affine — a shear, growing with depth.

A centre at infinity has no such member and cannot have one. An affine map takes points at infinity to points at infinity. The whole job here is to bring one in from infinity to a finite place, so no affine map does it, and every member of the family is properly projective.

That is a real difference and it is worth saying what it costs. When two finite eyes are absorbed, parallel walls stay parallel and every midpoint stays a midpoint; the room comes out skewed and remains, in a recognisable sense, the same room. When a parallel half is absorbed, midpoints move, so a wall’s centre is no longer its centre and a floor’s tiles are no longer evenly spaced in the absorbed reading.

The two cases swap the roles of the map’s own centre #

The absence of an affine member is stated above as an impossibility and it has a positive form, which is worth writing down because it says exactly what the map here is.

The absorbing map is $T = I + \mu\,\mathbf{O}\,\boldsymbol{\pi}^{\mathsf T}$ — an identity plus one outer product, with $\mathbf{O}$ the map’s own centre and $\boldsymbol{\pi}$ the picture plane. Its bottom row is the identity’s plus $\mu\,O_{4}\,\boldsymbol{\pi}^{\mathsf T}$, so the map is affine exactly when $O_{4} = 0$ — when its centre is at infinity.

Now the requirement. Carrying a centre $\mathbf{C}_{2}$ onto $\mathbf{C}_{1}$ needs $\mathbf{C}_{1}$ to be a combination of $\mathbf{C}_{2}$ and $\mathbf{O}$, so $\mathbf{O}$ lies on the line joining the two centres — and if $\mathbf{C}_{2}$ is at infinity while $\mathbf{C}_{1}$ is not, $\mathbf{O}$ must be finite, or the combination stays at infinity.

So the two cases exchange roles exactly:

That is a tidier statement than “no affine map does it”, and it identifies the map with an object this collection already has three numbers for. A homology is an axis of fixed points, a centre off it, and one ratio — which is what a floor anamorph turns out to be and what the census of fixed structures sorts maps by. Here the axis is the glass, the centre is a point on the ray running from the eye along the parallel half’s own direction, and the ratio is what moves the midpoints.

Two consequences follow, and the second is the sharper one.

The seam’s size follows from the same object. A homology’s displacement grows with the cross-ratio along each ray from its centre, so the disagreement between the two halves is largest where the ray is longest — deep in the picture — and vanishes on the glass, which is the axis. That is why the composite shows itself as a discrepancy about size rather than about position, and why the discrepancy is a factor rather than a distance: the two halves are related by a map with a centre, and a map with a centre multiplies.

The five per cent is the homology’s ratio, not a fixed cost. A homology’s departure from an affine map is set by that one number, so quoting a midpoint shift is quoting a member of the family rather than a property of the arrangement. It is the same distinction the second eye is a shear makes when it identifies its own distinguished member before claiming affineness — there the identification was possible and the claim was a measurement, and here the identification is the thing that is missing.

And there is no distinguished member at all. In the finite–finite case the family is singled down by one condition: take the member that becomes the identity as the two eyes come together. A centre at infinity has no coincidence limit — the two centres cannot approach each other, because one of them is not anywhere — so that condition is unavailable and nothing replaces it. Every member of the family absorbs the same picture, draws the identical marks, and hands back a different room, with a different midpoint shift and a different set of tile spacings.

So the absorbed reading of a mixed picture is worse off than the absorbed reading of a two-eyed one in two separate ways. It is projective rather than affine, so it spends midpoints; and it is not canonical, so which midpoints it spends is a choice nobody in the picture made. The composite reading — two rules, one sheet, no single eye — is the one with fewer arbitrary decisions in it, which is an unusual position for this collection to end up in and is worth stating plainly.

It also places the convention against its neighbours on the field’s one axis. Two finite eyes cost an angle and keep every ratio; a carpet drawn with its people upright needs axes ninety degrees apart, which is the limit no absorption reaches; and this one costs the ratios themselves. The ordering is by how much of the affine structure survives, and it runs from all of it, through most of it, to none — which is the same ladder the parallel systems sit at the top of and a picture with no size–distance signal prices from the reader’s side.

Which is the same statement as the convention’s own point #

The convention exists because a parallel floor is measurable. Its tiles are evenly spaced on the page, so a reader can count them and know how far into the room something is.

The absorbed reading destroys exactly that. Its midpoints have moved, so the evenly spaced tiles of the parallel half are unevenly spaced in the room it is a picture of — which is precisely the property the convention was drawn to have.

So the two descriptions are not equivalent in what they make available to a reader, even though they are equivalent in what they put on the page. The composite reading says “the tiles are even and there is no single eye”; the absorbed reading says “there is a single eye and the tiles are uneven”. Both are true of the same marks.

The two halves, as maps #

It helps to write the two halves side by side, because the difference between them is one entry in a matrix.

A projection through a finite centre divides by depth: the drawn position of a point is its lateral offset over its distance along the axis. A parallel projection does not divide at all: the drawn position is the lateral offset, full stop.

In the matrix form this collection uses for a rendering pipeline, that is a single row. Set the bottom row of a projection matrix to a row of zeros ending in one and the same matrix draws a parallel projection; leave it alone and it draws a perspective one.

So the mixed picture is a picture made by two matrices differing in one row, applied to different parts of the same scene, drawn onto the same glass. Said that way it is a small thing to do and the arithmetic of undoing it is correspondingly larger, which is a common enough pattern.

The seam, which is where a mixed picture shows #

A composite of two finite eyes has a seam whose disagreement is a parallax — it needs depth to show and it vanishes on a flat scene. A composite of a parallel half and a perspective half is different: the two halves disagree about diminution, which is a fact about size rather than about position.

So the discrepancy is much larger than a two-eye seam of comparable geometry, and it shows in a way a reader recognises without any geometry at all: a thing that ought to be smaller is not.

That is presumably why the convention is applied to floors and pavements and rarely to figures. A floor’s tiles have no expected size, so a reader has nothing to be surprised by; a person does.

What a reader can actually measure off it #

The interesting practical question is what a person holding such a picture is entitled to compute, and the two halves give different answers.

Off the parallel half: ratios of distances along the floor, exactly. That is what a system with no diminution buys and it is a strong result — a reader counting tiles is measuring the room, with no camera, no focal length and no station point.

Off the perspective half: cross-ratios, and everything the site’s ordinary single-view machinery extracts from them — which is more than ratios but needs a horizon, and the horizon of the parallel floor is at infinity and unavailable.

And across the seam: nothing. The two halves are pictures of the same room by different rules, so a length read from one and a length read from the other are not comparable, and there is no factor that makes them so — the ratio between them depends on depth, which is precisely what one half has deleted.

That last point is the practical content of the whole essay. A mixed picture is two measuring instruments on one sheet, each of them exact, with no conversion between them.

What the family looks like here #

The absorbed scene is still not one scene — the family is still one-parameter, for the same reason as before — and its members here are more different from each other than in the finite case.

The reason is the pole. A projective map of space has a plane it sends to infinity, and for the maps that absorb a parallel half that plane sits somewhere in the room rather than out beyond it. Points near it are thrown a long way, so members of the family that differ slightly in parameter differ enormously in where they put the far parts of the scene.

Which is a practical warning as much as an observation: the absorbed reading of a mixed picture is well behaved near the glass and badly behaved far from it, and quoting a single absorbed room without saying which member of the family is doing the quoting more work than it can bear.

The scroll is the continuous version #

One system this collection already models does exactly this along one axis and continuously.

A handscroll is orthographic along the roll and perspective across it: a mile of river holds its scale as the picture unrolls while a single pavilion still recedes. That is a parallel projection in one direction glued to a central one in the other — not two halves of a picture but two axes of the same map.

So the mixed floor and the handscroll are the same trade made along different cuts of the picture. The scroll cuts by direction and keeps every column a projection; the floor convention cuts by region and keeps every part a projection. Both buy measurability in one place with a station point everywhere.

What this does not say #

It says nothing about what any tradition intended. The convention is named as the source of a rule being modelled, and everything computed is a computation on a map.

It says nothing about the parallel half being a bad choice. It is a good one for what it is for, and this collection’s own table says why: a system with no centre keeps true measure, and a system with a centre gives up true measure to get a station point. There is no system with both.

And it does not say the mixed picture is a worse composite than the two-eye one. It is a more distorted one when absorbed, which is a different claim: a bigger seam and a projective rather than affine map. Whether that matters depends on what the picture is for, and the geometry prices it without preferring.

The vanishing plane, and where it sits #

The map that absorbs a parallel half is projective rather than affine, which means it has a vanishing plane — a plane of space it sends to infinity — and where that plane sits decides how well behaved the absorbed reading is.

For an affine map there is no such plane in the room: the plane at infinity goes to itself, which is what affine means. For the maps here it is somewhere finite, and points near it are thrown arbitrarily far.

That gives a practical rule about which part of the picture the absorbed reading can be trusted for. Near the glass it is tame; far from it, and especially near wherever the vanishing plane falls, the absorbed positions run away and small changes in which family member is used produce large changes in the answer.

Where else a mixed system appears #

Worth listing, because the pattern is more common than the floor convention alone suggests.

A removed roof. The interior is drawn as though from above and the walls as though from the side, which is the same trade with the parallel half applied to the plan.

And a rendering pipeline that mixes projections. A parallel projection for a shadow map and a perspective one for the camera is exactly this composite, done by a machine, with the two matrices differing in one row. Nothing about that is a convention and the geometry is identical.

So the mixed picture is not a historical curiosity. It is what happens whenever one part of a picture is wanted for measuring and another for looking at, and that want is old and current at once.

The transferable form #

A limit case is usually the same construction with an ordinary input, and where it is not, the reason is a property the limit does not have — which is more informative than the construction working.

The absorbing map takes a centre at infinity without a special case. What it does not take is the affine member, and the reason is one sentence: an affine map cannot bring a point in from infinity. That sentence explains the whole difference between the two rungs — why the two-eye composite keeps midpoints and this one does not, why the family here is worse behaved, and why the seam is larger.

This collection has met the same pattern before, and it is worth noticing that it usually goes the other way. The eye taken to infinity, the parallel projection as a limit of a perspective one, the infinite far plane of a frustum: in each of those the limit turned out to be an ordinary member of a family rather than a separate case. Here it is a separate case, and the thing that separates it is not the limit itself but a property — affineness — that the family only has when nothing is at infinity.

The comparison that makes the direction plain is the eye taken to infinity, where the limit is approached deliberately and everything about the picture stays continuous as it is reached. Here the limit is arrived at by accident, in one part of a drawing whose other parts did not take it, and the discontinuity is between the parts rather than in either of them.