The floor is a choice of coordinates

Four rungs of the light field have been about what a curved receiver breaks. The floor that is not a plane measures a four-point map mispredicting by millimetres. The curvature a shadow reports fits a curvature to the mispredict and finds it returns one for a floor made of two planes. The residual has a shape reads the mispredict as a field and separates three floors the scalar calls identical.

All three take the mispredict as a fact about shadows on curved floors.

It is not. It is a fact about plans.

The one-line argument #

A shadow mark is the point where the ray from the lamp through the occluder’s edge meets whatever is in the way.

That sentence contains the whole thing. The mark is on the ray. What the floor decides is where along the ray the mark sits — nothing else, because the ray was fixed by the lamp and the occluder before the floor was consulted.

So run the mark back up the ray it is on, and the occluder comes back. The floor never enters, because the parameter along the ray was never used.

Measured on all four receivers with the same lamp and the same 72-point outline: the recovered outline is out by $3.4\times10^{-16}$ metres on the plane, $3.4\times10^{-16}$ on the dish, $3.5\times10^{-16}$ on the ridge, $3.3\times10^{-16}$ on the step. Those are not four small numbers; they are four instances of the same arithmetic noise, and the step is the smallest of them.

It is worth being clear about what this does and does not assume. The ray route needs the lamp’s position and the occluder’s plane — two pieces of prior knowledge, and neither of them is about the floor. That is the entire content of the claim: the prior knowledge a shadow reading needs is knowledge about the light and the object, and the floor’s shape was never on the list.

A reader who does not know the occluder’s plane is not stuck either. The ray alone is recoverable without it, and the ray is a complete answer to “which direction from the lamp did this mark come from” — which is the question a great many shadow constructions are actually asking.

What the plan does instead #

The reading the earlier rungs use is not this one. It works in the floor’s own two coordinates.

Take the mark’s plan — its $x$ and $z$, with the height thrown away and replaced by zero — and run that back through the lamp. On a flat floor the substitution is not a substitution, because the height was zero. On any other floor it moves the point along the ray to wherever a flat floor would have caught it, and the error is the whole of the floor’s height translated through the lamp’s projection.

Measured: 75 mm on the dish, 45 mm on the ridge, 216 mm on the step. Those are the same three orders of magnitude the four-point predictor has been reporting, arriving from a route that makes the mechanism obvious.

The homography of the earlier rungs is a map between two plans — the occluder’s $(x,z)$ and the mark’s $(x,z)$ — and a plan is precisely the operation that discards where along the ray the mark was. The curvature it reports is the height it discarded, coming back as an error in something else.

Why anybody works in the plan at all #

This is not a criticism of four rungs of work, and it would be a poor essay if it read as one. The plan reading exists for a reason and the reason is good.

A photograph gives image points. To place a mark in space, its ray from the camera has to be intersected with something, and that something is usually the ground plane, because a ground plane is the one piece of scene geometry a reader can usually assume.

So the plan is not a mistake; it is a substitute for depth. What the earlier rungs measured is the cost of that substitute on floors where the assumption is wrong, and the cost is real.

The correction is that the mark’s depth is available from somewhere else. The lamp is the second eye establishes it: the lamp’s ray and the camera’s ray meet at the mark, and two lines in space determine a point. No floor, no assumption, and it is machinery the field already had.

That last point is the one worth dwelling on. The machinery that removes the floor from the problem is not new machinery; it is the field’s own rung on the lamp as a second eye, which was written to make a point about depth and turns out to make this one as well. The two questions — how far away is this mark and what shape is the floor — are the same question, and answering the first makes the second unnecessary rather than easier.

What the floor-free route still owes #

The trade is worth counting, because the floor does not disappear entirely and knowing what is left decides whether the route is usable on a real photograph.

The triangulation needs the lamp’s position — three numbers, exactly as many as the plane it replaces. And those three are not independently available: the ray family through the objects’ tops and their shadow tips gives the lamp’s image with no floor at all, but converting that image into a position in space needs its distance, and the usual way to get it is the ground construction, which needs the floor. So the bootstrap does not close.

What does close is weaker and more useful than it first looks. With the lamp’s image known and its distance unknown, every mark’s depth comes back up to one common scalar — the lamp’s own distance — because scaling the lamp’s distance scales the whole reconstruction along the camera’s rays. So:

The floor-free route determines the scene’s shape and not its size. One number is missing, it is the same number for every mark, and it is exactly the freedom a single view can never supply by any means at all.

Which sharpens the essay’s own claim rather than qualifying it. The floor was doing two jobs — supplying depth and supplying scale — and only the second is irreplaceable. Every ratio in the recovered scene, every angle, every shape, is available without knowing anything about the floor’s flatness; only the metre needs it, and a metre can come from any known length anywhere in the picture instead.

So the corrected reading of the earlier rungs is not that the plan reading was wrong but that it was paying for scale with an assumption about shape. A reader who has any known length in the scene can drop the floor entirely and lose nothing; a reader who has none can keep the floor and know that its flatness is buying a single scalar rather than the geometry.

What the receiver actually costs #

Not accuracy. Something else, and it takes a step to find it.

A vertical riser stands over one line of the plan. Every ray that lands on the riser produces a mark whose plan is on that line, however far up the riser it struck — so an arc of the occluder images to a straight segment of the plan, and along that segment the marks differ in height and in nothing else.

Measured on a fifty-centimetre step with a 144-point outline: 33 of the marks land on the seam. Their plan spread across it is $2\times10^{-13}$ millimetres, which is zero, and their spread up the riser is 323 millimetres.

So one whole direction of the plan collapses. Not distorted — collapsed, to a set of measure zero. And the information that direction carried is sitting in the marks, in the coordinate the plan does not record.

A collapse is not a large error #

The distinction matters for what to do about it.

A large error is a wrong number, and a better model or a better estimator improves it. The dish’s 75 millimetres is of this kind: model the dish, and the error goes.

A collapse is not a number at all. Two distinct occluder points map to the same plan mark, so the plan map is not injective there, and no estimator inverts a non-injective map. A model of the step would not help, because the step is not what went wrong — the projection chosen to read the marks is.

That is the sharpest form of the essay’s claim. The plan reading does not merely lose accuracy on a creased floor; over part of the shadow it loses the ability to be inverted at all, and it loses it silently, because a collapsed plan looks like a shadow with a straight bit in it.

The same distinction sorts a great deal of this collection. A drawn fold’s two readings are a discrete ambiguity — two answers, both consistent — and no refinement of the drawing chooses between them. The scale a single view cannot give is a continuous one, a whole family. And a curved floor’s mispredict is neither: it is a plain error, wrong by a computable amount, fixable by knowing more.

Three different things, and a fit reports all three as a residual.

Why the step’s collapse turns on abruptly #

There is a threshold and it is worth naming, because it is the same non-monotonicity the matching search in the residual’s shape had to be written around.

Below about twelve centimetres of step, the entire shadow lands on one side of the seam. The floor under it is then a plane at a constant height, the map is exactly a homology, and the plan reading is exact — the step might as well not be there.

Past that, part of the shadow crosses. The marks that cross land on the riser, their plans pile onto the seam line, and the collapse begins. It grows from 15 marks at a quarter of a metre to 33 at half a metre, which is roughly linear in the step’s height because the riser’s angular extent from the lamp is.

So the failure has a switch in it rather than a slope. That is unusual in this collection and it is worth flagging: nearly every departure measured here is smooth in its parameter, and a reader used to that will extrapolate a small step’s harmlessness straight past the point where it stops being harmless.

That last figure is the one to keep. A floor with a twelve-centimetre step in it is not a flat floor by any description, and a plan reading of a shadow on it is exact. What matters is not whether the floor is flat; it is whether the shadow ever meets the part that is not, and that is a fact about the lamp and the occluder rather than about the room.

Which is a general and slightly uncomfortable result for any diagnostic. The same floor is harmless or ruinous depending on where the lamp is standing, so a survey of the room does not answer the question and only the shadow itself does.

What a photograph can and cannot supply #

Everything above assumes the marks are known in space. A photograph does not hand them over that way, so it is worth walking the chain from a real picture to a real answer.

The photograph gives an image point for each mark. That is a ray from the camera and nothing more — a one-parameter family of possible marks, one of which is the true one.

To place the mark, that ray has to meet something. Three candidates, and they are the three readings above in a different order.

Meet it with the ground plane and the answer is the plan reading, with all of its costs, and it needs only that the ground plane is known.

Meet it with the lamp’s ray and the answer is exact, and it needs the lamp’s position — which comes out of the same photograph when there are posts in it, so this is not the imposition it sounds.

Meet it with a second photograph’s ray and the answer is exact and needs neither, at the price of a second camera position.

So the ranking by accuracy and the ranking by convenience are opposite, which is the ordinary situation and is worth stating plainly rather than implying that the plan reading is a mistake anybody makes out of carelessness.

The three readings, sorted #

The field now has three ways to read a shadow and they sort cleanly by what they need.

In rays, needing the lamp and the occluder’s plane. Exact on every receiver. This is what un-casting should always have been.

In rays with two centres, needing the lamp and the camera. Exact on every receiver, and it does not need the occluder’s plane either, because the mark’s position comes from the intersection rather than from an assumption.

In the plan, needing a ground plane. Exact on a plane, wrong by centimetres on a smooth floor, and non-invertible over part of a creased one.

The third is the cheapest and it is the one every drawing manual teaches, which is why it is the one this field measured first.

What this does to the earlier rungs #

It does not retract them; it explains them, and the difference is worth being precise about.

The curvature a shadow reports is still true. A fit of a curvature to a plan-reading’s mispredict does return 0.482 for a floor of two planes, with a residual of ten microns, and that is still a warning worth having about estimators. What changes is the diagnosis: the phantom curvature is not a fact about shadows on stepped floors, it is a fact about plans of shadows on stepped floors, and a reading in rays never produces it.

The residual has a shape is also still true, and it acquires a use rather than losing one. If a reader is going to work in the plan — and often they must, because the lamp’s position is not known — then reading the residual’s shape tells them which case they are in, and the spiky case is the one where the reading is not merely inaccurate but partly non-invertible.

So the row’s three essays stack rather than cancel. Measure the departure; read its shape; and know that the departure was never necessary.

There is one place where the correction changes an answer rather than an explanation, and it is worth naming. A reader who has measured a mispredict and fitted a floor to it has, on this reading, measured the lamp’s projection of the floor’s height field, not the floor. Those coincide up to a known transformation when the lamp’s position is known — so the fitted curvature is recoverable into a real curvature — and they do not coincide at all when it is not.

Which means the earlier rung’s number was never quite the number it was labelled. It was the right measurement of the right thing seen through one more projection than the label admitted, and the projection has a name and can be undone.

The general form #

The sentence worth carrying out of the field is not about shadows.

A projection loses one coordinate, and everything a reading of it gets wrong is that coordinate arriving somewhere else. A plan of a shadow loses the height, and the height comes back as a curvature. A photograph loses the depth, and the depth comes back as a scale that cannot be recovered. An outline loses the size, and the size comes back as a ratio.

The remedy is always the same and it is always the same shape: find a second thing that constrains the lost coordinate, or work in a representation that never discarded it. The second is cheaper when it is available, and here it was available all along.