The floor that is not a plane

A shadow on a flat floor is an exact record of the thing that cast it. The map from the occluder’s plane to the floor is a projection from a point between two planes, so it is a plane projectivity — a homology, as the census of this site’s maps found — and four correspondences determine it completely. Given four marks whose positions on the occluder are known, every other point of the outline comes back.

That sentence has a clause in it doing all the work, and this essay is about what the clause costs when it fails. Both surfaces have to be planes.

What is being measured, and what would have been easy to measure instead #

There is a way of asking this question that returns zero for every surface, and this site has already published it once.

Un-casting a shadow by running each shadow point back through the known lamp returns the occluder exactly, whatever the receiver does. Of course it does: it is the same ray, traversed backwards. A figure built that way prints 0.000 mm at every curvature, its slider moves the caption and not the picture, and the essay it sits in appears to be about nothing.

That was live on this site for two phases and was found by a sweep asking whether every slider reached the geometry. What is measured here instead is the question a reader with a photograph actually faces:

Fit the map from four marks whose positions are known, and see where it puts the marks it was not given.

On a plane that prediction is exact, because the map really is a homography and four correspondences really do determine one. On anything else the four still fit perfectly — a homography through four points always exists — and everything between them is wrong.

Four floors, one lamp #

The four surfaces are chosen because they fail differently, not because they are a spread of severity.

A plane is the case every homology argument on this site assumes.

A dish — height $k(x^2+z^2)$ — curves the same way in both directions. 5.67 mm.

A ridge — height $kx^2$ — curves one way and is flat the other, so it is developable: it can be unrolled onto a plane without stretching. 9.07 mm, which is worse than the dish.

A step — two horizontal planes at different heights — is flat everywhere except along one line. 74.95 mm, an order of magnitude worse than either curve.

The developable surface is not a special case, and the step is worse than the curve #

Two of those results are worth arguing about, because both are places a reader can reasonably expect the geometry to be kinder than it is.

The ridge is developable, and that does not help. A developable surface can be flattened without distortion, so the temptation is to say the shadow on it is “really” a plane shadow rolled up. It is — the map from the occluder to the unrolled ridge is a homography. What a photograph records is the shadow in place, and the map to the surface as it sits in space is not, because the unrolling is not a projective map of anything. The 9.07 mm is that distinction, priced.

The step is two planes and is the worst of the four. Each half of it is exactly a homology of the occluder, with its own exact four-point determination. What there is not is a single map covering both, and the fitted homography — forced to compromise between marks on the upper half and marks on the lower — is wrong on both.

That is the general lesson and it is not about curvature at all. Piecewise projective is not projective, and a surface that is flat almost everywhere can be further from admitting a homography than one that is curved everywhere.

Why the four numbers come out in that order #

The ordering — plane 0, dish 5.67, ridge 9.07, step 74.95 — is not a scale of severity, and the three factors behind it are separable.

A shadow point lands where the ray meets the surface. Raise the surface by $\Delta h$ at that point and the mark slides along the ray by $\Delta h \cot\varphi$, for $\varphi$ the angle the ray makes with the floor. That is the same grazing factor every ground measurement in this collection pays. Then the fitted homography absorbs whatever part of the resulting displacement field it can reach, and what is left is the reported number:

$$\text{error} \;=\; \Delta h \;\cot\varphi \;\times\; (1 - \text{absorbed}).$$

The step settles the middle factor. A discontinuity has no linear part, no quadratic part, no smooth part at all — a homography can absorb none of it, so its absorbed fraction is zero and the error is the step height times the cotangent outright. Six centimetres of step giving 74.95 mm puts $\cot\varphi$ at 1.25, or the light at about 39° above the floor, which is a fact about the arrangement read out of the measurement rather than a parameter fitted to it.

And the smooth surfaces are mostly absorbed. A dish or a ridge departs from its own best-fitting plane by a sagitta of order $ka^{2}$ over a patch of half-extent $a$, and an eight-parameter map fitted to four marks takes up most of it — the linear part exactly, and a good deal of the rest. What survives is a small fraction of a departure that was already smaller than the step’s.

So the step is worst not because it is the largest departure but because it is the one departure a projective map cannot touch, and the two curves are gentler because they are largely representable. That is the same accounting the vault reports as a percentage of the marks’ own spread, and it explains the essay’s headline result without appealing to curvature at all.

The practical form is a ranking of what to worry about on a real floor:

A discontinuity is unabsorbable and costs its full height. A threshold, a kerb, a step between two levels, the join between a floor and a wall — each contributes $\Delta h\cot\varphi$ and nothing removes it.

A smooth departure costs a fraction of its sagitta, which for a floor of ordinary flatness over a few metres is millimetres before the fit and less after.

And a tilt costs nothing at all, because a tilted plane is still a plane and the homography absorbs it exactly. A floor with a uniform fall to a drain is, for this measurement, flat.

Which is why un-casting a shadow works so well in practice on floors nobody has surveyed, and why the one thing worth measuring before trusting it is not the floor’s flatness but its joins. It is also why the tolerance on knowing a surface is stated the way the anamorph field states it: the quantity that matters is the departure from the best-fitting plane over the region actually used, and the same expression serves whether the surface is being cast onto or recovered from — which is one projection composed with another seen twice.

One caution on the middle factor, since it is the one a reader might try to use directly. The absorbed fraction is not a constant of the surface — it depends on where the four marks were taken, because the fit is what absorbs. Four marks spread across the whole shadow absorb more than four bunched at one end, and four nearly collinear ones absorb almost nothing and produce a wild map, which is the failure mode this collection has recorded more than once. So “mostly absorbed” is a statement about a well-chosen four, and the number to quote alongside any of these millimetres is where the marks were.

What the number means, and what it does not #

The millimetres above are the worst error of the prediction, over sixty-eight points, for one lamp and one occluder at one scale. Three things about them are worth pinning down before they are quoted anywhere.

They scale with the scene. Doubling everything doubles them. The dimensionless version is the error as a fraction of the outline’s own extent, which is the form breadth-02 settled on for the vault — 529.4 mm there, 7.7% of the marks’ own spread — and it is the form to use when comparing surfaces of different sizes.

They depend on which four marks are chosen. A different four gives a different map and a different prediction, exact at its own four. What does not change is that no four make the prediction exact: the surface admits no homography at all, so there is nothing for a lucky choice to find.

And they are not a tolerance. A residual of 5.67 mm from a fit would mean the model is nearly right. This is a prediction from an exactly-determined map, so the number says the map is not the map, and refining the fit cannot reduce it.

There is no threshold #

The slider on the un-casting figure runs the floor’s curvature from flat to sharply dished, and the shape of the resulting curve settles a question a reader is entitled to ask: is there an amount of curvature small enough to ignore?

There is not, in the sense that matters. The misprediction is proportional to the curvature over the whole range — 2.28 mm at 0.02, 4.49 at 0.04, 6.64 at 0.06, 12.82 at 0.12, 14.80 at 0.14 — with the constant of proportionality drifting only from about 114 to 106 mm per unit as the dish deepens.

So the answer to “how flat does the floor need to be” is not a threshold but a conversion: however flat the measurement needs, divided by about a hundred. A floor dished by a centimetre over the region a shadow covers costs roughly a millimetre of recovered outline, and there is no regime in which the homology starts holding.

That is the same shape as the rolled print’s behaviour, and the phrasing that essay used is the right one here too: any curvature at all breaks the theorem, and the only question is by how much relative to whatever precision the measurement needs.

Why no homography exists at all #

The stronger statement — not that the fitted map is wrong, but that no map of that kind exists — has a one-line argument, and it is the argument breadth-02 found for the vault.

A homography takes lines to lines. Take four points lying on a straight line in the occluder’s plane; their shadows on a plane receiver are four points on a straight line, because a projectivity between planes preserves collinearity. Cast the same four onto a curved receiver and their shadows lie on a curve — the intersection of the plane containing the lamp and the line with the receiving surface, which for a quadric receiver is a conic.

Four points on a conic are not four points on a line, and no homography can take a line to a conic. So the map from the occluder’s plane to a curved receiver is not a homography, and the failure is not a matter of degree: it is a failure of the class.

Which also says why the four-point fit behaves the way it does. It is not approximating a nearby homography badly; it is fitting a homography to data that does not come from one, and the fit is exact at the four points it is given because four points can always be matched, by anything.

What the fit is doing when it compromises #

Worth one paragraph on where the error lands, because it is not uniform and the pattern is informative.

The prediction is exact at the four marks, small near them, and worst at the points furthest from all four — which, for four marks spread evenly around a closed outline, is between them rather than beyond them. The error is a smooth function over the outline with four zeros in it, and it grows with the departure of the surface from the plane through the four contact regions rather than with the curvature at any one place.

That is why the step is so much worse than either curve. Its departure from any single plane is a fixed offset over half the region, so the four fitted marks cannot be placed to make it small: whichever four are chosen, one part of the outline is on the wrong plane by the full height of the step.

The same theorem, met in three fields #

This is the third time the site has arrived at one statement from a different direction, and putting the three together is the point of having fields at all.

Rephotography. Photograph a photograph and the composite map is a homography, because a projection composed with a projection through a plane is a projection. Roll the print and four marks mispredict the other sixteen by 30.5 px.

The vault. Cast an anamorph onto a barrel vault instead of a floor and no homography fits at all — 529.4 mm against 1.4e-15 m.

And the shadow. Cast onto a dished, ridged or stepped floor and the same collapse, at 5.67, 9.07 and 74.95 mm.

Three fields, three phases, one clause: the middle surface has to be a plane. Nothing about light, printing or anamorphosis enters any of the three arguments, which is why they are one argument.

What still works when the floor is not flat #

The negative result is narrow, and being precise about how narrow it is matters for anyone trying to use a shadow as a measurement.

Knowing the surface is enough. If the floor’s shape is known, every shadow point can be un-cast through the lamp exactly — the same ray, backwards — and the occluder comes back to arithmetic noise on any surface at all. What has been lost is not the recovery; it is the ability to do the recovery from four correspondences and nothing else.

Straightness survives on each planar piece. A straight edge’s shadow is straight wherever the receiver is flat, and bends only where the receiver does. So the shadow’s own shape is evidence about the floor, which is the observation a raking light in a survey photograph is used for.

And the lamp’s position is still recoverable, from shadow tips on any part of the floor whose shape is known — the lamp comes out of the picture by intersecting drawn lines, and those lines are straight in the picture whether or not the floor is.

The rule, stated for use #

Compressed to what a reader would actually apply:

Four correspondences determine a shadow’s map only between two planes. Both surfaces, not one. The occluder can be a flat stencil and the floor a bowl, and the four-point method fails.

A curved receiver needs its shape, not more marks. Adding a fifth and sixth correspondence improves nothing structural — it converts an exact fit into a least-squares one, whose residual then reports the surface rather than the measurement.

And a stepped floor is a harder case than a curved one, which is the counter-intuitive half and the one worth remembering on site: the surface that looks flattest in a photograph can be the one that most thoroughly breaks the recovery.

Why the flat case is the one everybody has met #

Worth a paragraph, because the flat case’s ubiquity is a fact about rooms rather than about geometry.

Floors are flat, walls are flat, tables are flat, and paper is flat. Nearly every shadow anybody has reasoned about lands on a plane, so the homology is a fair model of ordinary experience, and the four-point determination is genuinely useful — it is how a shadow can be used to measure, and it needs no camera calibration at all.

The cases where it fails are the ones the eye finds interesting: shadows across a stair, across a folded cloth, into a bowl, over a rutted road at sunset. Those are exactly the situations in which a shadow stops being a copy of the object and starts being a picture of the surface, and the numbers above say the transition is not gentle. A step of a few centimetres in the floor costs more than dishing the whole floor by the same amount.