A floor is read along curves

Three rungs of this row have measured what a shadow says about a floor. The residual has a shape reads the shape of the departure. The curvature a shadow reports fits a number to its size. The floor is a choice of coordinates argues that most of it need never have been an error at all.

None of them asks the prior question, which is how much of the floor a shadow has been anywhere near.

The answer bounds every one of them, and it is not a small correction.

A curve is not a surface #

A shadow mark is a point where a ray met the floor. What that mark constrains is the floor at that point — how high it is there, and nothing about anywhere else.

The set of marks from one lamp and one outline is a closed curve. Add lamps and there are more curves. A curve has no area, so no number of them covers a surface, and the whole business of “measuring the floor from a shadow” is a measurement along a one-dimensional set inside a two-dimensional one.

That is obvious once stated and it is completely absent from the three earlier rungs, each of which reports a number as though it described the floor.

It is worth resisting the obvious reply, which is that a floor is smooth and so a reading along curves interpolates to the rest of it.

That reply assumes what is in question. A floor is smooth over most of a room, and the interesting parts are exactly the ones where it is not — the crease, the step, the loose board, the sunken drain. Interpolating across a region no shadow visited assumes the region is like the ones that were visited, which is the assumption whose failure this row has spent three rungs measuring.

The same reply is available and equally weak for the sagitta of an anamorph’s design, and that rung handles it properly: it computes how far the surface can wander between marks and quotes the number. Doing the same here is the whole of what follows.

The arithmetic of coverage #

Draw the shadows of a 240-point outline from a ring of lamps and count what fraction of a three-metre-square patch comes within two centimetres of any mark.

One lamp gives 3.75 metres of curve and covers 1.2%. Two give 7.44 metres and 2.2%. Four, 14.94 and 4.4%. Eight, 29.87 and 8.5%. Sixteen, 59.74 and 16.8%. Thirty-two, 119.47 and 29.2%.

The curve length grows exactly linearly — a fitted exponent of 0.999 over five doublings, which is one lamp drawing one curve and thirty-two drawing thirty-two.

The coverage does not. Its fitted exponent is 0.94, and the gap between the two is the whole finding of the second half of this essay.

Why the returns fall away #

A band of width $2\epsilon$ along a curve of length $L$ has area $2\epsilon L$, so if the curves never overlapped the coverage would be exactly proportional to the length and both exponents would be one.

They overlap. Shadows from lamps at similar positions run close together; every shadow of the same outline crosses every other one; and as the count rises, more of each new curve’s band falls on floor that has already been read.

The measured exponent gap says how fast. At 0.94 against 0.999, the thirty-second lamp is contributing about eighty per cent of what the first did — which sounds mild until the extrapolation is done. Continuing at that rate, reaching half the patch takes about seventy lamps, and reaching ninety per cent takes several hundred.

That is the practical ceiling and it is a hard one. A shadow reading is not a measurement of a floor that is merely coarse; it is a measurement along curves whose union is being asked to stand in for a surface.

One thing the coverage number does not say, and it matters for what a reader should ask of a shadow survey. Coverage counts floor that has had a shadow on it, not floor whose height is known — a band read once carries one measurement of the surface there, and a band read twice by two lamps carries a check. So the crossings are the only places the reading is over-determined, and their number grows as the square of the lamp count while the coverage grows as the first power. The redundancy arrives faster than the coverage does, which is the opposite of what a reader hoping for area would want.

The law behind both exponents #

The falling exponent is not a second measurement; it is the same law read at a different place, and the law is the one any covering problem has. If $n$ bands of width $2\epsilon$ and length $L$ fall on a patch of area $A$ with no preference for where, the fraction still uncovered is $e^{-2\epsilon Ln/A}$, so

$$\text{coverage} \;=\; 1 - e^{-x}, \qquad x = \frac{2\epsilon L n}{A}.$$

At thirty-two lamps the measured coverage of 29 per cent puts $x = 0.343$, and the model then predicts 65 lamps for half the patch and 215 for ninety per cent — against the “about seventy” and “several hundred” the essay’s extrapolation gives. Two independent readings of one exponential.

Three things follow that the fitted exponent cannot say.

The exponent is not a property of the arrangement. It is $x e^{-x}/(1-e^{-x})$, which falls continuously from one at $x = 0$ toward zero, so “0.94” is the exponent at thirty-two lamps and would be 0.83 at sixty-five and 0.4 at two hundred. Quoting it as a rate invites an extrapolation the curve does not support.

Tolerance and lamp count are exactly interchangeable. Both enter only through the product $\epsilon n$, so doubling the tolerance and doubling the lamps do the same thing to the coverage — which is what the two figures above show side by side, and it means a reader loosening their tolerance is buying lamps rather than accuracy.

And each decade of the remaining gap costs the same again. Going from 90 per cent to 99 takes $x$ from 2.3 to 4.6, a doubling; from 99 to 99.9, another doubling. So the floor is never covered, it is approached, and the approach costs a constant number of lamps per factor of ten in what is left — which is the honest form of “a hard ceiling” and is a good deal more useful than a number of lamps, because it says what any target costs rather than what one target costs.

What the exponents are worth as a pair #

Neither exponent means much alone and the pair means a great deal, which is worth saying because it is the general form of a great many of this collection’s measurements.

The length exponent of 0.999 is a control. Curve length must grow linearly in the lamp count — one lamp draws one curve — so a fitted value near one is a check that the machinery is doing what it says, not a result. Had it come back at 0.8 the coverage figures would be measuring something else entirely.

The coverage exponent of 0.94 is the result, and it is a result only against the control. On its own, 0.94 is a number close to one and reads as “roughly linear”. Against 0.999 it is a measured deficit, and the deficit is the overlap.

That pairing is the same one an error law’s two terms need — a falling term is not interpretable until something says what falling would have looked like if nothing were in the way.

The tolerance is doing more work than the lamps #

There is a second lever and it is much stronger than the first, which is worth knowing before anybody buys lamps.

The band’s area is proportional to $\epsilon$, so doubling the tolerance doubles the coverage — the same effect as doubling the lamp count, for free. What a reader gives up is the resolution of the answer, which is the whole content of the trade.

At two centimetres, eight lamps cover 8.5%. At four centimetres, they cover roughly seventeen. At ten, roughly forty. The question a reader has to answer first is not how many lamps they can afford; it is how finely they need the floor.

And the phrasing of that question is the useful part. “How flat is this floor” has no answer without a length scale attached, because a floor that is flat to a centimetre over a metre and a floor that is flat to a millimetre over a metre are different claims requiring different amounts of evidence. The coverage arithmetic makes the dependence explicit rather than leaving it in a footnote.

The crossings, which are the only over-determined places #

The overlaps are a loss for coverage and a gain for something else, and it is worth separating the two.

Where two shadow curves cross, the floor’s height at that point is constrained twice — once by each lamp’s ray. Two constraints on one number is an over-determination, and an over-determination is where a consistency check lives.

So a set of shadows carries two different kinds of information. Along the curves, height measurements. At the crossings, checks — and a disagreement at a crossing is evidence that something in the arrangement is wrong, in exactly the way the lamp recovery’s residual is evidence.

The number of crossings grows as the square of the lamp count while the coverage grows as slightly less than the first power. So more lamps buy checks much faster than they buy coverage, which is a genuinely useful asymmetry and the opposite of what a reader would guess.

That last figure is worth reading carefully, because it puts a limit on the crossings argument. On a flat floor the two shadows are related by a known map, so the second is entirely predicted by the first and the crossings are not over-determinations of anything — they confirm flatness and nothing else.

The crossings become informative exactly when the floor is not flat, which is when the homothety fails. So the checks a second lamp buys are checks against the flat-floor hypothesis, and their value grows with how wrong that hypothesis is. That is the right behaviour for a diagnostic and it is worth noticing that it comes out of the geometry rather than being designed in.

What a design that covers a surface looks like #

For contrast, it is worth naming the arrangement in this collection that does read a surface rather than a curve.

The wall under the paint recovers a painted surface from an anamorph’s design, and it recovers it at every mark of the design — which is a grid, two-dimensional, covering a patch. It gets the wall back at $10^{-14}$ metres on a dished floor, a ridged one, a flat one, and a floor with a step in it.

The difference is not the solver and it is not the precision. It is that a design is a grid and a shadow is a curve, and the recovery inherits the dimension of whatever it was given.

That essay also reports the right accompanying number, which is what happens between the marks: the sagitta, 4.3 mm at seven marks across and 1.1 mm at thirteen. That is the two-dimensional version of exactly the question this essay is asking, and it has the same answer in a different currency.

Where the lamps should stand #

The ring used above is the obvious arrangement and it is not the best one, which is worth measuring rather than asserting.

Lamps at similar azimuths cast shadows that run close together, so a ring spreads them as widely as one height allows. What it does not vary is the elevation, and elevation is what moves a shadow radially — a low lamp throws a long shadow far from the occluder and a high one throws a short one near it.

So a ring at one height reads an annulus and a set of lamps at several heights reads a disc. That is a real difference in what the curves can cover, and it costs nothing beyond putting the lamps in different places.

The general form is the one every sampling problem has. Coverage is decided by how much the family of curves varies, not by how many of them there are, and a family that varies in only one parameter covers a one-parameter neighbourhood of a curve however many members it has.

What can and cannot be claimed #

The results of the earlier rungs need one qualification each, and it is the same qualification.

A fitted curvature from a shadow’s mispredict is a curvature fitted to the floor along the shadow. On a floor that is a paraboloid of revolution it is the floor’s curvature, because a paraboloid has one. On a floor that varies, it is a weighted average over a curve, weighted by how much of the mispredict each part of the curve produced — and nothing in the fit says which parts.

A residual’s shape likewise reads the departure along the curve. The step’s spikiness comes from where the curve crossed the crease; a crease the shadow never reached leaves no spike, and the floor is exactly as creased as before.

Neither of those is a defect of the measurements. They are the honest scope of a reading taken along a one-dimensional set, and the useful move is to state the scope rather than to widen it.

More lamps, or a bigger occluder #

There are two ways to draw more curve and they are not equivalent.

More lamps multiply the curves, and every new curve is a shadow of the same outline, so the family is highly redundant — the curves are all projectively related to one another and cluster where the outline’s own shape puts them.

A bigger or more complicated occluder lengthens each curve, and a curve with more structure in it visits places a simpler one does not. A grid of rods casts a shadow whose curve length grows with the number of rods, and it covers the floor the way a net does rather than the way a family of nested loops does.

The second is much the better buy per unit of curve, and it is the one nobody thinks of because the lamp is the thing that seems to be the instrument. It is not; the caster is, and the lamp only decides which projection of it lands where.

The number a reader should quote #

The honest summary of a shadow-based floor measurement has three parts, and the field has been quoting one.

The departure, in millimetres — which is what the earlier rungs report.

The tolerance, in millimetres — how close a mark has to be for the floor there to count as read, which sets what “measured” means.

And the coverage, as a fraction — how much of the region of interest was within that tolerance of any mark.

A measurement quoted with all three is a claim a reader can act on. A measurement quoted with only the first reads as a statement about the floor and is a statement about a curve.

The same three-part form applies well outside this field. A silhouette’s reconstruction has an error, a view count, and a part of the object no view ever reaches — and that third number is a coverage in exactly this sense, reported because somebody asked what the first two left out.

The sentence to carry #

The row has now said three things about reading a floor from a shadow and this is the fourth.

Read it in rays rather than in the plan, or the floor’s height comes back as an error in something else. Read the residual’s shape, not only its size. Split the recovery’s two residuals, because they diagnose different faults.

And know that all three are statements about the floor where the shadow was. A shadow is a curve. A floor is a surface. Everything the first says about the second is said along a set of measure zero, and no amount of lamps changes the dimension of what is being asked.