The arrangement the count cannot see

How many lamps a drawing has counts the lights in a picture from the bundle of lines through each post’s top and its shadow’s tip. Every one of those lines passes through one lamp’s image and through no other, so cutting the bundle into pencils counts the lamps.

The count is read against a leverage: how far a pixel of the reader’s clicking travels by the time it reaches the centre the lines are supposed to meet at. The drawing does not run out of lines established that the number of lines cancels out of the criterion, which is why the count does not simply improve with more posts.

That was swept against the post count. It was measured for one post arrangement, and the round that measured it wrote in its own shortfall that the arrangement had not been swept.

The arrangements #

Five layouts a photograph actually contains, and a sixth built to break the reading.

A row across the view — bollards along a kerb. A colonnade running away — columns receding. A tight clump — chair legs. A ring — posts round a space. A scatter, which is what every earlier figure in the field used.

And a row along the shadows: posts strung out along the direction their own shadows run, which is the degenerate case the lamp, out of the picture recorded and never drew. It is built here from the lamp’s own position rather than guessed at, so it is exactly degenerate rather than nearly so.

Every arrangement has the same number of posts at the same height under the same lamp, so anything that differs between them is the arrangement and nothing else.

The answer, which is nothing #

The leverage runs from 5.53 to 6.40 across the six — a spread of sixteen per cent. The separation at which two lamps stop being counted as one runs from 0.067 to 0.082 metres, a spread of twenty-two per cent, with no ordering that tracks the leverage.

Neither moves. The earlier essay’s claim about the count of lines extends to their arrangement, and the sweep that was supposed to find a dependence found none.

Including the one built to break it #

The sixth arrangement is the point of the sweep and it is no worse than the others: leverage 6.40 against a mean of about 6.0, separation limit 0.068 metres against a mean of 0.072.

That is a surprising result and it is the finding, because the degeneracy is real. A row of posts along the shadow direction gives collinear shadow segments, no vanishing point exists for them, and the least-squares fit is singular — which is exactly what the earlier essay recorded and warned about.

The resolution is that the degeneracy belongs to a different family of lines.

Two families, one drawing #

Every post in a drawing supplies two lines, not one.

The ray line goes through the post’s top and its shadow’s tip. It passes through the lamp’s image, and it is what the count uses.

The ground line goes through the post’s foot and the same tip. It passes through the image of the point on the floor beneath the lamp, and it is what the taught horizon reading uses — where shadows vanish is the construction, and it is genuinely useful because it needs no calibration.

For posts strung out along the shadow direction, the feet and the tips are all on one line. So the ground family is not five lines, it is one line drawn five times, and its meet is undefined.

The ray family is not degenerate at all. The tops are above the line and the tips are on it, so the five ray lines are five distinct lines lying in one vertical plane through the lamp, and their meet is perfectly well determined.

The conditioning, measured #

The claim deserves a number rather than a description, and the number is unambiguous.

For a bundle of lines, the least-squares fit for their common point has a two-by-two normal matrix, and the ratio of its two eigenvalues says how well the point is pinned: one for a bundle that fixes it equally in every direction, zero for a bundle that fixes it in one direction and not at all in the other.

Five ordinary layouts give between 0.039 and 0.412. The row along the shadows gives exactly zero.

The statistic is more legible than it looks, because for two lines it has a closed form. Two unit normals at angle $\alpha$ give a normal matrix with eigenvalues $1 \pm \cos\alpha$, so the ratio is

$$\tan^{2}\frac{\alpha}{2},$$

zero for parallel lines and one for perpendicular ones. Read back through it, the five ordinary layouts correspond to effective crossing angles of about 22° to 66° — a comfortable spread, and the reason none of them is near the degenerate case.

That translation is worth making because it says what “well conditioned” costs in the units a reader can see. The ratio is quadratic near zero, so an arrangement whose lines cross at 10° scores 0.008 and one crossing at 5° scores 0.002 — the statistic falls four times as fast as the angle does, which makes it a sensitive alarm and a poor thermometer. A reader wanting to know how bad an arrangement is should take the square root and read an angle; a reader wanting to know whether it is degenerate can watch the ratio.

It also explains why the degenerate case is so isolated. Reaching exactly zero needs every line parallel, which needs the feet and the tips exactly collinear; a degree of departure from that puts the ratio at $7.6\times10^{-5}$ — small, and not zero, and enough to determine the point badly rather than not at all. Which is the distinction the essay’s next sections are about, and it is the difference between a measure-zero set and a neighbourhood of it.

So the sweep of five plausible arrangements finding nothing is not a sweep that failed to find an effect. It is a sweep whose effect is confined to a family it does not use, and the control that shows this is a single arrangement that has to be constructed on purpose.

The separation limit, and what it is #

The number the sweep is scored on deserves a paragraph, because thirty per cent of variation in something is a lot or a little depending on what it is.

Two lamps a stated distance apart are drawn, the reader’s clicking is added at a stated precision, and the criterion decides how many centres the bundle needs. The separation limit is the distance at which it gets the answer right half the time — bisected on the success fraction over sixteen drawings at each trial separation rather than read off a table.

So it is a threshold in a noisy classification, and its own sampling error over sixteen trials is comparable to the spread between arrangements. Twenty-two per cent is therefore not a signal at all; it is what the measurement’s own noise looks like, which is what a null result should look like when it is reported honestly.

The absolute value is worth having too. Seven centimetres at a metre of clicking precision, for lamps three metres up and posts two metres across the view — which is to say the count separates lamps a hand’s breadth apart, which is a great deal better than a reader would guess and is the reason the distance at which two lamps part is a rung rather than a footnote.

Why the leverage does not absorb the arrangement #

There is a subtlety worth pulling out, because the shortfall’s own wording suggested a different answer.

The criterion divides the observed residual by the expected one, and the expected one is computed from the leverage. So the leverage is where any arrangement effect would be absorbed — the reasoning being that a layout with more leverage produces a larger residual for the same clicking, the criterion knows it, and the division removes it.

That reasoning is right and it turns out to be unnecessary. The leverage barely moves across arrangements, so there is almost nothing for the division to absorb. The criterion is arrangement-independent because its input is, not because its normalisation is doing work.

That is a better outcome than the alternative and it is worth distinguishing. A reading that was arrangement-independent because it divided by the leverage would depend on the leverage being computed correctly for unusual layouts; a reading whose input does not vary does not care.

What would actually break the count #

Having established that the arrangement does not, it is worth naming the things that do — because a reader wanting to know when the count fails deserves the list.

Two lamps close together. Below the separation limit the bundle is one pencil to within the clicking, and the criterion returns one. That is not a failure, it is the measurement’s resolution.

Too few posts per lamp. Each pencil needs at least two lines of its own, so m lamps need at least 2m posts’ worth of lines. The drawing does not run out of lines shows that the count of lines is never the binding constraint — every post supplies one line to every lamp — and that what runs out is the partition: assigning each line to the right pencil gets harder as the lamps crowd, and the count of unknowns cannot see it.

And a lamp behind the camera, which comes out anyway at the point the reversed divide puts it, so it is on the list of things that look like failures and are not.

The arrangement is not on the list, and after this sweep it can be taken off the list of things to worry about.

Why the ordinary arrangements are so alike #

The null result wants an explanation as much as a positive one would, and it is short.

The leverage is set by where the lamp is relative to the posts, not by how the posts are laid out among themselves. A line through a post’s top and its shadow’s tip is extended to a lamp’s image that is far off the top of the frame, and how far it has to be extended depends on the lamp’s height and distance — which are the same for every arrangement here.

Rearranging the posts moves each line’s own endpoints by a metre or two and moves the lamp not at all, so the extension factor is nearly unchanged. Sixteen per cent is what a metre of post movement is worth against a lamp three metres up.

That also predicts what would move it, and the prediction is checkable: a lamp much lower, so that its image is nearer the posts, would give a smaller leverage and a smaller spread; a lamp near the horizon would give an enormous one. The arrangement is not the parameter — the lamp’s own place is.

What a measure-zero degeneracy means for a reader #

The degenerate arrangement is a set of measure zero in the space of layouts — posts exactly along a ray from the lamp’s foot — and it is worth saying what that does and does not imply.

It does not mean it never happens. A colonnade running away from a lamp behind the camera, a row of fence posts along a path lit from one end, an avenue of trees at sunset: these are ordinary arrangements and they are close to the degenerate one. Near-degeneracy is what a reader meets, and it is a continuum rather than a switch.

It does mean a sweep will not find it. Five layouts chosen for plausibility all miss it, and adding a hundred more would too, because the set has no volume. A defect that only appears on a measure-zero configuration has to be constructed deliberately or it will not appear at all.

That is the methodological point and it is why the control exists. A sweep of plausible cases returning a null result is evidence of nothing until one implausible case has been tried, and this collection’s habit — that an assertion which has never rejected anything proves nothing — is the same rule stated for gates.

What this says about the two readings #

The asymmetry between the families is not new here and it is the third time this collection has arrived at it, which is what makes it worth naming as a property rather than as three observations.

The lamp comes out in rays and not in plan found that the ray family is exact on any floor and the ground family assumes a plane. A floor cannot fake a second lamp found the same asymmetry in a different guise: the residual is unchanged across thirteen floors because a line through a point and another point that has slid along it is the same line.

And here the ray family is well conditioned on an arrangement that destroys the ground family entirely.

Three findings, one cause. The ray family is a statement about lines through a point in space, and the ground family is a statement about a point on an assumed plane — so anything that damages the plane, or the arrangement’s relation to it, damages the second and leaves the first alone.

A reader running the count is running the robust half without knowing it. A reader running the taught horizon test is running the fragile half, and the arrangement that breaks it is one they will meet.

What a reader should look at instead #

The practical output of a null result is a shortened checklist, and here it is.

Do not look at the posts’ layout. It does not enter the count, and a photograph whose posts are awkwardly arranged is not a photograph the count works less well on.

Do look at where the lamp is. The leverage — and therefore how much a pixel of clicking is worth at the answer — is set by how far the lamp’s image is from the drawn lines. A lamp high and close gives a small leverage and a confident count; a lamp near the horizon gives an enormous one, and the count is then a statement about the reader’s steadiness rather than about the picture.

And do look at which family is being used. A reader following the taught rule — ground lines, meeting on or below the horizon — is running the fragile half, and the arrangement they will meet on a path lit from one end is the one that destroys it. A reader running the count is running the robust half.

That third item is the one worth carrying out of this essay, because it is a correction to advice rather than a number. The taught construction and the count look like two uses of the same drawing and they use different lines, and only one of the two has a configuration that kills it.

The shortfall this discharges, and what it was for #

Worth a closing paragraph on the process, because the finding exists only because the earlier round wrote down what it had not done.

The round that measured the leverage measured it for one arrangement. Nothing failed, no gate complained, and every number it reported was correct. What it did was record in its own shortfall queue that the arrangement was a parameter and had not been swept — a sentence with no consequence at the time and the whole reason this sweep exists.

The sweep found nothing, which is the outcome a shortfall entry most often has, and then found something by including a case nobody would have swept: the degenerate arrangement that had been described in an earlier essay and never drawn. That case is what turns a sweep that did not matter into a claim that it cannot matter, with the family where it does named beside it.

So the entry earned its keep twice over — once for the sweep and once for making somebody ask what the sweep would have missed.

The short version #

The count of lamps in a drawing does not depend on how the posts are arranged. Five plausible layouts of the same five posts give the same leverage to within a sixth and the same separation limit to within a quarter, with no ordering between them.

The sixth arrangement — posts along their own shadows, built to be exactly degenerate — is no worse, because the degeneracy belongs to the ground family and the count uses the ray family. The ground family’s conditioning there is exactly zero and between 0.039 and 0.412 everywhere else.