A floor cannot fake a second lamp

The lamp comes out in rays and not in plan leaves the field with a diagnosis. The lamp recovery produces two residuals, and the pair reads cleanly: a large ground residual beside a zero ray residual means the floor is not flat, and both large means a second light.

That diagnosis has an obvious hole in it, and a reader who has followed the field this far will already have found it. If a curved floor can make one half of the construction fail, could a sufficiently peculiar floor make the other half fail as well — and so imitate a second lamp that is not there?

The answer is no, and the reason is better than the demonstration.

The argument comes before the search #

The ray family is built from two points per post: the image of the post’s top and the image of its shadow’s tip.

Now ask what the floor does. A shadow’s tip is where the lamp’s ray over the post’s top met a surface — and the floor decides how far along that ray the meeting happened, and nothing else. Dish the floor and the tip moves; it moves along the ray, because the ray is where it was and the surface is what changed.

The image of a ray is a line. A point sliding along a ray has an image sliding along that line. And a line through two points, one of which has slid along the line, is the same line.

So the floor is not an argument to the ray construction. It is not that its effect is small, or second order, or bounded: it is not in the computation at all, and no shape a floor could take enters anywhere.

The search that had to return one number #

An argument like that is exactly the kind this collection distrusts on its own, because it is short, it is about an absence, and absences are what every gate here is blind to. So it is run as a search anyway.

Take a drawing made by two lamps two and a half metres apart. Cast it onto a plane, a dish, a ridge and a step — the four receivers the field’s own receiver table uses, chosen because they fail differently — at four curvatures each, and ask each of the thirteen resulting drawings how well a single centre explains it.

Every one returns 138.32765289494785 pixels. The spread over all thirteen is $6\times10^{-14}$, which is two parts in $10^{16}$: the last bits of the bisection that found each tip, and nothing else.

The drawings themselves are not the same. The shadows land in visibly different places, the ground family’s residual moves by two orders of magnitude, and a reader shown the four pictures side by side would not confuse them. What is identical is the one number the count is read from.

Why the control matters more than the result #

The row above would be worth very little on its own. A measurement that returns the same number on thirteen inputs might be a measurement that returns the same number on everything — a constant with an argument list.

So the same sweep is run with one lamp, and there the answer is $3\times10^{-13}$ pixels on every floor: exact, including on the step, which is two planes and is the receiver that breaks every other construction in the field.

Two rows, thirteen floors each. The two-lamp row is 138 pixels everywhere and the one-lamp row is zero everywhere. The construction sees the lamps and does not see the floor, and it takes both rows to say so.

That pairing is the shape this collection’s gate insists on: a claim that something has no effect needs a case where the same instrument does show an effect, or it is a claim about the instrument.

And the other half, which does see the floor #

The ground family is the mirror image, and running it on the same drawings completes the diagnosis.

Lines through each post’s foot and the same tips meet at the lamp’s foot only if the feet and the tips are coplanar, which is exactly the assumption a floor that is not flat breaks. On the flat floor that family’s residual is $2\times10^{-13}$ pixels; on a dished floor at a curvature of 0.12 it is 122.

So the pair separates the two causes because each half is blind to one of them. The ray residual counts lights and cannot see the floor. The ground residual sees the floor and cannot count lights. Two numbers the recovery has always computed, kept apart.

What the 138 is #

The number that comes back thirteen times is worth understanding, because a reader is entitled to ask what a residual of 138 pixels is a residual of.

It is structural rather than statistical. With two pencils and one centre allowed, the fitted point is a least-squares compromise between two places, and every line misses it by something like the distance from that compromise to its own pencil’s centre — so the residual is a fraction of the separation of the two lamps’ images in the picture.

The fraction can be written down. With $n_1$ lines through one centre and $n_2$ through the other, separated by $S$ on the page, the compromise sits at the count-weighted mean and the root-mean-square miss comes to about

$$S\,\frac{\sqrt{n_1 n_2}}{n_1 + n_2},$$

which is $S/2$ for a balanced pair — the case the sweep uses, and the reason 138 pixels reads as roughly half the lamps’ image separation.

Two consequences follow, and the second is a failure mode worth naming.

The residual is proportional to $S$, not to the lamps’ separation in the room. Two lamps far away, or close together, or nearly in line with the camera, all give a small $S$ and a small residual — which is what the distance at which two lamps part measures directly, and it means the number cannot be quoted as a threshold without the arrangement attached.

And an unbalanced arrangement hides the second lamp. The factor $\sqrt{n_1 n_2}/(n_1+n_2)$ falls as $1/\sqrt{n}$ when one lamp casts $n$ shadows and the other casts one: ten against one gives 0.29 rather than 0.5, and twenty against one gives 0.21. A scene lit mostly by one lamp with a single object catching a second is therefore the hardest two-lamp scene to detect, and it is also the commonest — a room with one main light and one small one in a corner. The fall is only a square root, so the second lamp does not disappear; it becomes a residual a reader might attribute to careless marking.

Measured across separations, with no clicking noise at all, that fraction is about 0.42 while the two images are close and falls to 0.20 as they move far apart: 29 pixels of residual for 67 of image separation, 185 for 538, 212 for 1,077. The falling fraction is the compromise point drifting toward the pencil with the better-conditioned lines rather than sitting between them.

So the residual is a signal rather than an error: it is the two-lamp geometry expressed in one number, and it is large for the same reason the count is easy. What makes it useless on its own is that a reader cannot tell 138 pixels of signal from 138 pixels of noise without the leverage, which is the previous rung’s subject and is why every criterion here is stated against a computed expectation.

What can imitate a second lamp #

The floor cannot. Five other things can, and a reader with a photograph should have the list, because every one of them is more likely than the case the previous rung was worrying about.

A mirror. A reflecting surface produces a second centre of projection exactly — a mirror is a second camera — and the image of a lamp in a mirror is a lamp as far as any shadow construction can tell. This is not a false positive: there really are two centres casting light into the room, and the count is right.

A window. The same thing with the sun on the other side of it, and with the added complication that one of the two sources is at infinity, so one pencil is a family of parallels rather than a pencil at all.

A wall. Light reflected diffusely from a large surface is not a point source and does not produce a pencil; it produces a soft, spatially extended illumination whose shadow edges are gradients. What that does to a count is decided entirely by where the reader clicks on the gradient.

A moved lamp. Two exposures of the same scene with the lamp moved between them, combined — which is a real photographic technique and produces a drawing with two genuine pencils in it, correctly counted and wrongly interpreted.

And a mis-assigned tip. If a post’s shadow is matched to the wrong post’s top, the resulting line belongs to no pencil at all. One such error puts a line through the middle of the drawing and inflates every residual; the count then reports whatever the partition can make of it. This is the failure mode with no geometric signature, and the defence against it is the same as it has always been: more posts than the construction needs, so that the extra ones can disagree.

The taxonomy this completes #

Four rungs of this field have now measured which constructions have the floor in them, and it is worth setting the list out in one place, because the pattern is sharper than any single result.

In rays, and blind to the floor: the lamp’s image from tops and tips; un-casting a shadow back to its occluder, which is exact on all four receivers; the corners of a wire’s shadow, because a corner is where the wire’s tangent runs along the ray and no surface appears in that statement; and the count of lamps, which is this essay.

In the plan, and about the floor: the lamp’s foot; the four-point homology; the curvature a shadow reports; and every reading that begins by dropping a mark to the ground.

The division is not a coincidence and it is not really about shadows. A statement in rays is a statement about the light and the occluder; a statement in the plan is a statement about where things landed, and where things landed is what a surface decides. A floor is read along the curves a shadow touched is the same division stated from the floor’s side: the marks are where the rays landed, so what a shadow says about a floor is limited to the curves it happened to reach.

Four rungs, one rule, and the rule tells a reader which of their measurements a wrong assumption about the floor will damage before they make it.

The same statement in three other fields #

A construction that uses only rays cannot see the surface the rays landed on. Stated that way, the result stops being about lamps, and this collection has it in three other places without having noticed that they are one place.

In the anamorph field, the marks a design makes on a floor determine the eye’s position on the ground and leave its height free — the marks name the place, not the height — and putting six centimetres of corrugation into the floor buys the height back. That is the same division with the sign reversed: what a flat surface cannot supply, a shaped one can, because the shape enters the construction only when the construction reads where along the ray the mark is.

In the metrology field, one picture of a ball returns a direction and a ratio and refuses the size, because a pinhole is scale-free and nothing in the arrangement breaks the similarity.

And in the two-view field, two views give shape and no size, for the same reason again.

Four fields, one rule: a quantity that is not in the construction cannot be recovered by it, and cannot corrupt it either. The second half of that sentence is this essay, and it is the half that is usually left unsaid — a reader worrying about an unmodelled floor is worrying about a variable that either appears in the formula or does not, and looking is cheap.

What this does and does not license #

The result is narrow and should be stated narrowly.

It licenses a count of lamps from a photograph of a room whose floor is unknown, uneven, sloped, stepped, carpeted or curved. None of those enters, so none of them has to be measured, assumed or even looked at.

It does not license anything about where the lamps are. The count and the two image points are facts about the drawing; turning an image into a place in the room needs the horizon and the ground plane, and that half is exactly the half a floor which is not flat destroys. A reader can therefore be certain there are two lamps and quite wrong about where either of them is.

And it says nothing about how bright they are, which is the quantity most people asking “how many lights” actually want. The construction reads geometry off drawn lines; a lamp contributing a hundredth of the illumination casts a shadow with the same lines as one contributing half of it, provided anything can be seen at all.

What a floor can do to the count, indirectly #

The result is exact and it has one indirect route that a careful reader should know about, because it is the difference between “the floor cannot affect this” and “the floor cannot affect this by any path”.

The floor decides where the tips are in the picture, and therefore how long each drawn segment is and where it sits. That does not move the line — the tip slides along it — but it does change the leverage: a tip that lands further from its post’s top makes a longer segment, and a longer segment pins its line better against the reader’s clicking.

So a floor that pushes the shadow tips further away improves the count’s resolution, and one that pulls them in worsens it, without either of them touching the ray family’s residual by a bit. Measured on the thirteen floors above the effect is small, because the four receivers move the tips by tens of centimetres in a scene metres across.

It matters at the extremes. A shadow falling onto a wall a short distance behind the post gives a very short segment and enormous leverage; a shadow stretching away across a floor gives a long one and very little. The floor cannot change what the drawing says, and it can change how well the drawing is measured — which is a second-order effect on the noise rather than a first-order one on the geometry, and is the only route by which the surface enters at all.

The short version #

A floor decides where along a ray a shadow’s tip landed. A line through a post’s top and its shadow’s tip is the same line wherever along that ray the tip is. So the family of lines a lamp count is computed from does not contain the floor, and thirteen floors return one number to fifteen digits.

The control is a single lamp on the same thirteen floors, meeting exactly on every one; the other half is the ground family, which sees the floor and cannot count lights.

What can imitate a second lamp is a mirror, a window, a bright wall, a second exposure, or a shadow matched to the wrong post — and of those, only the last is a mistake.