Counting shadows is not counting lamps

There is an obvious way to find out how many lights a room has, and everyone who has ever looked at a photograph has used it: count the shadows.

Two dark patches, two lamps. One patch, one lamp. It is quick, it needs nothing drawn, and it is right often enough to feel like a rule.

It is not a rule, and it fails in both directions on arrangements that are entirely ordinary.

Two lamps, one shadow #

Put two lamps a quarter of a metre either side of a point three metres up, and a card between them and the floor.

Each lamp casts its own shadow of the card. The two shadows overlap, because the lamps are close and the card is not; the union of the two is a single connected region; and the floor shows one dark patch with slightly darker middle.

A reader counting shadows says one lamp. The drawn lines say two, without any difficulty at all: the ray family leaves 39 pixels at a single centre against an expectation of four, which is the count’s ordinary verdict and is not near any boundary.

So the patch count is wrong, and it is wrong in the direction that matters: it under-reports, and a reader who trusts it concludes there is one source and reasons about the room from there.

One lamp, two shadows #

Now the other direction, which is easier still to arrange: one lamp, two cards.

The floor shows two dark patches, separated by lit floor between them. A reader counting shadows says two lamps.

The drawn lines say one, exactly — every line in the drawing passes through a single point to $10^{-13}$ pixels, because there is only one place for them to pass through.

This failure is even more common than the first, since most rooms contain more than one object. Any scene with two separated objects in it produces two separated shadows from a single source, and the patch count reports the object count rather than the light count.

The control, which is the reason to keep the reading #

Two counter-examples on their own would say the patch count is useless, and it is not.

Move the two lamps 1.6 metres either side of the same point. Now the two shadows of the one card land in different places, the region between them is lit by both, and the floor shows two patches for two lamps. The reading is right, and it is right for the reason everybody’s intuition supplies: the shadows separated because the sources did.

So the honest statement is not that counting shadows fails. It is that the patch count answers a different question — how many separated dark regions are there — and that question has an answer that depends on the sources, the objects, and their arrangement together. It coincides with the light count when the sources are far apart and the objects are few, which is the case a reader has in mind when the rule feels obvious.

And the same floor gives a third answer #

There is a further reading of the same three arrangements, and it makes the ambiguity of the word rather than of the method.

“Shadow” can mean the floor a reader sees as dark — blocked from at least one lamp — or the fully dark part, blocked from every lamp. The second is the umbra, and under one source the two are the same region.

Under two they are not, and the control above is where they part most sharply. Two lamps 1.6 metres apart produce two visible patches and no umbra at all: there is no point of the floor hidden from both lamps, because the card is not wide enough or the lamps not close enough for their two shadows to overlap anywhere.

So one arrangement gives three answers to “how many shadows”: two, if a reader counts the visible patches; zero, if they count the fully dark regions; and two lamps, if they draw the lines. The close pair gives one, one and two.

The condition for an umbra to exist at all is one inequality. A card of width $w$ with the lamps $L$ behind it and the floor $D$ in front casts shadows of width $w(L+D)/L$, displaced from each other by $sD/L$ for a lamp separation $s$. They overlap — and an umbra exists — when

$$s \;<\; w\left(1 + \frac{L}{D}\right).$$

Two readings, and both are checkable on the figures.

A card near the floor always has an umbra. As $D \to 0$ the right-hand side runs away, so an occluder resting on the receiving surface is fully dark beneath itself whatever the lamps are doing. That is why the shadow under a book on a table has an umbra and the shadow of a hand held above it may not.

And far from the floor the rule is simply $s < w$. For $D \gg L$ the bracket tends to one, so the umbra survives only while the lamps are closer together than the occluder is wide. The 1.6-metre pair that produces no umbra is that inequality failing on a card a few tens of centimetres across.

The same inequality is the extended-source rule from the neighbouring rung wearing different clothes. There a source of angular diameter $\alpha$ swallows the umbra once $\alpha D$ exceeds the occluder’s size; here two lamps $s$ apart at distance $L$ subtend $s/L$, and the two conditions are the same one. Two point lamps behave, as far as the umbra is concerned, exactly like one source of angular size $s/L$ — which is worth knowing because it says a reader cannot distinguish the two cases by looking at whether there is an umbra. Only the number of patches separates them, and the patches are the reading this essay has just shown to be unreliable.

That is not a paradox. It is a word doing three jobs, and the collection’s own habit of naming quantities separately would have caught it earlier: the penumbra is the lamp’s image already establishes that a shadow’s edge is a structured object rather than a boundary, and the umbra-and-patch distinction is the same structure arriving as a count.

How far apart is far enough #

The control works and the close pair fails, so there is a separation between them where the patches divide, and it can be computed rather than found by eye.

The card here is 0.8 m across at 1.2 m up, with the lamps at 3.0. Each lamp’s shadow of it is magnified by $3.0/(3.0-1.2) = 1.67$, so each shadow is 1.33 m wide; and moving a lamp sideways by $s$ moves its shadow by $s \times 1.2/1.8 = 0.67s$. The two shadows stop overlapping when the displacement exceeds the width, which is at $s = 2.0$ metres.

Measured, the patches divide between 2.00 and 2.05 metres. The prediction and the measurement agree, which is the point of doing both.

Two metres. And the drawn lines resolve the same two lamps at 7.2 centimetres.

So there is a factor of twenty-eight between the two methods on the same arrangement, and it is a factor rather than a matter of taste: over that whole band — from seven centimetres to two metres of lamp separation — the floor shows one patch and the lines show two centres, and the lines are right.

Two events that are one event #

The sweep above contains something the essay did not set out to find, and it is worth a paragraph because it explains the third answer from earlier.

The separation at which the two visible patches divide and the separation at which the fully dark region disappears are the same number — 2.0 metres on both, to the resolution of the sweep.

They have to be. The visible patch is the union of the two shadows and the umbra is their intersection; two convex regions of the same size sliding apart divide their union at exactly the moment they empty their intersection. One event, two descriptions, and a reader watching either quantity is watching the other.

That also explains why the control arrangement — the one where counting patches gives the right answer — is precisely the one with no umbra. The patch count only becomes reliable at the separation where the fully dark shadow ceases to exist, so a reader who has two clean patches to count has, necessarily, no fully dark region anywhere in the picture. The two facts are not independent observations to be weighed against each other; they are the same fact.

Why a region is the wrong invariant #

The failures above have a common shape, and this field has met it before with the object rather than the light.

A hole is not preserved shows that the shadow of an object with a hole in it need have no hole: the topology of the shadow is not the topology of the caster, because a projection can close a gap that a solid keeps open. Counting features of a shadow region therefore says very little about the object.

This essay is the same statement about the source. The connected components of a shadow region are not a function of the number of centres, because the union of two overlapping regions is one region and the union of two disjoint shadows of two objects is two.

What is an invariant is the structure of the drawn lines: each source contributes a pencil, and pencils do not merge no matter how the regions overlap. Two lamps a centimetre apart still contribute two pencils; whether a reader can tell them apart is a question about noise with a number attached, and it is a different failure from the region count’s, which is exact and structural rather than statistical.

The general lesson is one the collection keeps arriving at from different directions: count the thing the geometry has, not the thing the picture shows. A shadow’s region is what the picture shows; the pencil is what the geometry has.

What a forensic reader should do instead #

Counting shadows in a photograph is not a hypothetical activity: it is one of the standard checks applied to images whose authenticity is in question, and it is applied by eye.

The measurements above suggest a small revision to the procedure, and none of it is expensive.

Count pencils, not patches. Draw the line from each object’s top to the tip of its own shadow and see how many points those lines pass through. That is the count, and it is exact when the sources are further apart than the resolution limit.

Expect one object to give one shadow and several objects to give several, whatever the light count. Two patches under one lamp is the normal case rather than evidence of anything.

And check the umbra separately if the argument turns on the darkness of a region rather than on its position, because the fully dark part can be empty in an arrangement whose visible shadows are perfectly clear.

The lamp, out of the picture sets out what the construction needs to be worth using in that setting — two posts to determine, a third to test — and this rung adds what to count once the lines are drawn.

The three arrangements, side by side #

The whole essay is three floors, and they are worth seeing together.

Two lamps, one card, close. One patch. One umbra. Two pencils.

One lamp, two cards. Two patches. Two umbrae. One pencil.

Two lamps, one card, far. Two patches. No umbra. Two pencils.

A fourth column could be added and would make the same point again: the area of floor in shadow, which is 1.6 square metres for the close pair, 1.9 for one lamp and two cards, and 2.7 for the far pair. It is a smooth function of the arrangement with no discontinuity anywhere, so it says nothing about counts at all — and it is the quantity a reader’s eye is actually most sensitive to, because it is how dark the room looks.

The pencil column is the only one that is a fact about the lights. The other two columns are facts about the arrangement, and they are perfectly good facts — a set designer, a photographer or anybody choosing where to put a lamp cares about them a great deal — but they are answers to a question about the floor rather than about the room.

What changes the threshold #

The two-metre figure is a fact about one card at one height under lamps at another, and the arithmetic above says exactly which quantities move it.

The occluder’s size, in proportion. A wider card has a wider shadow and the lamps must be further apart to separate them, in direct proportion — so a person casting a shadow, half a metre across, needs a metre and a quarter of lamp separation where a pencil needs centimetres.

The occluder’s height, through the magnification. An object close to the floor casts a small, sharp shadow that separates readily; one close to the lamp casts a large one that takes a wide separation to divide. That is the same magnification a shadow’s length curve is built on, appearing as a threshold rather than as a size.

And nothing else. The camera does not enter, the floor’s shape does not enter unless it is steep enough to fold the region, and the lamps’ brightness does not enter at all — which is worth saying because brightness is what a reader is actually looking at when they count patches, and it has no effect on where the boundary between one patch and two falls.

That last point is the practical one. A reader can compute their own threshold from two lengths they can see in the picture — how big the object is, and how high — and can then say whether the patch count in front of them is in the regime where it means anything.

What the count is for, if not for lights #

The patch count has been given a hard time here and it does answer a real question, which is worth stating so that the essay is a correction rather than a dismissal.

How dark is the room, and where. The area in shadow, the number of separate dark places, and how deep each of them is are the quantities a lighting designer works in, and none of them is improved by knowing how many sources there are. Two lamps arranged to leave no umbra is a good lighting design and the patch count is how it is judged.

And whether a picture is consistent with itself. A forensic reader looking at a composite image is often not asking how many lights there were; they are asking whether the shadow of one object is consistent with the shadow of another. That question is answered by the pencils, but a gross inconsistency — one object with two shadows and its neighbour with one — shows up in the patch count first and is worth noticing.

The distinction to keep is between a count that is evidence and a count that is a measurement. Patches are evidence: they suggest where to look. Pencils are a measurement: they answer. This collection’s habit is to insist on the second, and the reason is on display here, since the first gives a different answer to the same room depending on which of three darknesses is being counted.

The short version #

The number of dark patches on a floor is not the number of lights in the room, in either direction: two lamps close together give one patch, one lamp and two objects give two, and the arrangement where the count is right gives two patches and no fully dark region at all.

What survives every one of those rearrangements is the pencil structure of the drawn lines, which is why the count is done there.