Two lamps and one map

Light a flat object from two lamps and it casts two shadows. They are different sizes, they point in different directions, and the region where they overlap is darker than either.

The question worth asking is not what each shadow is — each is a homology of the object, and that is settled — but what the map between them is. Given one shadow, what does the other one have to be?

A shadow between parallel planes is a scaling #

The answer comes out of one observation about the geometry, and the observation is that the two planes involved are parallel.

A flat object lying at height $h$ and a floor at height $0$ are two parallel planes. A homology’s axis is where the two planes meet — and two parallel planes meet at infinity, so the axis is the line at infinity.

A central collineation whose axis is the line at infinity is a homothety: a scaling about a point. That is not a special case to be checked for, it is what the classification says as soon as the axis is named. So the map from the object to its shadow is

$$\mathbf{p} \mapsto k\,\mathbf{p} + (1-k)\,\mathbf{a}, \qquad k = \frac{H}{H - h}$$

for a lamp at height $H$ over the floor point $\mathbf{a}$ — a scaling about the lamp’s own foot, by a ratio set by the two heights and nothing else.

Two lamps give two such scalings. Composing one with the inverse of the other gives a third, with ratio $k_B/k_A$, and that is the map between the two shadows.

Which means the two shadows of a flat object are similar figures. Same shape, different size, no rotation, and corresponding points joined by lines through a common centre.

The ratio is a measurement of the lamps #

The ratio $k_B/k_A$ depends on the two lamps’ heights and on the object’s height, and on nothing else — not on where the lamps are horizontally, not on the object’s shape, not on where it sits on the floor.

Two consequences that a reader looking at a photograph can use.

Equal ratios means equal heights. Set the two lamps at the same height and $k_A = k_B$, so the composition has ratio exactly 1 — a pure translation. The two shadows are then congruent: same size, same orientation, offset. This is a statement about the lights read off their shadows, and it needs nothing about the object at all.

And the ratio is not a similarity of the object. Both shadows are enlargements of the object, so “the shadows are similar” is expected. What is worth the slider is that the map between them stays a pure scaling with no rotation, at every pair of lamp positions — including lamps on opposite sides of the object, where the shadows point in opposite directions.

No rotation is available, and that is the surprising part #

The absence of a rotation in the map is worth dwelling on, because it is the claim a reader is most likely to doubt after looking at a real pair of shadows.

Put the two lamps on opposite sides of the object. One shadow falls left and the other falls right; they point in opposite directions and look like mirror images of each other. The map between them still has no rotation in it, and no reflection either — it is a scaling about a point, with a positive ratio, and the “opposite directions” are entirely accounted for by where the centre of the scaling sits relative to the two outlines.

The reason is the axis. A central collineation’s angular behaviour lives in what it does to the line at infinity, and a map fixing that line pointwise cannot move a direction at all: every direction is its own image, so no line changes its angle and no rotation is possible. A scaling about a point far from the object drags the whole outline across the floor while leaving every edge parallel to where it was.

So the visual impression of two shadows “pointing different ways” is an impression about position rather than about orientation, and the two outlines are exactly parallel edge for edge. That is checkable in a photograph with a straightedge, and it fails immediately if the object is not parallel to the floor — which makes it a test of the flatness assumption rather than of the lamps.

The ratio is positive, always #

A homology’s characteristic ratio carries a sign, and the sign says where the projection centre is: negative when the centre lies between the two planes, positive when it does not.

For a lamp above an object above a floor, the centre is above both planes, so $k = H/(H-h)$ is positive and greater than one — the shadow is an enlargement, never a reversal. The composition of two such maps has a positive ratio too, which is why the two shadows are similar rather than inverted.

The contrast with the viewing field is exact and is the reason the sign is worth mentioning. A floor anamorph’s homology has ratio $-1.4815$: negative, because the eye sits between the design’s plane and the ground, so the map carries points across the axis. That is what makes an anamorph read as inside out rather than merely stretched, and a shadow never does it.

What the ratio measures, and how well #

The composed ratio is $k_2/k_1$ with $k_i = H_i/(H_i - h)$, and reading it as a measurement is worth doing carefully because it gives less than it appears to and gives it with a stated conditioning.

One pair gives one relation. With the object’s height $h$ known, the observed ratio is one equation in the two lamp heights, so it fixes neither — it says only how the two compare. Equal heights give exactly one, which is the pure translation the figure above shows, and that is the one reading available without any further information.

Three lamps do not close it either. The three pairwise ratios multiply to one, so only two of them are independent — enough to fix the three $k_i$ up to a common factor and therefore the three heights up to one unknown. The shadows give the lamps’ arrangement and not their scale, which is the same shape of answer a single view gives about a scene, arriving one level up.

And the conditioning is a height ratio. Differentiating, a relative error in a lamp’s height appears in the shadow’s size amplified by $h/(H-h)$, so inverted, the recovered height’s relative error is $(H-h)/h$ times the shadow’s. An object half the lamp’s height reads one for one; a card a tenth of the lamp’s height amplifies the shadow’s error ninefold. So a tall object under a low lamp is the well-conditioned arrangement and a flat card under a ceiling light is the badly-conditioned one — which is the opposite of the arrangement a demonstration would naturally use.

And the one case where the ratio does go negative #

The claim that the ratio is always positive carries a hypothesis: $H > h$, the lamp above the object’s top. Drop the lamp below it and $H - h$ changes sign, so

$$k \;=\; \frac{H}{H-h} \;<\; 0,$$

and the map is a homology carrying points across its axis — the shadow inverted, thrown backwards past the object rather than enlarged in front of it.

That is not an exotic case. It is a lamp on a table beside a taller object, which is an ordinary room, and it is exactly the condition a ball’s shadow becomes unbounded at: the lamp lower than the top of the caster. The two results are the same boundary read on two different objects — a flat card’s shadow reverses where a ball’s shadow opens — and both are the moment the tangent rays stop all descending.

So the positive ratio is a statement about the usual arrangement rather than about shadows. Above the object’s top the shadow is an enlargement and two lamps give similar outlines; below it the shadow is a reversal and the composed map between two shadows can have a negative ratio, which reads as one outline being inside out relative to the other. A photograph with that in it is not inconsistent; it is a photograph of a low lamp, and the sign of the ratio says so.

Three lamps, and the loop that closes #

The two-lamp result makes a prediction, and it is the kind of prediction a wrong derivation fails.

Three lamps give three pairwise maps: A to B, B to C, and C back to A. Their ratios are $k_B/k_A$, $k_C/k_B$ and $k_A/k_C$, so they multiply to exactly 1, and going round the loop returns every point of the shadow to itself.

Checked rather than assumed: the product is 1 to the last bit, and every point of a sampled outline comes back to 2e-16 m after all three maps. A map with a spurious rotation or shear in it — the kind of error a plausible-looking derivation produces — would not close, and the residual would say so in metres rather than in argument.

The loop closing is what makes this a group statement rather than a coincidence about two lamps. Any number of lamps, any order, and the composition of the round trip is the identity.

What breaks it, and it is not the second lamp #

The result is narrower than it first appears, and every clause is load-bearing.

The object has to be flat, and parallel to the floor. If the object is not a plane figure, or is a plane figure tilted relative to the floor, the two planes are no longer parallel, the axis is a real line rather than the line at infinity, and each shadow is a general homology instead of a homothety. The composition is then a central collineation with that axis rather than a scaling.

The floor has to be flat. A dished, ridged or stepped floor gives no homology at all, so there is nothing to compose. The composed map’s existence is exactly as fragile as each individual map’s.

And the lamps have to be above the object’s plane. A lamp below it does not cast a shadow downward onto the floor at all, and the machinery refuses it rather than returning a negative ratio — a refusal paired, as the site’s habit requires, with the acceptance one step above.

None of those is about the second lamp. Adding lights does not weaken any of the geometry; it composes maps that were already there, which is why the composition is so much simpler than the individual maps look.

What happens when the object is not flat #

The flatness clause is the one a reader will most often be standing outside of, so it is worth saying what the result degrades into rather than only that it fails.

A plane object tilted relative to the floor still casts a homology from each lamp — two planes that are not parallel still meet in a line, and that line is a real axis rather than the line at infinity. Each shadow is then a general homology, and the map between them is a central collineation with that same axis. Which is still a strong statement: corresponding points of the two shadows are joined by lines through a common centre, and the line where the object’s plane meets the floor is fixed pointwise by both maps and therefore by the composition.

What is lost is the similarity. A homology with a finite axis does not preserve directions, so the two shadows are no longer parallel edge for edge, and the ratio no longer reads directly as a comparison of lamp heights.

A solid object has no single map at all. Its shadow is the outline of its silhouette from the lamp, and the silhouette is a different curve on the body for each lamp — so the two shadows are outlines of two different curves, and no map between them is defined even in principle. This is the same fact the parallel field meets as the visual hull: a silhouette is a boundary, and which boundary it is depends on where it is being looked at from.

So the result’s reach is precisely: flat object, parallel to a flat floor, lamps above it. Which sounds narrow and covers the case anybody actually reasons about — a cut-out, a leaf, a sheet of paper, a stencil, a flat sign — and, more importantly, is the case where the shadow is being used as a measurement.

The four regions, and why “the shadow” is a compound #

The picture with two lamps has four kinds of place in it, and separating them says what a soft-edged shadow actually is.

Lit by both. Neither lamp is blocked.

Lit by A only and lit by B only. One lamp is blocked; the region is the difference between one shadow and the other.

Lit by neither — the intersection of the two shadows, and the only part that is fully dark.

Which is the discrete version of a penumbra. An extended source is the limit of many point sources, and the graded edge of a real shadow is the region where some of the source is blocked. Two lamps give a two-step version of the same structure, and the geometry is the same geometry: the penumbra’s width is the source’s own image through the occluder’s edge, and here the “source” is two points.

So the region a reader identifies as “the shadow” is the intersection of two shapes that differ by one scaling. It is not a shadow of anything — no single projection produces it — and its outline is made of arcs of two different homothetic copies of the object.

Composing maps is a group statement #

There is a tidier way to say why the composition is well behaved, and it is the reason the census of this site’s plane maps was worth doing.

The maps of a plane fixing one line pointwise form a group. So two central collineations with a common axis compose to a third with the same axis, and the composed map’s class follows from the two without any case analysis.

Here the common axis is the line at infinity, shared because both object-to-floor maps are between parallel planes. The group of collineations fixing the line at infinity pointwise is the group of homotheties and translations — so the composition is one of those, necessarily, and the only question is which. The ratio answers it: not 1 gives a homothety with a finite centre; exactly 1 gives a translation.

That is the same argument the viewing field uses for what a wrong eye does to an anamorph: two anamorphs laid out for two different eyes share the ground line as their axis, so reading one from the other’s position composes two maps with a common axis and lands inside the same group, for every wrong eye and with no computation.

One group statement, two fields, and neither needed a derivation.

What a photograph of two shadows gives up #

Reading the result backwards is where the measurements are, and there are three of them.

The ratio gives the height comparison. Measure the two shadow outlines, fit the scaling that carries one to the other, and its ratio is $k_B/k_A$ — a function of the two lamp heights and the object’s height. With the object’s height known, the ratio of the two lamps’ heights follows.

The centre gives a line. The homothety’s centre is a computable point of the floor, and it lies on the line joining the two lamps’ feet. So the two shadows locate that line without either lamp being in the picture.

And congruence gives equality. Two shadows of the same size are two lamps at the same height, exactly, with no scale, no calibration and no camera needed — the cleanest measurement in this essay and the one that survives the most missing information.