A hole is not preserved

A shadow is a picture of a thing, and it is fair to ask what it keeps. Length goes, angle goes, area goes, straightness survives — those are settled by the fact that a shadow on a flat floor is a projection, and the foundations field has said what a projection does to each of them.

There is a coarser set of properties that a projection also acts on, and it is not usually asked about: how many pieces the shadow is in, and how many holes it has. They are the crudest facts about a shape and they are exactly the ones a person reads off a shadow at a glance.

One of them survives and one does not.

Connectedness survives, and the proof is a sentence #

The shadow of a connected caster is connected. Always. There is no lamp position, no floor, no shape and no arrangement that breaks one object’s shadow into two.

The reason has nothing to do with light. The shadow is the image of the caster under the map that sends a point to where the lamp’s ray through it meets the floor, and that map is continuous on the region where it is defined. The continuous image of a connected set is connected. That is the whole argument.

It survives every complication one might try to introduce. A caster with a hole in it, a caster that is nearly two objects joined by a thread, a lamp almost inside the caster, a caster half of whose shadow runs off to the horizon: all still one piece, because none of them breaks the continuity.

So the count of pieces in a shadow is a lower bound on the count of pieces of the caster, and nothing more. Two shadows means at least two objects; one shadow means nothing at all.

And a hole does not #

A ring lying flat, lit from above, casts an annulus. Tip it and the hole shrinks. Tip it far enough and the hole is gone, and what is left is a solid dark patch the shape of a squashed washer.

The transition has a closed form and it is short.

Project the tube’s core circle along the direction of the light. It is a circle of radius R seen obliquely, so its image is an ellipse with semi-axes R and R·cos θ, where θ is the angle between the light and the ring’s axis. A line along the light clears the tube exactly while its distance from that ellipse exceeds the tube’s radius r, and the largest clear disc that fits inside an ellipse is centred and has the semi-minor axis for its radius. So the hole survives while

$$R\cos\theta > r$$

and closes at θ = arccos(r/R), which for the ring in the figures — half a metre across the core, sixteen centimetres of tube — is 71.34°.

Getting that wrong is instructive. The first version used R − r rather than R, on the reasoning that the opening starts at the tube’s inner edge. That is the right picture face-on and the wrong one at every other angle: the quantity that shrinks with the obliquity is the distance from the core circle, and the tube’s thickness enters as a constant rather than as part of what is foreshortened. The wrong version put the transition at 61.9° and the measurement said 71.

The lamp cannot close it — only the tilt can #

The first attempt at this measurement asked the wrong question, and the wrong question is the one most people would ask: how high does the lamp have to be before the hole in a flat ring’s shadow closes?

It never does.

A flat ring lit from anywhere above it casts a shadow with a hole, at every lamp height, at every horizontal offset, and at every distance. The bisection that went looking for the transition reported that the hole was present at both ends of every bracket it was given, which is a function politely saying the question has no answer.

The reason is one line and it is worth having, because it is the sort of thing an intuition about “steep light” gets backwards. The segment from a lamp above the ring’s plane to any point of the floor crosses that plane exactly once, somewhere. As the floor point ranges over the whole floor, the crossing point ranges over the whole plane — including the disc inside the ring’s inner edge. So there are always floor points whose ray goes through the hole, and the hole in the shadow is exactly the set of them.

What that argument uses is that the ring’s own plane is being crossed transversally by every ray. Tip the ring and the argument still holds for the plane — but the tube is no longer a thin annulus in the plane the rays are crossing, it is a foreshortened one, and the clear opening in it is what shrinks. So the closing is a fact about the obliquity of the light to the ring, and a lamp’s height only matters through the angle it makes with the ring’s axis.

Which means the tilt in the figures could equally have been a lamp moved sideways, and the figures use a sun straight down and a tipped ring because that makes the obliquity exactly the tilt rather than approximately it. A lamp at a finite distance sees different parts of a large ring at different obliquities, and the transition then happens raggedly, one part of the opening at a time.

Measuring a count is not like measuring a length #

The measurement is where this essay turned into a gate.

The shadow region is computed cell by cell: a patch of floor is dark when the segment from it to the light meets the ring. That is the definition, and it has to be the definition rather than an outline, because an outline presumes the shadow is the inside of one closed curve and half the point here is that it need not be.

Then the holes are counted by flood-filling the complement and keeping the components that do not touch the edge of the sampled patch. Which is correct, and which reported seven holes at forty degrees of tilt on a grid four hundred cells across, where the true answer is one.

The six extra were single cells. A ray that grazes the tube can take a long time to resolve — the marching gets closer and closer to the surface without crossing it — and a handful of cells ran out of steps and were reported as lit. A flood fill calls an isolated lit cell inside a dark region a hole, and it is right to; the count was correct arithmetic on the wrong region.

Two things fixed it and both are worth stating, because neither is a tolerance in the geometry.

A grazing ray is counted as a hit once it is within a hundredth of a millimetre of the surface. That converts an unresolvable case into a decision, and the decision is the one a physical ray would make.

A hole has to be bigger than a stated number of cells to count as a hole. Otherwise the answer is a fact about the sampler, and the way that shows up is unmistakable once looked for: the count was stable at a coarse grid, wrong at a fine one, and wronger at a finer one — the opposite of what a converging measurement does.

The ladder in the figure is what makes the threshold honest. The measured transition rises with the grid — 70.6°, 70.8°, 71.0° — and approaches 71.34° from below, which is the only direction the sampling can err in, because a hole a fraction of a cell across is invisible to any grid and no grid can invent one. If the threshold were doing the work, the ladder would flatten somewhere short of the closed form or wander. It does neither.

How the hole goes, on the way #

The transition is a threshold in the count and not in anything else, and watching the area of the hole approach it is the more informative picture.

The clear opening’s two semi-axes are R·cos θ − r and R − r, so the hole’s area falls linearly in cos θ near the transition and reaches zero at a definite angle rather than tapering. There is no long tail: a degree short of the closing the opening is a slot about a centimetre and a half across, and five degrees short it is about eight centimetres.

The grid in the figures is fine enough to resolve the first of those and not much finer, which is exactly why the measured transition lands a third of a degree below the closed form rather than on it. The sampling loses the hole when the hole is a few cells wide, and a few cells is a few millimetres.

A reader watching a real ring being tipped therefore sees the hole narrow into a slot, hold as a visible line for a degree or two, and go. What they do not see is a slow fade, and that is worth saying because the arithmetic of a threshold and the arithmetic of a limit look the same on a plot until the axis is chosen. Plotted against the tilt the curve looks like a fade; plotted against cos θ it is a straight line running into the axis.

And a caster with no hole at all #

The other direction is the surprise.

Take a helical ramp of a turn and a quarter — a solid tube swept along a rising spiral, its top end above its bottom end and the two not joined. There is nothing to put a finger through. It is a thickened arc, which is topologically a ball, and it has no hole in any sense.

Its shadow has one.

The reason is easy once seen. The ramp’s shadow is the union of the shadows of all its points, and the two ends of the ramp — a full turn apart — cast onto overlapping parts of the floor. Their shadows meet and join up, and the region between them, which no part of the ramp is above, is enclosed. A projection can glue two ends of an arc together, and the moment it does, the arc’s shadow is a loop.

So the count of holes in a shadow bounds the caster’s holes in neither direction. A shadow with a hole may come from a solid with none. A shadow with no hole may come from a ring.

The quantity that behaves itself #

Beside the count, the same computation gives an area, and the contrast is worth drawing because it is the reason a count needed a gate and an area did not.

The shadow’s area is a sum over cells, and a cell that is resolved wrongly contributes one cell’s worth of error to a total of thousands. Refine the grid and the errors get smaller and more numerous in exactly the proportion that keeps the total converging; the area at a hundred and forty cells across and at three hundred and eighty agree to well under a per cent, and neither is anywhere near a decision.

The count is not a sum. It is a decision about connectivity, and a single wrongly resolved cell changes it by a whole unit. There is no averaging, no cancellation and no smallness: the error is one hole, and one hole is the entire quantity.

That is the distinction the collection’s convergence gate was built around, and it generalises past shadows. A quantity that is an integral over a sampled region is robust to the sampler in a way that a quantity that is a predicate over the same region is not, and a measurement of the second kind needs its ladder shown rather than assumed. Both were computed here from the same array of ones and zeros.

What that leaves #

Two properties, one of which is preserved and one of which is not, is a small yield from a whole class of question. It is worth being explicit about why the yield is small, because the reason is the same one that makes this field work at all.

A shadow is a projection, and a projection is a map that is continuous and generally not injective. Continuity is what preserves connectedness, and it is essentially the only topological property a non-injective continuous map preserves in the direction one wants. Holes are destroyed because points that were apart can be brought together; and they are created for exactly the same reason. Everything else in this vicinity — the number of boundary curves, the number of self-intersections of the outline, whether the dark region is simply connected — follows from the projection’s failure to be injective and can go either way.

There is one thing more that can be said, and it is the practical form:

A shadow’s topology is a joint fact about the object and the lamp, and the lamp’s contribution is not small. One ring gives an annulus or a disc depending on nothing but an angle.

The same ring, the same floor, the same light — one number changed, and the shape a person would describe from across the room has changed category. That is the shape of the claim the edge of a shadow is drawn on the object makes about the boundary curve and a wire with a corner in its shadow makes about a cusp, arriving a third time about the crudest property a shape has.

And what a reader can do with it #

The negative results are the useful ones here, because they are the ones that stop an inference.

A photograph of a shadow with a hole in it does not license the conclusion that the object has a hole. A photograph of a shadow without one does not license the conclusion that it does not. What a photograph does license is: at least this many objects, and no more than this much extent. The rest needs a second lamp, a second view, or a walk round.

That last remark connects this to the twoviews field rather than to anything in this one, and the connection is real: a second lamp is a second centre of projection, and two shadows of one object from two lamps are exactly as informative about it as two photographs from two cameras. The ring lit twice, from two directions far enough apart, has a hole in one of its shadows and that settles it. Two lamps and one map works out what the pair of shadows is as a map; this is one thing the pair can decide that neither can.