The corners a floor cannot add

A wire with a corner in its shadow establishes one of the collection’s better results. A smooth curve with no corner anywhere on it casts a shadow that has one — and the lamps that do it are neither an accident nor a rare coincidence, but a surface in the room, the curve’s own tangent developable, which a lamp carried across the room passes through.

That essay casts onto a plane. This one asks what a floor that is not a plane does to the count.

The answer is nothing, and the reason has no floor in it.

Where a corner comes from #

The shadow of a point is the lamp’s ray through it, continued to whatever is in the way. Differentiate along the curve and the shadow’s velocity has two terms — one from the point moving, one from the ray’s landing parameter changing — and they cancel exactly when the curve’s tangent runs along the ray.

That condition names the lamp and the curve. It does not name a surface, because no surface appears anywhere in it.

So the natural way to say what a shadow’s corners are is not in terms of the floor at all. Project the curve from the lamp onto a sphere around the lamp. The resulting spherical curve has corners exactly where the tangency condition holds. The floor then maps that spherical curve to itself — one to one, and smoothly, everywhere a ray meets the floor transversally.

A smooth one-to-one map does not create corners and does not remove them. The count survives.

It is worth being careful about what “transversally” is doing, because it is the whole hypothesis and it is easy to skate over. A ray meets a floor transversally when it is not tangent to it — when the ray crosses rather than grazes. Where a ray grazes, the landing parameter is infinitely sensitive to the direction, the map is singular, and nothing above applies.

Grazing is not exotic. A ray nearly parallel to a rising ridge grazes it, and a long shadow on a sloping floor is full of such rays. So the theorem’s reach is real but bounded, and the boundary is a property of the arrangement rather than of the floor alone.

The measurement #

Four receivers, one wire, one lamp placed on the wire’s tangent developable at a stated point.

The tangency test predicts one corner. Every receiver draws one — plane, dish, ridge, and the creased floor too — and each is a place where the shadow’s step between consecutive samples collapses to about half a per cent of the median step.

The shadows themselves are emphatically different. The dish’s and the plane’s differ by metres over the sprawl of the helix’s shadow, so the count is not surviving by the shadows being the same shadow.

The developability is worth a note precisely because it is not the reason. A ridge can be unrolled without stretching, and a reader might reasonably think that is why its shadow keeps its corners — roll the floor flat, and the shadow is a plane shadow. That reasoning is available and it is not the argument here, because it fails on the dish, which cannot be unrolled at all and which keeps the corner just as exactly.

The argument uses only that the map from the sphere to the floor is a local diffeomorphism, which every smooth surface met transversally supplies. Developability is a much stronger property and it buys other things — the drawing and the development is about what — but not this.

The exception, and it is a crease #

The creased floor draws six corners, not one, and finding that was the point at which the essay changed shape.

The prediction had been that a crease could do this. What was not predicted is that the two kinds of corner are cleanly separable, by a quantity neither of them is about.

At a real corner the shadow stops dead. The step between consecutive samples collapses — measured at 0.005 of the median step — because the shadow map’s derivative genuinely vanishes there.

At a crease the shadow does not stop. It carries on at full speed and merely turns, because nothing has happened to the wire: the mark is running up a riser while its plan stalls on the seam line. The least such gap is 0.49 of the median.

That is a factor of about a hundred between two corners that look identical on the page — and the hundred is not a constant, which is worth knowing before it is quoted as one.

At a genuine corner the shadow map’s derivative vanishes, so the curve has a stationary point in its parameter and the step between consecutive samples falls as the square of the sample spacing while the median step falls as the first power. The reported 0.005 is therefore about $1/N$ for $N$ samples, and refining the sampling drives it toward zero without limit. The crease’s 0.49 does not move with $N$ at all, because nothing there is stationary: the shadow crosses the seam at full speed and merely changes direction.

So the classifier improves without bound as the sampling is refined, and the separation between the two kinds of corner is really a factor of about $0.5N$ rather than a factor of a hundred. That is an unusually comfortable position for a threshold — most of the thresholds in this collection are properties of the arrangement and this one is a property of how carefully the question is asked.

The crease’s own number is the one with a geometric meaning, and it is not universal either. A mark crossing a seam changes speed abruptly from its rate on the near face to its rate on the far one, and the statistic picks up the smaller of the two — so 0.49 is the ratio of the shadow’s speeds on the two faces, which is set by the fold’s angle to the rays and by nothing else. A steeper riser slows the shadow more and pushes the number down.

Which names the classifier’s failure mode exactly. A crease nearly parallel to the rays slows the shadow almost to a stop and imitates a real corner, and that is the same grazing condition the transversality hypothesis excludes two sections above — so the theorem’s boundary and the classifier’s boundary are the same boundary, met from two directions. A floor met transversally cannot add a corner and cannot fake one; a floor grazed by the rays can do both, and the arrangement rather than the floor decides which.

How the classification was found #

The first version of the classifier got this backwards, and the way it did is worth recording because the correct answer is counter-intuitive.

The expectation was that a crease would make the shadow jump — a riser is a discontinuity in the floor’s height, so the mark ought to leap across it and leave a gap in the drawn curve. The classifier was written to call a corner a crease if the step between samples spiked.

It does not spike. It falls to about half the median, and the reason is that a riser is a vertical wall, so the marks that meet it do not skip past — they climb it. Their plan stalls on the seam line while their height runs up the riser, and the three-dimensional step between consecutive marks is perfectly ordinary.

So the classifier had the sign of its test wrong and reported six real corners on the creased floor. What fixed it was measuring the actual distribution rather than reasoning about it — 0.005 and 0.49, two decades apart, and the small one is the real corner.

That is worth generalising. A discontinuity in a surface is not a discontinuity in what lands on it, because the rays that would have crossed the gap land on the wall instead. A shadow is continuous across a crease for the same reason a shadow is continuous across the join between a floor and a wall — the two pieces of surface share the crease, and a point of the crease is in both.

Why a crease is allowed to and a curve is not #

The argument above says the floor’s map is a diffeomorphism “everywhere a ray meets the floor transversally”, and a crease is precisely where that qualification bites.

A smooth floor met transversally gives a smooth invertible map from the sphere around the lamp to the floor, so corners transfer intact and nothing new appears. A crease is not a smooth surface at the seam; the map has a corner in it, and a corner composed with a smooth curve gives a corner.

So the honest statement of the theorem carries its hypothesis. A smooth receiver cannot add a corner to a shadow. A receiver with an edge in it adds one wherever the shadow crosses the edge.

That reading is worth keeping because it puts a crease and a curvature in different categories, which is where the rest of this row has been putting them too: a crease collapses one direction of the plan where a curve merely distorts it, and a crease’s residual is spiky where a curve’s is smooth. Three rungs, three measurements, one distinction.

The refusal #

A lamp two metres off the tangent developable gives no real corner at all. The tangency test finds none, and the drawn count on a curved floor is none as well.

That is the check that makes the rest of the essay a measurement. A corner-counter that found corners everywhere would be counting sampling noise, and on a curve sampled at fourteen hundred points there is plenty of noise to count. The creased floor’s own corners are still there in that case, which is the second half of the same check: the two kinds are genuinely independent, and removing the cause of one leaves the other untouched.

What survives a receiver, in general #

This is now the third quantity in the row that turns out to be receiver-independent, and they are worth listing together because the pattern is the whole point.

The un-cast occluder survives, exactly, on all four floors.

The lamp’s own image survives, exactly, on all four floors.

The corner count survives, on all smooth floors.

And the common thread is that each of them is a statement about rays. An un-cast is a ray run backwards. A lamp’s image is where a family of rays’ images meet. A corner is where a ray is tangent to the caster. None of the three mentions a surface, so none of them can depend on one.

What does not survive is anything stated in the floor’s own coordinates: the plan of the shadow, its length, its curvature, the four-point map’s residual. Those depend on the floor completely, and the field spent four rungs measuring how much.

A note on counting corners at all #

The count above is made on the drawn curve rather than from the tangency condition, and that is deliberate rather than incidental.

The tangency condition predicts where a corner should be. Counting them on the drawn polyline asks whether one is actually there, which is a different question and the one a reader with a photograph is in a position to ask. Two routes to one number is this collection’s habit, and here the two routes disagree in an informative way — the condition predicts one corner on every floor, and the drawing shows one on three floors and six on the fourth.

The disagreement is the crease, and it would have been invisible to a check that only ran the condition. A prediction that is never compared against a drawing is a prediction about a model.

The topological cousin #

There is one shadow property that is receiver-independent for a different reason and it is worth separating from these, because it looks like the same kind of statement and is not.

A hole is not preserved shows that the number of holes in a shadow is not a property of the occluder — a ring can cast a shadow with no hole in it, depending on the direction of the light. That is a fact about the projection, not about the receiver, and it holds on a plane.

So the two results sit either side of the same line. Corners are preserved by the receiver and not by the light. Holes are destroyed by the light and would be preserved by any receiver the rays meet transversally. Which quantity survives which operation is not something intuition supplies; it has to be worked out one quantity at a time.

Where the threshold is #

The classification above uses a threshold — a corner is “real” if the shadow’s step collapses below a tenth of the median — and a threshold is a choice, so it is worth saying how arbitrary it is.

It is not very. The measured gaps cluster at 0.005 and at 0.49 with nothing between them, so any threshold between about 0.02 and 0.3 gives the identical answer. That is a two-decade gap in a quantity whose scale is set by the sampling, and a classifier with a two-decade margin is not a classifier anybody has to tune.

What would narrow the gap is a crease so shallow that the shadow barely turns at it. In that limit the crease’s corner is not a corner at all — it is a gentle bend, no different from the shadow’s ordinary curvature — and the count returns to one. Which is correct: a crease of zero height is a smooth floor, and the theorem’s hypothesis is satisfied again.

What moving the lamp does #

One more measurement rounds out the picture, and it is the one that turns the result into a way of finding something.

Move the lamp along the tangent developable and the corner moves along the caster, in a determined way — the tangency condition names which point of the curve is involved, and it varies smoothly with the lamp’s position on the surface. Move the lamp off the developable and the corner vanishes, not gradually but at a definite place: the tangency condition either holds or does not.

Both of those are true on every smooth floor, because both are statements about the tangency condition rather than about the shadow. So a reader who can move a lamp and watch a shadow has an instrument, and it is one whose readings are not contaminated by whatever the floor is doing.

What the floor does affect is where on the floor the corner appears. That is the ordinary situation — the geometry of the caster and the light decides what happens, and the floor decides where it is drawn.

What a reader can use #

A photograph of a shadow with a sharp corner in it poses a question, and the two answers are very different objects.

If the corner is a place where the shadow stops and reverses, it is a property of the caster and the light. Something in the scene has its tangent along the light’s ray, and moving the light a little moves the corner along the caster in a determined way. That is a measurement of the caster.

If the corner is a place where the shadow turns and carries on, it is the floor, and there is an edge under the shadow at that point. That is a measurement of the room.

The way to tell them apart on paper is the shadow’s own density of detail near the corner — a real corner has the shadow crowding into it, because the shadow is moving slowly there. A crease has it passing through at the same rate as everywhere else. It is visible without any arithmetic once a reader knows to look for it, which is the most a geometrical result can usually offer.